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+arXiv:2301.13234v1  [cond-mat.str-el]  30 Jan 2023
+Field Theoretic Aspects of Condensed Matter Physics: An
+Overview⋆
+Eduardo Fradkin
+Department of Physics and Institute for Condensed Matter Theory,
+University of Illinois at Urbana-Champaign, 1110 West Green St, Urbana Illinois 61801-3080, U.S.A.
+Abstract
+In this chapter I discuss the impact of concepts of Quantum Field Theory in modern Condensed
+Physics. Although the interplay between these two areas is certainly not new, the impact and
+mutual cross-fertilization has certainly grown enormously with time, and quantum Field Theory
+has become a central conceptual tool in Condensed Matter Physics. In this chapter I cover how
+these ideas and tools have inﬂuenced our understanding of phase transitions, both classical and
+quantum, as well as topological phases of matter, and dualities.
+Keywords:
+Contents
+1
+Introduction
+3
+2
+Early Years: Feynman Diagrams and Correlation Functions
+3
+3
+Critical Phenomena
+4
+3.1
+Classical Critical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3.2
+Landau-Ginzburg Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3.3
+The Renormalization Group
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+3.3.1
+Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+3.3.2
+The Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . .
+6
+3.3.3
+Fixed Points
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+3.3.4
+Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.3.5
+RG ﬂows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.3.6
+Asymptotic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4
+Quantum Criticality
+11
+4.1
+Dynamic Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+4.2
+The Ising Model in a Transverse Field . . . . . . . . . . . . . . . . . . . . . . .
+12
+4.3
+Quantum Antiferromagnets and Non-Linear Sigma Models . . . . . . . . . . . .
+14
+4.3.1
+Spin coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+4.3.2
+Path integral for a spin-S degree of freedom . . . . . . . . . . . . . . . .
+15
+4.3.3
+Quantum Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+4.3.4
+Quantum Antiferromagnet . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+5
+Topological Excitations
+17
+5.1
+Topological Excitations: Vortices and Magnetic Monopoles . . . . . . . . . . . .
+18
+5.1.1
+Vortices in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+5.1.2
+Magnetic monopoles in compact electrodynamics . . . . . . . . . . . . .
+22
+5.2
+Non-Linear Sigma Models and Antiferromagnetic Quantum Spin Chains . . . . .
+25
+5.3
+Topology and open integer-spin chains . . . . . . . . . . . . . . . . . . . . . . .
+26
+⋆Work was supported in part by the US National Science Foundation through grant No. DMR 1725401 at the Uni-
+versity of Illinois.
+Preprint submitted to Encyclopedia of Condensed Matter Physics 2e
+February 1, 2023
+
+6
+Duality in Ising Models
+27
+6.1
+Duality in the 2D Ising Model
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+6.2
+The 3D duality: Z2 gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+7
+Bosonization
+31
+7.1
+Dirac fermions in one space dimensions . . . . . . . . . . . . . . . . . . . . . .
+31
+7.2
+Chiral symmetry and chiral symmetry breaking . . . . . . . . . . . . . . . . . .
+34
+7.3
+The chiral anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+36
+7.4
+Bosonization, anomalies and duality . . . . . . . . . . . . . . . . . . . . . . . .
+38
+8
+Fractional Charge
+42
+8.1
+Solitons in one dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+42
+8.2
+Polyacetylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+43
+8.3
+Fractionally charged solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+43
+9
+Fractional Statistics
+45
+9.1
+Basics of fractional statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+46
+9.2
+What is a topological ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . . . .
+46
+9.3
+Chern-Simons Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+47
+9.4
+BF gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+48
+9.5
+Quantization of Abelian Chern-Simons Gauge Theory . . . . . . . . . . . . . . .
+48
+9.6
+Vacuum degeneracy a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+50
+9.7
+Fractional Statistics and Braids . . . . . . . . . . . . . . . . . . . . . . . . . . .
+51
+10 Topological Phases of Matter
+54
+10.1 Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+54
+10.1.1 Dirac fermions in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . .
+54
+10.1.2 Dirac Fermions and Topological Insulators
+. . . . . . . . . . . . . . . .
+55
+10.1.3 Chern invariants
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+56
+10.1.4 The quantum Hall eﬀect on a lattice . . . . . . . . . . . . . . . . . . . .
+58
+10.1.5 The Anomalous Quantum Hall Eﬀect . . . . . . . . . . . . . . . . . . .
+61
+10.1.6 The Parity Anomaly
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+62
+10.2 Three-dimensional Z2 topological insulators . . . . . . . . . . . . . . . . . . . .
+67
+10.2.1 Z2 Topological Invariants
+. . . . . . . . . . . . . . . . . . . . . . . . .
+67
+10.2.2 The Axial Anomaly and the Eﬀective Action
+. . . . . . . . . . . . . . .
+68
+10.2.3 Theta terms, and Domain walls: Anomaly and the Callan-Harvey Eﬀect .
+71
+10.3 Chern-Simons Gauge Theory and The Fractional Quantum Hall Eﬀect . . . . . .
+73
+10.3.1 Landau levels and the Integer Hall eﬀect . . . . . . . . . . . . . . . . . .
+73
+10.3.2 The Laughlin Wave Function . . . . . . . . . . . . . . . . . . . . . . . .
+77
+10.3.3 Quasiholes have fractional charge . . . . . . . . . . . . . . . . . . . . .
+78
+10.3.4 The Jain States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+79
+10.3.5 Quasiholes have fractional statistics . . . . . . . . . . . . . . . . . . . .
+80
+10.3.6 Hydrodynamic Eﬀective Field Theory . . . . . . . . . . . . . . . . . . .
+80
+10.3.7 Composite Boson Field Theory
+. . . . . . . . . . . . . . . . . . . . . .
+82
+10.3.8 Composite Fermion Field Theory
+. . . . . . . . . . . . . . . . . . . . .
+85
+10.3.9 The Compressible States . . . . . . . . . . . . . . . . . . . . . . . . . .
+90
+10.3.10 Fractional Quantum Hall Wave Functions and Conformal Field Theory
+.
+91
+10.3.11 Edge States and Chiral Conformal Field Theory . . . . . . . . . . . . . .
+99
+11 Particle-Vortex Dualities in 2+1 dimensions
+105
+11.1 Electromagnetic Duality
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
+11.2 Particle-Vortex Duality in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . . 105
+11.2.1 The 3D XY Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
+11.2.2 Scalar QED in 3D
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
+11.2.3 The duality mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
+11.3 Bosonization in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 109
+11.3.1 Bosonization of the Dirac theory in 2+1 dimensions
+. . . . . . . . . . . 110
+11.3.2 Bosonization of the Fermi Surface . . . . . . . . . . . . . . . . . . . . . 114
+12 Conclusions
+116
+2
+
+1. Introduction
+Many (if not most) puzzling problems in Condensed Matter Physics involve systems with
+a macroscopically large number of degrees of freedom often in regimes of large ﬂuctuations,
+thermal and/or quantum mechanical. The description of the physics of systems of this type
+requires the framework provided by Quantum Field Theory. Although quantum ﬁeld theory has
+its origins in high-energy physics, notably in the development of Quantum Electrodynamics, it
+has found a nurturing home in Condensed Matter Physics.
+There is a long history of of cross-fertilization between both ﬁelds. Since the 1950’s in many
+of the most signiﬁcant developments in Condensed Matter Physics, Quantum Field Theory has
+played a key role if not in the original development but certainly in the eventual understanding the
+meaning and the further development of the discoveries. As a result, many of the discoveries and
+concepts developed in Condensed Matter have had a reciprocal impact in Quantum Field theory.
+One can already see this interplay in the development of the Bardeen-Cooper-Schrieﬀer theory
+of superconductivity [1] and its implications in the theory of dynamical symmetry breaking in
+particle physics by Nambu [2].
+The close and vibrant relationship between both ﬁelds has continued to these days, and it
+is even stronger today than before. Many textbooks have been devoted to teaching these ideas
+and concepts to new generations of condensed matter physicists (and ﬁeld theorists as well).
+The earlier texts focused on Green functions which are computed in perturbation theory using
+Feynman diagrams [3] [4] [5], while the more modern ones have a broader scope, use path
+integrals and attack non-perturbative problems [6, 7] [8] [9, 10] [11]. In two recent books I have
+discussed many aspects of the interrelation between condensed matter physics and quantum ﬁeld
+theory in more depth than I can do in this chapter [9, 10].
+2. Early Years: Feynman Diagrams and Correlation Functions
+Quantum ﬁeld theory played a key role in the development of the Theory of the Fermi Liquid
+[12, 13]. The theory of the Fermi liquid was ﬁrst formulated by Landau using the framework of
+hydrodynamics and the quantum Boltzmann equation. Landau’s ideas were later given a micro-
+scopic basis using Green functions and Feynman diagrams [3], including the eﬀects of quantum
+ﬂuctuations at ﬁnite temperature and non-equilibrium behavior [14, 15]. Linear response theory
+was developed which allowed the computation of response functions (such as electrical conduc-
+tivities and magnetic susceptibilities) from the computation of correlation functions for a given
+microscopic theory [16]. In turn, correlation functions can be computed in terms of a set of Feyn-
+man diagrams. These concepts and tools borrowed many concepts from ﬁeld theory including
+the study of the analytic structure of the generalized susceptibilities and the associated spectral
+functions (together with the use of dispersion relations). These developments led to the deriva-
+tion of the ﬂuctuation-dissipation theorem. These ideas were widely applied to metals [13] and
+superconductors [1], as well as to quantum magnets [17].
+The spectrum of an interacting system has low energy excitations characterized by a set of
+quantum numbers associated with the symmetries of the theory. These low energy excitations are
+known as quasiparticles. In the case of the Landau theory of the Fermi liquid the quasiparticle is
+a “dressed” electron: it is a low energy excitation with the same quantum numbers (charge and
+spin) and an electron but with a renormalized eﬀective mass. There are many such quasiparticles
+in condensed matter physics. The correlation functions (the propagators) of a physical system
+has a speciﬁc analytic structure. In momentum (and frequency) space, the quasiparticle spectrum
+is given by the poles of the correlators.
+The role of symmetries and, in particular of gauge invariance, in the structure of correlation
+functions was investigated extensively. A direct consequence of symmetries is the existence of
+Ward identities which must be satisﬁed by all the correlation functions of the theory. Ward iden-
+tities are exact relations that relate diﬀerent correlation functions. Such identities contain a host
+of important results. For example, in a theory with a globally conserved charge, the Hamiltonian
+(and the action) have a global U(1) symmetry associated with the transformation of the local ﬁeld
+operator φ(x) (which can be fermionic or bosonic) to a new ﬁeld φ′(x) = eiθφ(x) (where θ is a
+constant phase). Theories with a global continuous symmetry have a locally conserved current
+(and satisfy a continuity equation). The Ward identity requires the correlators of these currents
+(and densities) to be be transverse (i.e. they should have vanishing divergence). In the absence
+of so-called quantum anomalies (which we will discuss below) global symmetries can be made
+local and become gauge symmetries.
+3
+
+In many circumstances a global symmetry can be spontaneously broken. If the global sym-
+metry is continuous, then the Ward identities imply the existence of gapless excitations known
+as Goldstone bosons. For example in the case of a superﬂuid, which has a spontaneously broken
+U(1) symmetry the Goldstone boson is the gapless phase mode. Instead, a the N´eel phase of a
+quantum antiferromagnet has two gapless Goldstone bosons, the magnons of the spontaneously
+broken SO(3) global symmetry of this state of matter. Another example is the Ward identity of
+quantum electrodynamics, which relates the electron self-energy to the electron-photon vertex
+function, which also holds in non-relativistic electron ﬂuids. In addition, these identities implied
+the existence of sum rules that the spectral functions must satisfy. All of these results became part
+of the standard toolkit of condensed matter experimentalists in analyzing their data and for the-
+orists to make predictions. Ward identities and sum rules also imply restrictions on the allowed
+approximations which are often needed to obtain predictions from a microscopic model.
+3. Critical Phenomena
+3.1. Classical Critical Phenomena
+The late 1960s and particularly 1970s brought about an intense back and forth between con-
+densed matter physics and ﬁeld theory in the context of the problem of classical critical phenom-
+ena and phase transitions. This was going to become a profound revolution on the description of
+macroscopic physical systems with large-scale ﬂuctuations. The problem of continuous (“sec-
+ond order”) phase transitions has a long history going back to the work of Landau [18, 19] who
+introduced the concept of an order parameter ﬁeld. This turned out to be a powerful concept of
+broad applicability in many physical systems sometimes quite diﬀerent from each other at the
+microscopic level.
+A simple example is that of a ferromagnet with uniaxial anisotropy in which the spins of the
+atoms in a crystal are strongly favored to be aligned (or anti-aligned) along certain directions of
+the crystal. The simplest microscopic model for this problem is the Ising model, a spin system in
+which the individual spins are allowed to take only two values, σ = ±1. The partition function
+of the Ising model (in any dimension) is
+Z =
+�
+[σ]
+exp
+− J
+T
+�
+⟨r,r′⟩
+σ(r)σ(r′)
+
+(1)
+where J is the exchange coupling constant and T is the temperature (measured in energy units);
+here [σ] denotes the sume over the 2N spin conﬁgurations (for a lattice with N sites), and ⟨r, r′⟩
+are nearest neighbor sites of the lattice. The order parameter of the Ising model is the local mag-
+netization ⟨σ(r)⟨ which, in the case of a ferromagnet, is uniform. The partition function of the
+Ising model can be computed trivially in one dimension. The solution of the two-dimensional
+Ising model by Onsager constituted a tour-de-force in theoretical physics [20]. Its actual meaning
+remained obscure for some time. The work of Schultz, Mattis and Lieb [21] evinced a deep con-
+nection between Onsager’s solution and the problem if the spectrum of one-dimensional quantum
+spin chains [22] (speciﬁcally the one-dimensional Ising model in a transverse ﬁeld [23]). One
+important result was that the Ising model was in fact a theory of (free) fermions which, crudely
+speaking, represented the conﬁgurations of domain walls of the magnet. However, even this sim-
+ple model cannot be solved exactly in general dimension, and approximate mean ﬁeld theories
+of various sorts were devised over time to understand its physics.
+3.2. Landau-Ginzburg Theory
+Landau’s approach assumed that close enough to a phase transition, the important spin con-
+ﬁgurations are those for which the local magnetization varies slowly on lattice scales. In this
+picture the local magnetization, on long enough length scales, becomes an order parameter ﬁeld
+that takes values on the real numbers, and can be positive of negative. Thus, the order parameter
+ﬁeld is eﬀectively the average of local magnetizations on some scale large compared to the lattice
+scale, which we will denote by a real ﬁeld φ(x). The thermodynamics properties of a system of
+this type in d dimensions can be described in terms of a free energy
+F[φ] =
+�
+ddx
+�κ
+2 (▽φ(x))2 + a(T − Tc)φ2(x) + uφ4(x) + . . .
+�
+(2)
+4
+
+which is known as the Ginzburg-Landaufree energy. Here κ is the stiﬀness of the order parameter
+ﬁeld, Tc is the (mean-ﬁeld) critical temperature; a and u are two (positive) constants. This
+expression make sense if the transition is continuous and hence that the order parameter is small
+near the transition. The energy of the Ising model is invariant under the global symmetry [σ] �→
+[−σ]. This is the symmetry of the group Z2. Likewise, the Ginzburg-Landau free energy has the
+global (discrete) symmetry [φ(x)] �→ [−φ(x)], and also has a Z2 global symmetry.
+In Landau’s approach, which was a mean ﬁeld theory, the equilibrium state is the global
+minimum of this free energy. The nature of the equilibrium state depends on whether T > Tc
+or T < Tc: for T > Tc the global minimum is the trivial conﬁguration, ¯φ(x) = 0 (this is the
+paramagnetic state), whereas for T < Tc the equilibrium state is two fold degenerate, ¯φ(x) =
+±(a(Tc − T)/2)β, with the two degenerate states being related by the Z2 symmetry (this is the
+ferromagnetic state). In the Landau theory the critical exponent of the magnetization is β = 1/2
+and the critical exponent of the correlation length is ν = 1/2. However, in the case of the 2D Ising
+model the order parameter exponent is β = 1/8 [24] and the correlation length exponent is ν = 1.
+These (and other) apparent discrepancies led many theorists for much of the 1960s believe that
+each model was diﬀerent and that these behaviors reﬂected microscopic diﬀerences. In addition,
+Landau’s theory was regarded as phenomenological and believed to be of questionable validity.
+3.3. The Renormalization Group
+This situation was to change with the development of the Renormalization Group, due pri-
+marily to the work of Leo P. Kadanoﬀ [25, 26, 27] and Kenneth G. Wilson [28, 29, 30, 31, 32, 33].
+The renormalization Group was going to have (and still has) a profound eﬀect both in Condensed
+Matter Physics and in Quantum Field Theory (and beyond).
+3.3.1. Scaling
+Several phenomenological theories were proposed in the 1960s to describe the singular be-
+havior of physical observables near a continuous phase transition [34, 35, 36]. These early works
+argued that in order to explain the singular behavior of the observables the free energy density
+had to have a singular part which should be a homogeneous function of the temperature, mag-
+netic ﬁeld, etc. A function f(x) is homogeneous if it satisﬁes the property that it transforms
+irreducibly under dilations, i.e. f(λx) = λk f(x), there λ is a real positive number (a scale) and
+k is called the degree. These heuristic ideas then implied that the critical exponents should obey
+several identities. In 1966 Kadanoﬀ wrote and insightful paper in which he showed that the ho-
+mogeneity hypothesis implied that in that regime these systems should obey scaling. He showed
+that this can be justiﬁed by performing a sequence of block-spin transformations in which the
+conﬁgurations that vary rapidly at the lattice scale a become averaged at the scale of a larger sized
+block of length scale ba > a which resulted in a renormalization of the coupling constants from
+{K} at scale a to {K′} at the new scale ba [25, 27]. In other words, the block spin transformation
+amounts to a scale transformation and a renormalization of the couplings (and operators). From
+this condensed matter/statistical physics perspective the important physics is in the long distance
+(“infrared”) behavior.
+A signiﬁcant consequence of these ideas was that close enough to a critical point, if the
+distance |x − y| between two local observables O(x) and O(y) is large compared to the lattice
+spacing a but small compared to the correlation length ξ, their correlation function takes the
+form of a power law
+⟨O(x)O(y)⟩ ∼
+const.
+|x − y|2∆O
+(3)
+where ∆O is a positive real number known as the scaling dimension of the operator O [34, 25, 37].
+These conjectures were known to be satisﬁed in the non-trivial case of the 2D Ising model [26],
+as well as in the Landau-Ginzburg theory once the eﬀects of Gaussian ﬂuctuations were included
+[37]. For example, in the 2D Ising model, the scaling dimensions of the local magnetization σ is
+∆σ = 1/8 and of the energy density ε is ∆ε = 1, which were suﬃcient to explain all the singular
+behaviors known at that time.
+The concept of renormalization actually originated earlier in quantum ﬁeld theory as part
+of the development of Quantum Electrodynamics (QED). In QED the notion of renormaliza-
+tion was used to “hide” the short distance (“ultraviolet”) divergencies of the Feynman diagrams
+needed to compute physical processes involving electrons (and positrons) and photons, i.e. their
+strong, divergent, dependence of an artiﬁcially introduced short-distance cutoﬀ or regulator. In
+particular the sum of the leading diagrams that enter in the electron-photon vertex amounted to
+5
+
+a redeﬁnition (renormalization) of the coupling constant. It was observed by Murray Gell-Mann
+and Francis Low that this renormalization was equivalent to the solution of a ﬁrst order diﬀer-
+ential equation that governed the inﬁnitesimal change of the coupling, the ﬁne structure constant
+α = e2/4π, under an inﬁnitesimal change of the UV cutoﬀ Λ [38]
+Λ dα
+dΛ ≡ β(α) = 2
+3πα2 + O(α3)
+(4)
+where β(α) is the Gell-Mann-Low beta function. Except for the work by Nikolai Bogoliubov
+and coworkers [39], this reinterpretation by Gell-Mann and Low was not actively pursued, partly
+because it predicted that the renormalized coupling became very large at short distances, α → ∞,
+and, conversely, it vanished in the deep long distance regime, α → 0 (if the electron bare mass is
+zero). In other terms, QED is strongly coupled in the UV and trivial in the IR. The same behavior
+was found in the case of the theory of a scalar ﬁeld φ(x) with an φ4 interaction which is relevant
+in the theory of phase transitions. In addition to these puzzles, the1960s saw the experimental
+development of the physics of hadrons which involve strong interactions. For these reasons, for
+much of that decade most high-energy theorists had largely abandoned the use of quantum ﬁeld
+theory, and explored other, phenomenologically motivated, approaches (which led to an early
+version of string theory.) At any rate the notion that the physics may depend on the scale was
+present as was the notion that in some regimes ﬁeld theories may exhibit scale-invariance at least
+in an approximate form.
+3.3.2. The Operator Product Expansion
+The next stage of the development of these ideas was the concept of the operator product
+expansion (OPE). If we denote by {O j(x)} the set of all possible local operators in a theory (a ﬁeld
+theory or a statistical mechanical system near criticality), then the product of two observables on
+this list closer to each other than to any other observable (and to the correlation length ξ) obeys
+the expansion
+lim
+x→y O j(x)Ok(y) = lim
+x→y
+�
+l
+C jkl
+|x − y|∆j+∆k−∆l Ol
+� x + y
+2
+�
+(5)
+where this equation should be understood as a weak identity, valid inside an expectation value.
+Remarkably, this concept was derived independently and simultaneously by Leo Kadanoﬀ [40]
+(who was working in critical phenomena), by Kenneth Wilson [41] (who was interested in the
+short distance singularities arising in Feynman diagrams), and by Alexander Polyakov [42, 43]
+(also working in critical phenomena). In Eq.(5) {∆j} are the scaling dimensions of the operators
+{O j}. The coeﬃcients C jkl are (like the dimensions) universal numbers. In a follow up paper
+Polyakov showed that if the theory has conformal invariance, i.e. scale invariance augmented by
+conformal transformations which preserve angles, then, provided the operators O j are suitably
+normalized, the coeﬃcients C jkl of the OPE are determined by a three point correlator
+⟨O j(x)Ok(y)Ol(z)⟩ =
+C jkl
+|x − y|∆jk|y − z|∆kl|z − x|∆l j
+(6)
+where ∆jk = ∆j + ∆k − ∆l. These results constitute the beginnings of Conformal Field Theory.
+In a nontrivial check, Kadanoﬀ and Ceva showed that the OPE holds for the local observables of
+the 2D Ising model [44].
+3.3.3. Fixed Points
+The next and crucial step in the development of the renormalization Group was made by
+Kenneth Wilson. Wilson was a high-energy theorist who wanted to know how to properly deﬁne
+a quantum ﬁeld theory and the physical meaning of renormalization.
+In a Lorentz invariant quantum ﬁeld theory one is interested in the computation of the ex-
+pectation value of time-ordered operators. In the case of a self-interacting scalar ﬁeld φ(x) in
+D-dimensional Euclidean space-time, obtained by analytic continuation from Minkowski space-
+time to imaginary time, the observables are computed from the functional (or path) integral by
+functional diﬀerentiation of the partition function
+Z =
+�
+Dφ exp
+�
+−S (φ, ∂µφ) +
+�
+dDx J(x)φ(x)
+�
+(7)
+6
+
+with respect to the local sources J(x). For a scalar ﬁeld the Euclidean action is
+S =
+�
+dDx
+�1
+2(∂µφ(x))2 + m2
+2 φ2(x) + λ
+4!φ4(x)
+�
+(8)
+which has the same form as the free energy of the Landau-Ginzburg theory of phase transitions
+shown in Eq.(2). It is apparent that the Landau-Ginzburg theory is the classical limit of the theory
+of the scalar ﬁeld whose partition function is a sum over all histories of the ﬁeld. It is easy to
+see that en expansion of the partition function (or of a correlator) in powers of the the coupling
+constant λ can be cast in the form of a sum of Feynman diagrams. To lowest order in λ a typical
+Feynman diagram involves a one-loop integral in momentum space of the form
+I(p) =
+�
+dDq
+(2π)D
+1
+(q2 + m2)((q − p)2 + m2)
+(9)
+As noted by Wilson in his Nobel Lecture [33], this integral has large contributions from the IR
+region of small momenta q ∼ 0, but for any dimension D ≥ 4 has a much larger contribution
+form large the UV region of large momenta, which requires the introduction of a UV cutoﬀ Λ
+in momentum space (or a lattice spacing a in real space by deﬁning the theory on a hypercubic
+lattice). In quantum ﬁeld theory one then has to require that somehow one takes the limit a → 0
+(or Λ → ∞). To take this limit in the ﬁeld theory is very much analogous to the deﬁnition of
+a conventional integral in terms of a limit of a Riemann sum. The diﬀerence is that this is a
+functional integral. While for a function of bounded variation in a ﬁnite interval (a, b) the limit
+of a partition of the interval into N steps each of length ∆x, such that N∆x = b − a, exists and
+deﬁnes the integral of the function
+lim
+∆x→0 lim
+N→∞
+N
+�
+j=1
+f(x j)∆x j =
+� b
+a
+dx f(x),
+(10)
+the analogous statement does not obviously exists in general for a functional integral, i.e. an
+integral over a space of functions which is what is required. In fact, although thousands of
+integrals of a function are known to exist, there are extremely few examples for a functional
+integral. Moreover, in order to take the continuum limit the lattice spacing must approach zero,
+a → 0. This means that physical scales, such as the correlation length ξ, must diverge in lattice
+units so that they can be ﬁxed in physical units. But to do that one has to be asymptotically
+close to a continuous phase transition! Hence, the problem of deﬁning a quantum ﬁeld theory is
+equivalent to the problem of critical phenomena at a continuous phase transition!
+Wilson gave a systematic formulation to the Renormalization Group by generalizing the ear-
+lier ideas introduced by Kadanoﬀ and the earlier work by Gell-Mann and Low. Wilson’s key
+contribution was the introduction of the concept of a ﬁxed point of the Renormalization Group
+transformation [28, 29, 30, 31]. As we saw, the block-spin transformation is a procedure for
+coarse graining the degrees of freedom of a physical system resulting in a renormalization of the
+coupling constants. Upon the repeated action of the RG transformation its eﬀect can be pictured
+as a ﬂow in the space of coupling constants. However, in addition of integrating-out short dis-
+tance degrees of freedom one needs to restore the units of length which have changed under that
+process. This requires a rescaling of lengths. Once this is done, Wilson showed that the resulting
+RG ﬂows necessarily have ﬁxed points, special values of the couplings which are invariant (ﬁxed)
+under the action of the RG transformation. He then deduced that at a ﬁxed point the theory has
+no scales, aside from the linear size L of the system and the microscopic UV cutoﬀ (the lattice
+spacing a in a spin system).
+This analysis means that for length scales long compared to a → 0 but short compared to
+L → ∞ the theory acquired a new, emergent, symmetry: scale invariance. Therefore, at a ﬁxed
+point the correlators of all local observables must be homogeneous functions (hence, must scale).
+3.3.4. Universality
+A crucial consequence of the concept if the ﬁxed point is that phase transitions can be clas-
+siﬁed into universality classes. Universality means that a large class of physical systems with
+diﬀerent microscopic properties have ﬁxed points with the same properties, i.e. the same scaling
+dimensions, operator product expansions and correlation functions at long distances independent
+on how they are deﬁned microscopically. Although the renormalization group transformation is
+a transformation scheme that we deﬁne and, because of that the location in coupling constant
+7
+
+space of the ﬁxed point itself does depends on the scheme we choose, its universal properties are
+the same. Thus, universality classes depend only on features such as the space (and spacetime)
+dimension and the global symmetries of the system. But the systems themselves may be quite
+diﬀerent. This we speak if the Ising universality class in 2D, on the superﬂuid (or XY) transi-
+tion class in 3D, etc. This concept, which originated in the theory of phase transition, has been
+adopted and generalized in the development of conformal ﬁeld theory.
+3.3.5. RG ﬂows
+Combined with the condition that the correlators decay at long separations, homogeneity
+implies that the correlators must have the form of Eq.(3). In addition, this equation also implies
+that at a ﬁxed point the operators (the local observables) have certain scaling dimensions. Let
+us consider a theory close to a ﬁxed point whose action we will denote by S ∗. Let {O j(x)} be
+a complete set of local observables whose scaling dimensions are {∆j}. The total action of the
+theory close to the ﬁxed point then can be expanded as a linear combination of the operators with
+dimensionless coupling constants {g j}
+S = S ∗ +
+�
+dDx
+�
+j
+g ja∆j−DO j(x)
+(11)
+Under a change of length scale x → x′ = bx, with b > 1, the operators (which must transform
+homogeneously)change as O j(bx) = b−∆jO j(x). Since the phase space changes as dDx′ = bDddx,
+we can keep the form of the action provided the coupling constants also change to compensate
+for these changes as g′
+j = bD−∆jg j. Let b = |x′|/|x| = 1+da/a, where da is an inﬁnitesimal change
+of the UV cutoﬀ a. Then, if we integrate-out the degrees of freedom in the range a < |x| < a+da,
+the rate of change of the coupling constants {g j} under this rescaling is
+adg j
+da ≡ β(g j) = (D − ∆j)g j + . . .
+(12)
+which we recognize as a Gell-Mann Low beta function for each coupling constant.
+This result says that if the scaling dimension ∆j < D, then the renormalized coupling will
+increase as we increase the length scale, g′
+j > g j, and along this direction in coupling constant
+space the RG ﬂows away from the ﬁxed point. Conversely, if ∆j > D the renormalized coupling
+ﬂows to smaller values, g′
+j < g j and the RG ﬂows into the ﬁxed point. We then say that an
+operator is relevant if its scaling dimension satisﬁes ∆j < D, and that it is irrelevant if ∆j > D.
+If ∆j = D then we say that operator is marginal.
+To go beyond this simple dimensional analysis one has to include the eﬀects of ﬂuctuations.
+To lowest orders in the couplings one ﬁnds [45]
+adg j
+da = (D − ∆j)g j +
+�
+k,l
+C jkl gkgl + . . .
+(13)
+where {C jkl} are the coeﬃcients of the OPE shown in Eq.(6). This expression is the general form
+of a perturbative renormalization group and it is valid close enough to a ﬁxed point.
+Wilson and Fisher [30] used a similar approach to analyze how ﬂuctuations alter the results
+of the Landau-Ginzburg theory. They considered the partition function of Eq.(7) with the action
+of Eq.(8). Instead of working in real space they considered the problem in momentum space and
+partitioned the ﬁeld conﬁgurations into slow and fast modes
+φ(x) = φ<(x) + φ>(x)
+(14)
+where φ>(x) are conﬁgurations whose Fourier components have momenta in the range bΛ < |p| <
+Λ, where Λ is a UV momentum cutoﬀ and b < 1. Hence, if we choose b → 1, the fast modes φ >
+(x) have components in a thin momentum shell near the UV cutoﬀ Λ. Conversely, the slow modes
+φ<(x) have momenta in the range 0 ≤ |p| < bΛ. One can then use Feynman diagrams to integrate
+out the fast modes and derive an eﬀective low-energy action with renormalized couplings. In the
+case of a free ﬁeld theory (with λ = 0) the scaling dimension of the φ4 operator is ∆4 = 2(D − 2)
+whereas the φ2 operator has dimension ∆2 = D − 2. Upon deﬁning a dimensionless mass and
+coupling constant by m2 = tΛ2 and λ = gΛ4−D, the beta functions are found to be [10]
+β(t) = − Λ dt
+dΛ = 2t + g
+2 − gt
+2 + . . .
+(15)
+β(g) = − Λ dg
+dΛ = (4 − D)g − 3
+2g2 + . . .
+(16)
+8
+
+(where we absorbed an uninteresting factor if the deﬁnition of the coupling constant).
+The RG ﬂows of Eq.(16) show that the free-ﬁeld (Gaussian) ﬁxed point at g = 0 is stable for
+D > 4 and the asymptotic IR behavior is the same as predicted by the Landau-Ginzburg theory.
+However, for D < 4, the free-ﬁeld ﬁxed point becomes unstable and a new ﬁxed point arises at
+g∗ = 2
+3ǫ + O(ǫ2), where we have set ǫ = 4 − D. This is the Wilson-Fisher ﬁxed point. At this
+ﬁxed point the correlation length diverges with an exponent ν = 1
+2 + ǫ
+12 + O(ǫ2), which deviates
+from the predictions of the Landau-Ginzburg theory. The small parameter of this expansion is ǫ,
+and this is known as the ǫ expansion.
+The Wilson-Fisher (WF) ﬁxed point is an example of a non-trivial ﬁxed point at which the
+correlation length is divergent. It has only one relevant operator, the mass term, which in the IR
+ﬂows into the symmetric phase for t > 0 and ﬂows to the broken symmetry for t < 0. Conversely,
+in the UV it ﬂows into the WF ﬁxed point. For these reasons condensed matter physicists say
+that this is an IR unstable (or critical) ﬁxed point while high-energy physicists say that it is the
+UV ﬁxed point. At this ﬁxed point a non-trivial ﬁeld theory can be deﬁned with non-trivial
+interactions. UV ﬁxed points also deﬁne examples of what in high-energy physics are called
+renormalizable ﬁeld theories and can be used to deﬁne a continuum ﬁeld theory.
+The D = 4-dimensional theory is special in that the φ4 operator is marginal. As can be seen
+in Eq.(16), at D = 4 the beta function for the dimensionless coupling constant g does not have
+a linear term and is quadratic in g. In this case the operator is marginally irrelevant, and its
+beta function has the same behavior as the beta function of Gell-Mann and Low for QED. Such
+theories are said to have a “triviality problem” since, up to logarithmic corrections to scaling,
+there are no interactions in the IR and, conversely, become large in the UV.
+There are also ﬁxed points at which the correlation length ξ → 0. These ﬁxed points are
+IR stable (and in a sense trivial). These stable ﬁxed points are sinks of the IR RG ﬂows. Such
+ﬁxed points deﬁne stable phases of matter, e.g. the broken symmetry state, the symmetric (or
+unbroken state), etc. However in the UV they are unstable and in high-energy physics such ﬁxed
+points correspond to non-renormalizable ﬁeld theories.
+More sophisticated methods are needed to go beyond the lowest order beta functions of
+Eq.(13), and the computation of critical exponents beyond the leading non-trivial order. Pos-
+sibly the record high-precision calculations have been done for φ4 theory for which the beta
+function is know to O(ǫ5). This has been achieved using the method of dimensional regulariza-
+tion [46, 47, 48] (with minimal subtraction). Special resummation methods (Borel-Pad´e) have
+been used to do these calculations in D = 3 dimensions [49]. Remarkably, these results are
+so precise that in the case of the superﬂuid transition, which is well described by a φ4 theory
+with a complex ﬁeld, the results could only be tested in the microgravity environment of the
+International Space Station!
+3.3.6. Asymptotic Freedom
+There are several physical systems systems of great interest whose beta function has the form
+β(g) = Ag2 + . . .
+(17)
+The coupling constant has a diﬀerent interpretation in each theory and the constant A > 0,
+opposite to the sign of the beta function found in QED and φ4 theory in D = 4 dimensions.
+This beta function means that while the associated operator is marginal, with this sign is actually
+marginally relevant. This also means that the ﬁxed point is unstable in the IR but the departure
+from the ﬁxed point is logarithmically small. Conversely, the in the UV the RG ﬂows into the
+ﬁxed point and the eﬀective constant is weak at short distances. This is the origin of the term
+asymptotic freedom [50]. The paradigmatic examples of theories with a beta function of this
+form are the Kondo problem, the 2D non-linear sigma model, and the D = 4 dimensional Yang-
+Mills non-abelian gauge theory.
+The Kondo problem is the theory of a localized spin-1/2 degree of freedom in a metal. the
+electrons of the metal couple to this quantum impurity through an exchange interaction of the
+impurity and the magnetic moment density of the mobile electrons in the metal with coupling
+constant J. This problem is actually one dimensional since only the s wave channel of the
+mobile (conduction band) electrons actually couple to the localized impurity. In 1970 Philip
+Anderson developed a theory of the Kondo problem in terms of the renormalization of the Kondo
+coupling constant g as a function of the energy scale [51]. Anderson used perturbation theory
+in J to progressively integrated out the modes of the conduction electrons close to an eﬀective
+9
+
+bandwidth Ec and found that the beta function has the form of Eq.(17)
+Ec
+dJ
+dEc
+= −ρJ2 + . . .
+(18)
+where ρ is the density of states at the Fermi energy of the conduction electrons. This work
+implied that the free-impurity ﬁxed point is IR unstable and that the eﬀective coupling constant
+J increases as the energy cutoﬀ Ec is lowered. He argued that at some energy scale, the Kondo
+scale, perturbation theory breaks down and that there is a crossover to a strong coupling regime
+which is not accessible in perturbation theory.
+Shortly thereafter, in 1973 Wilson developed a numerical renormalization group approach
+which showed that the Kondo problem is indeed a crossover from the free impurity ﬁxed point
+to the “renormalized” Fermi liquid [32]. In addition, Wilson use the numerical renormalization
+group to examine the approach to the strong coupling ﬁxed point and showed that it is charac-
+terized by a Wilson ratio, a universal number obtained from the low temperature speciﬁc heat
+and the impurity magnetic susceptibility (in suitable units). Wilson’s numerical RG predicted
+a number close to 2π for the this ratio. In 1980 N. Andrei and P. Wiegmann showed (indepen-
+dently) that the Kondo problem is an example of an integrable ﬁeld theory that can be solved
+by the Bethe ansatz [52, 53]. Their exact result was consistent with Wilson’s RG, including the
+numerical value of the Wilson ratio.
+In 1972 Gerard ’t Hooft and Martinus Veltman showed that Yang-Mills gauge theory is renor-
+malizable [46]. This groundbreaking result opened the door to use quantum ﬁeld theory to de-
+velop the theory of strong interactions in particle physics known as Quantum Chromodynamics
+(QCD). In 1973 David Gross and Frank Wilczek [50] and, independently, David Politzer [54]
+computed the renormalization group beta function of Yang-Mills theory with gauge group G and
+found it to be of the same form as Eq.(17),
+Λ dg
+dΛ = − g3
+16π2
+11
+3 C2(G) + . . .
+(19)
+where g is the Yang-Mills coupling constant and Λ is a UV momentum scale. Here C2(G) is the
+quadratic Casimir for a gauge group G. For S U(3), the case of physical interest, C2(S U(3)) = 3.
+This result implies that under the RG at large momenta (short distances) the Yang-Mills coupling
+constant ﬂows to zero (up to logarithmic corrections). This result holds in the presence of quarks
+provided the number of quark ﬂavors is less than a critical value. Hence at short distances the
+eﬀective coupling is weak. Gross and Wilczek called this phenomenon asymptotic freedom.
+This behavior was consistent with the observation of weakly coupled quarks in deep inelastic
+scattering experiments. However, the ﬂip side of asymptotic freedom is that at low energies
+(long distances) the coupling constant grows without limit, which implies that at low energies
+perturbation theory is not applicable. This strong infrared behavior suggested that in QCD quarks
+are permanently conﬁned in color neutral bound states (hadrons). However, unlike the Kondo
+problem we just discussed, QCD is not an integrable theory (so far as we know) and to show that
+it conﬁnes has required the development of Lattice Gauge Theory [55, 56]. To this date the best
+evidence for quark conﬁnement has been obtained using large-scale Monte Carlo simulations in
+Lattice Gauge Theory [57].
+We close this subsection with a discussion of an important case: the non-linear sigma models.
+The O(N) non-linear sigma model is the continuum limit of the classical Heisenberg model for a
+spin with N components. Historically, the non-linear sigma model is the eﬀective ﬁeld theory for
+pions in particle physics. We will discuss its role in the theory of quantum antiferromagnets in
+subsection 4.3 and especially in the case of quantum antiferromagnetic spin chains in subsection
+5.2.
+The simplest non-linear sigma model is a theory of an N-component scalar ﬁeld n(x) which
+satisﬁes the unite length local constraint, n2(x) = 1. The Euclidean Lagrangian is
+L = 1
+2g(∂µn(x))2
+(20)
+where g is the coupling constant (the temperature in the classical Heisenberg model). At the
+classical level, i.e. in the broken symmetry phase, where ⟨n⟩ � 0, this model describes the N − 1
+massless modes (Goldstone bosons) of the spontaneously broken O(N) symmetry. Dimensional
+analysis shows that the coupling constant has units of ℓD−2. Hence, we expect to ﬁnd marginal
+behavior at D = 2. In 1975 Alexander Polyakov used a momentum shell renormalization group
+10
+
+in D = 2 dimensions and showed that the beta function of this model is (here a is the short-
+distance cutoﬀ) [58]
+β(g) = adg
+da = N − 2
+2π g2 + O(g3)
+(21)
+Hence, in D = 2 dimensions also this theory is asymptotically free. As in the other examples we
+just discussed, asymptotic freedom here also implies that the coupling constant g grows to large
+values in the low-energy (long-distance) regime. In close analogy with Yang-Mills theory in
+D = 4 dimensions, Polyakov conjectured that the O(N) non-linera sigma model also undergoes
+dynamical dimensional transmutation [50], that the global O(N) symmetry is restored and that
+for all values of the coupling constant g the theory is in a massive with a ﬁnite correlation length
+ξ ∼ exp((N − 2)/2πg). Extensive numerical simulations, used to construct a renormalization
+group using Monte Carlo simulations [59], showed that there is indeed a smooth crossover from
+the weak coupling (low temperature) regime to the high temperature regime where the correlation
+length is ﬁnite. The non-linear sigma model is a renormalizable ﬁeld theory in D = 2 dimensions
+[60]. For D > 2 dimensions it can be studied using the 2 + ǫ expansion [60, 49], which predicts
+the existence of a nontrivial UV ﬁxed point and a phase transition from a Goldstone phase to a
+symmetric phase.
+It turns out that there is a signiﬁcant number of asymptotically free non-linear sigma models
+in D = 2 dimensions, many of physical interest [61], in particular non-linear sigma models
+whose target manifold is a coset space, a quotient of a group G and a subgroup H. The O(N) non
+linear sigma model is an example since the broken symmetry space leaves the O(N −1) subgroup
+unbroken (the manifold of the Goldstone bosons). In that case the quotient is O(N)/O(N − 1)
+which is isomorphic to the N −1 dimensional sphere S N−1. In later sections we will discuss other
+examples in which more general non-linear sigma models play an important role.
+Models on coset spaces arise in the theory of Anderson localization in D = 2 dimensions.
+Anderson localization is the problem of a fermion (an electron) in a disordered system in which
+the electron experiences a random electrostatic potential. In the limit of strong disorder Philip
+Anderson showed that all one-particle states are exponentially localized and the diﬀusion con-
+stant (and the conductivity) vanishes[62]. There was still the question of when it is possible
+for the electron to have a ﬁnite diﬀusion constant (and conductivity). In D = 2 dimensions
+the conductivity is a dimensionless number which suggests that this may be the critical dimen-
+sion for diﬀusion. Abrahams, Anderson, Licciardello and Ramakrishnan used a weak disorder
+calculation to construct a scaling theory that implied that in D = 2 dimensions the RG ﬂow
+of the conductivity at long distances (large samples) ﬂows to zero and all states are localized
+[63]. Shortly thereafter Wegner gave strong arguments that showed that the existence of diﬀu-
+sion implied that there are low-energy “diﬀusson” modes which behaved as Goldstone modes of
+a non-linear sigma model on the quotient manifold O(N+ + N−)/O(N+) × O(N−) in the “replica
+limit” N± → 0 [64]. A ﬁeld theory approach to this non-linear sigma model was developed by
+McKane and Stone [65] and by Hikami [66].
+4. Quantum Criticality
+Quantum criticality is the theory of a phase transition of a system (e.g. a magnetic system)
+at zero temperature that occurs as a coupling constant (or parameter) is varied continuously.
+Although not necessarily under that name, this question has existed as a conceptual problem for
+a long time, In particular, already in 1973 Wilson considered the problem of the behavior of
+quantum ﬁled theories blow four spacetime dimensions and their phase transitions [67].
+The modern interest in condensed matter physics stems from discoveries made since the late
+1980s. Since that time he behavior of condensed matter systems at a quantum critical point has
+emerged as a major focus in the ﬁeld. There were several motivations for this problem. One
+was (and still is) to understand the behavior of quantum antiferromagnets in the presence of
+frustrating interactions. Frustrating interactions are interactions which favor incompatible types
+of antiferromagnetic orders. The result is the presence of intermediate non-magnetic “valence
+bond” phases that favor the formation of spin singlets between nearby spins. these phases typ-
+ically either break the point group symmetry of the lattice or are spin liquids (which will be
+discussed below). Another motivation is that doped quantum antiferromagnets typically harbor
+superconducting phases (among others) whose high-temperature behavior is a “strange” metal
+that violates the basic assumptions (and behaviors) of Fermi liquids. The most studied version
+of this problem is the case of the copper oxide high temperature superconductors. It was con-
+jectured that there is a quantum critical point inside the superconducting phase at which the
+11
+
+antiferromagnetic order (or other orders) disappears and which may be the reason for the strange
+metal behavior above the superconducting critical temperature. Many of these questions are
+discussed in depth by Sondhi and coworkers [68] and in the textbook by Sachdev [69].
+4.1. Dynamic Scaling
+We will consider a general quantum phase transition and assume that it is scale invariance.
+However, except for the case of relativistic quantum ﬁeld theories, in condensed matter systems
+space and time do not need to scale in the same way. Let us assume that the system of interest has
+just one coupling constant g and that the system of interest has a quantum phase transition (at zero
+temperature) at some critical value gc between two phases, for instance one with a spontaneously
+broken symmetry and a symmetric phase. If the quantum phase transition is continuous then the
+correlation length ξ will diverge at gc and so will the correlation time ξt. However these two
+scales are in general diﬀerent and do not necessarily diverge at the same rate. So, in general,
+if some physical quantity is measured at the quantum critical point at some momentum p and
+frequency ω, the length scale of the measurement is 2π/|p| and at a frequency is ω ∼ |p|z, where
+z is the dynamic critical exponent. Let us say that we measure the observable O at momentum p
+and frequency ω at gc. Scale invariance in both space and time means that at gc the observable
+O(p, ω) at momentum p and frequency ω must scale as
+O(p, ω) = |p|−∆O ˜O(|p|z/ω)
+(22)
+where ∆O is the scaling dimension of the observable O.
+The situation changes at ﬁnite temperature T. A quantum ﬁeld theory at temperature T is
+described by a path integral on a manifold which along the imaginary time direction τ is ﬁnite
+of length 2π/T and periodic for a theory bosonic ﬁelds and anti-periodic for fermionic ﬁelds
+[10]. Since the imaginary time direction is ﬁnite, the behavior for correlation times ξτ < 2π/T
+and ξτ > 2π/T must be diﬀerent. Indeed, in the ﬁrst regime the behavior is essentially the same
+as at T = 0, while in the second it should be given by the classical theory in the same space
+dimension.At the quantum critical point gc there is only one time scale ξτ ∼ 2π/T and only one
+length scale ξ ∼ (2π/T)1/z.
+4.2. The Ising Model in a Transverse Field
+The prototype of the quantum phase transition is the Ising model in a transverse ﬁeld. This
+model describes a system of spin-1/2 degrees of freedom with ferromagnetic interactions with
+uniaxial anisotropy in the presence of a transverse uniform magnetic ﬁeld. The Hamiltonian is
+H = −J
+�
+⟨r,r′⟩
+σ3(r)σ3(r′) − h
+�
+r
+σ1(r)
+(23)
+where J and h are the exchange coupling constant and the strength of the transverse ﬁeld, re-
+spectively. Here σ1 and σ3 are the two Pauli matrices deﬁned on the sites {r} of a lattice with
+ferromagnetic interactions between spins on nearest neighboring sites. The Hilbert space is the
+tensor product of the states of the spins at each site of the lattice. At each site there are two nat-
+ural bases of states: the eigenstates of σ3, which we denote by | ↑⟩ and | ↓⟩ (whose eigenvalues
+are ±1), and the eigenstates of σ1, which we denote by |±⟩ (whose eigenvalues are also ±1).
+It is well known that the Ising Model in a Transverse Field on a hypercubic lattice in D
+dimensions is equivalent to a classical Ising model in D + 1 dimensions [70, 71]. These two
+models are related through the transfer matrix. Indeed, a classical Ising model can be regarded
+as a path integral representation of the quantum model in one dimension less. For simplicity
+we will see how this work for the 2D the classical ferromagnetic Ising model of Eq.(1), but the
+construction is general. We will regard the conﬁguration of spins on a row of the 2D lattice as
+a state of a quantum system, and the set of states on all rows as the evolution of the state along
+the perpendicular direction that we will regard as a discretized imaginary time. The contribution
+from two adjacent rows to the partition function deﬁnes the matrix element of a matrix between
+two arbitrary conﬁgurations. In Statistical Physics this matrix is known as the Transfer Matrix ˆT
+and the full partition function (with periodic boundary conditions) is
+Z = tr ˆT Nτ
+(24)
+where Nτ is the number of rows. For the case of the ferromagnetic Ising model (actually, for any
+unfrustrated model) the transfer matrix can always be constructed to be hermitian. This property
+12
+
+holds in fact for any theory that satisﬁes a property known as reﬂection positivity which requires
+that all (suitably deﬁned) correlation functions be positive. For theories of this type, and the Ising
+model is an example, the matrix elements of the transfer matrix can be identiﬁed with the matrix
+element of the evolution operator of a quantum theory for a small imaginary time step [70]. Also,
+the positivity of the correlators is equivalent to the condition of positivity of the norm of states in
+the quantum theory.
+For the classical models that satisfy these properties, all directions of the lattice are equiv-
+alent. Moreover, asymptotically close to the critical point, the behavior of all the correlators
+becomes isotropic, i.e. invariant under the symmetries of Euclidean space. This means that
+the arbitrary choice of the direction for the transfer matrix is irrelevant. Consequently, tin the
+quantum model its equal-time correlators behave the same way as its correlation functions in
+imaginary time. In other words space and time behave in the same way and the quantum the-
+ory is relativistically invariant. This implies at the quantum critical point the energy ε(p) of its
+massless excitations should behave as ε(p) = v |p|. In a relativistic theory the dynamical critical
+exponent must be z = 1 and the coeﬃcient v is the speed of the excitations (the “speed of light”).
+We should note that this is not necessarily always the case. There are in fact classical systems,
+e.g. liquid crystals [72], which are spatially anisotropic and map onto quantum mechanical theo-
+ries in one less dimension for which the dynamical critical exponent z � 1. One such example are
+the Lifshitz transitions of nematic liquid crystals in three dimensions and the associated quantum
+Lifshitz model in D = 2 dimensions, for which the dynamical exponent is z = 2 [73].
+Just as in the classical counterpart in D + 1 dimensions, the quantum model in D ≥ 1 has
+two phases: a broken symmetry ferromagnetic phase for J ≫ h and a symmetric paramagnetic
+phase for h ≫ J. In the symmetric phase the ground state is unique (asymptotically is the
+eigenstate of σ1 with eigenvalue +1), while in the broken symmetry phase the ground state
+is doubly degenerate (and asymptotically is an eigenstate of σ3) and there is a non-vanishing
+expectation value of the local order parameter ⟨σ3(r)⟩ � 0. In the symmetric phase the correlation
+function of the local order parameter decays exponentially with distance with a correlation length
+ξ, as does the connected correlation function in the broken symmetry phase. The model has
+a continuous quantum phase transition at a critical value of the ratio h/J. For general space
+dimensions D > 1 this model is not exactly solvable and much of what we know about it is due
+to large-scale numerical simulations.
+This problem was solved exactly in one-dimension [23] using the Jordan-Wigner transfor-
+mation that maps a one dimensional quantum spin system to a theory of free fermions [22]. The
+fermion operators at site j are
+χ1(j) = K(j − 1)σ3(j),
+χ2(j) = i ˆK(j)σ3(j)
+(25)
+where K(j) is the kink creation operator (i.e. the operator that creates a domain wall between
+sites j and j + 1 [70], and is given by
+K(j) =
+�
+n≤ j
+σ1(n)
+(26)
+The operators χ1(j) and χ2(j) are hermitian, χ†
+j(n) = χ j(n), and obey the anticommutation algebra
+{χ j(n), χ j′(n′)} = 2δ j j′δnn′
+(27)
+Hence, they are fermionic operators are hermitian, anti-commute with each other and square to
+the identity. Operators of this type are called Majorana fermions.
+Alternatively, we can use the more conventional (Dirac) fermion operators c(n) and its adjoint
+c†(n) which are related to the Majorana fermions as
+c(n) = χ1(n) + iχ2(n),
+c†(n) = χ1(n) − iχ2(n)
+(28)
+which obey the standard anticommutation algebra
+{c(n), c(n′)} = {c†(n), c†)n′)} = 0,
+{c(n), c†(n′)} = δ − nn′
+(29)
+In this sense, a Majorana fermion is half of a Dirac fermion.
+In terms of the Majorana operators the Hamiltonian of Eq.(23) becomes
+H = i
+�
+j
+χ1(j)χ2(j) + ig
+�
+j
+χ2(j)χ1(j + 1)
+(30)
+13
+
+where we have rescaled the Hamiltonian by a factor of h and the coupling constant is g = J/h.
+Here we have not speciﬁed the boundary conditions (which depend on the fermion parity). Qual-
+itatively, the Majorana fermions can be identiﬁed with the domain walls of the classical models.
+In the Ising model the number of domain walls on each row is not conserved but their parity is.
+Likewise, the number of Majorana fermions NF is not conserved either but the fermion parity,
+(−1)NF, is conserved.
+It is an elementary excercise to show that the spectrum of this theory has a gap G(g) which
+vanishes at gc = 1 as G(g) ∼ |g − gc|ν, with an exponent ν = 1. Since the Hamiltonian of
+Eq.(30) is quadratic in the Majorana operators, these operators obey linear equations of motion.
+In the scaling regime we take the limit of the lattice spacing a → 0 and the coupling constant
+g → gc = 1, while keeping the quantity m = (g − gc)/a ﬁxed. In this regime, the two-component
+hermitian spinor ﬁeld χ = (χ1, χ1), and χ† = χ, satisﬁes a Dirac equation
+(i/∂ − m)χ = 0
+(31)
+where we set the speed � = 1, and where deﬁned the 2 × 2 Dirac gamma-matrices γ0 = σ2,
+γ1 = iσ3, and γ5 = σ1. Upon deﬁning ¯χ = χTγ0, we ﬁnd that the Lagrangian of this ﬁeld theory
+is
+L = ¯χi/∂χ − 1
+2m ¯χχ
+(32)
+which indeed becomes massless at the quantum phase transition of the Ising spin chain. For these
+considerations, we say that the phase transition of the Ising model (2D classical or 1D quantum)
+is in the universality class of massless Majorana fermions where m → 0. In Eq.(32) we have
+used the standard Feynman slash notation, /a = γµaµ, where aµ is a vector.
+4.3. Quantum Antiferromagnets and Non-Linear Sigma Models
+As we noted above, the discovery of high temperature superconductors in the copper ox-
+ide compounds prompted the study of the behavior of these strongly correlated materials at low
+temperatures and of possible quantum phase transitions which they may host. The prototypical
+cuprate material La2CuO4 is a quasi two-dimensional Mott insulator which exhibits long-range
+antiferromagnetic order below a critical temperature Tc. A simple microscopic model is a spin-S
+quantum Heisenberg antiferromagnet on the 2D square lattice of the Cu atoms, whose Hamilto-
+nian is
+H = 1
+2
+�
+r,r′
+J(|r − r′|) S(r) · S(r′)
+(33)
+where S are the spin-S angular momentum operators. We will consider the case where the ex-
+change interaction for nearest neighbors J is dominant and a weaker J′ ≪ J for next nearest
+neighbors. In this section we do not consider the regime J′ ≃ J in which the interactions com-
+pete for incompatible ground states due to frustration eﬀects.
+4.3.1. Spin coherent states
+The simplest way to see the physics of this antiferromagnet is to construct a path-integral
+representation for a spin-S system using spin coherent states [74, 75, 76]. For details see Ref.[9]
+which we follow here. A coherent state of the (2S + 1-dimensional) spin-S representation of
+SU(2) is the state |n⟩, labeled by the spin polarization unit vector n
+|n⟩ = eiθ(n0×n)·S |S, S ⟩
+(34)
+where n2 = 1. The states of the spin-S representation are spanned by the eigenstates of S 3 and
+S2,
+S 3 |S, M⟩ = M|S, M⟩,
+S2|S, M⟩ = S (S + 1)|S, M⟩
+(35)
+and |S, S ⟩ is the highest weight state with eigenvalues S and S (S + 1). In Eq.(34) n0 is a unit
+vector along the axis of quantization (the direction e3), and θ is the colatitude, such that n · n0 =
+cos θ. Two spin coherent states, |n1⟩ and |n2⟩, are not orthonormal,
+⟨n1|n2⟩ = eiΦ(n1,n2,n0) S
+�1 + n1 · n2)
+2
+�S
+(36)
+where Φ(n1, n2, n0) is the area of the spherical triangle of the unit sphere spanned by the unit
+vectors n1, n2 and n0. However, there is an ambiguity in the deﬁnition of the area of the spherical
+14
+
+triangle since the sphere is a 2-manifold without boundaries: if the “inside” triangle has spherical
+area Φ, the complement (“outside”) triangle has area 4π − Φ. Thus, the ambiguity of the phase
+prefactor of Eq.(36) is
+ei4πS = 1
+(37)
+since S is an integer or a half-integer. So, the quantization of the representations of SU(2) makes
+the ambiguity unobservable. In addition, the spin coherent states |n⟩ satisfy the resolution of the
+identity
+I =
+�
+|n⟩⟨n|
+�2S + 1
+4π
+�
+δ(n2 − 1) d3n
+(38)
+and
+⟨n|S|n⟩ = S n
+(39)
+4.3.2. Path integral for a spin-S degree of freedom
+As an example consider problem of a spin-S degree of freedom coupled to an external mag-
+netic ﬁeld B(t) that varies slowly in time. The (time-dependent) Hamiltonian is given by the
+Zeeman coupling
+H(t) = B(t) · S
+(40)
+As usual, the path-integral is obtained by inserting the (over-complete) set of coherent states at
+a large number of intermediate times. The resulting path integral is a sum of the histories of the
+spin polarization vector n(t)
+Z = tr exp
+�
+i
+� T
+0
+dt H(t)
+�
+=
+�
+Dn exp (iS[n])
+�
+t
+δ(n2(t) − 1)
+(41)
+where the action is
+S = S SWZ[n] − S
+� T
+0
+dt B(t) · n(t)
+(42)
+where SWZ[n] is the Wess-Zumino action
+SWZ[n] =
+� T
+0
+dt A[n] · ∂tn
+(43)
+=
+� 1
+0
+dτ
+� T
+0
+dt n(t, τ) · ∂tn(t, τ) × ∂τn(t, τ)
+(44)
+where A[n] is the vector potential of a Dirac magnetic monopole (of unit magnetic charge) at
+the center of the unit sphere. The vector potential A[n] has a singularity associated with the
+Dirac string of the monopole. We can write an equivalent expression which is singularity-free
+using Stokes Theorem. We did this in the second line of Eq.(44) which required to extend the
+circulation of A on the closed path described by n(t) to the ﬂux of the vector potential through
+the submanifold Σ of the unit sphere S 2 whose boundary is the history n(t), i.e. the area of Σ.
+The smooth (and arbitrary) extension of conﬁguration n(t) to the interior of Σ is done by deﬁning
+n(t, τ) such that n(t, 1) = n0, n(t, 0) = n(t), and n(0, τ) = n(T, τ). Since SWZ is the area of the
+submanifold Σ of the unit sphere S 2, just as in Eq.(37), here too there is an ambiguity of 4π in
+the deﬁnition of the area. Here too, this ambiguity is invisible since the spin S is an integer or a
+half-integer.
+The path integral of Eq.(41) was derived ﬁrst by Michael Berry [77] (and extended by Barry
+Simon [78]). The ﬁrst term (which we called Wess-Zumino by analogy with its ﬁeld theoretic
+versions) is called the Berry Phase. The role of this term, which is ﬁrst order in time derivatives,
+is to govern the quantum dynamics of the spin which, in presence of a uniform magnetic ﬁeld,
+executes a precessional motion of the (Bloch) sphere. It is also apparent from this expression that
+in the large-S limit, the path integral can be evaluated by means of a semiclassical approximation.
+The coherent-state construction shows that this problem is equivalent to the path integral
+of a formally massless non-relativistic particle of unit electric charge on the surface of the unit
+sphere with a magnetic monopole of magnetic charge S in its interior! This is not surprising
+since the Hilbert space of a non-relativistic particle moving on the surface of a sphere with and
+radial magnetic ﬁeld (the ﬁeld of a magnetic monopole) has a Landau level type spectrum with
+a degeneracy given by the ﬂux. The condition of a massless particle means that only the lowest
+Landau level survives and all other levels have an inﬁnite energy gap.
+The coherent state approach has been used to derive a path integral formulation for ferromag-
+nets and antiferromagnets. A detailed derivation can be found in Ref. [9].
+15
+
+4.3.3. Quantum Ferromagnet
+We will consider ﬁrst the simpler case of a quantum ferromagnet and in Eq.(33) we will set
+J = −|J| < 0 for nearest neighbors and zero otherwise. The action for the path-integral for the
+spin-S quantum Heisenberg ferromagnet on a hypercubic lattice is
+S = S
+�
+r
+SWZ[n(r, t)] − |J|S 2
+2
+�
+⟨r,r′⟩
+� T
+0
+dt �n(r, t) − n(r′, t)�2
+(45)
+where we have subtracted the classical ground state energy. The oder parameter for this theory
+is the expectation value of the local magnetization, n = ⟨n(r)⟩, which is constant in space but
+points in an arbitrary direction in spin space.
+In the low energy regime the important conﬁgurations are slowly varying in space and we
+can simply approximate the action of Eq.(45) by its continuum version in d space dimensions
+S = S
+ad
+0
+�
+ddx SWZ[n(x, t)] − |J|S 2
+2ad
+0
+�
+ddx
+� T
+0
+dt (▽n(x, t))2
+(46)
+where a0 is the lattice spacing. As before, the path integral; is done for a ﬁeld which satisﬁes ev-
+erywhere in space-time the constraint n2(x, t) = 1. This action can be regarded as non-relativistic
+non-linear sigma model.
+It is straightforward to show that the classical equations of motion for this theory are the
+Landau-Lifshitz equations
+∂tn = |J|S a2
+0 n × ▽2n
+(47)
+subject to the constraint n2 = 1. Due to the constraint, the Landau-Lifshitz equation is non-linear.
+We will a decomposition of the ﬁeld into a longitudinal and two transverse components, σ and
+π, respectively
+n =
+�σ
+π
+�
+(48)
+subject to the constraint σ2 +π2 = 1. The linearized Landau-Lifshitz equations become (to linear
+order in π)
+∂tπ1 ≃ −|J|S a2
+0 ▽2 π2,
+∂tπ2 ≃ +|J|S a2
+0 ▽2 π1
+(49)
+The solution to these equations are ferromagnetic spin waves (magnons or Bloch waves) which
+satisfy the dispersion relation
+ω(p) ≃ |J|S a2
+0p2 + O(p4)
+(50)
+which shows that the dynamic exponent for a ferromagnet is z = 2. Notice that in this case the
+two transverse components are not independent (they are eﬀectively a dynamical pair). These
+are the Goldstone bosons of a ferromagnet.
+4.3.4. Quantum Antiferromagnet
+Formally, the quantum antiferromagnet has a coherent state path integral whose action is
+S = S
+�
+r
+SWZ[n(r, t)] − JS 2
+2
+�
+⟨r,r′⟩
+� T
+0
+dt n(r, t) · n(r′, t)
+(51)
+with J > 0. For a bipartite lattice, e.g. the 1D chain, and the square and cubic lattices, the classi-
+cal ground state is an antiferromagnet with a N´eel order parameter, the staggered magnetization.
+Let m(r) be the expectation value of the local magnetization. A bipartite lattice is the union of
+two interpenetrating sublattices, and the local magnetization is staggered, i.e. it takes values with
+opposite signs (with equal values) on the two sublattices. Thus, we make the change of variables,
+n(r, t) → (−1)rn(r, t) in Eq.(51) and ﬁnd
+S = S
+�
+r
+(−1)rSWZ[n(r, t)] − JS 2
+2
+�
+⟨r,r′⟩
+� T
+0
+dt (n(r, t) − n(r′, t))2
+(52)
+We want to obtain the low energy eﬀective action for the ﬁeld n(r, t). To this end, we decompose
+this ﬁeld into a slowly varying part, that we will call m(r, t), and a small rapidly varying part
+l(r, t) (which represents ferromagnetic ﬂuctuations)
+n(r, t) = m(r, t) + (−1)ra0l(r, t)
+(53)
+16
+
+Since n2(r, t) = 1, we will demand that the slowly varying part also obeys the constraint,
+m2(r, t) = 1, and require that the two components be orthogonal to each other, m · l = 0.
+Due to the behavior of the staggered Wess-Zumino terms of Eq.(52), the resulting continuum
+ﬁeld theory turns out to have subtle but important diﬀerences between one dimension and higher
+dimensions. Here we will state the results for two and higher dimensions. We will discuss in
+detail the one-dimensional below when we discuss the role of topology.
+It turns out that if the dimension d > 1, the contribution of the staggered Wess-Zumino terms
+for smooth ﬁeld conﬁgurations is [75, 79, 80]
+lim
+a0→0 S
+�
+r
+(−1)rSWZ[n(r, t)] = S
+�
+d3x l(x, t) · m(x, t) × ∂tm(x, t)
+(54)
+The continuum limit of the second term of Eq.(52) in the case of a two-dimensional system is
+lim
+a0→0
+JS 2
+2
+�
+⟨r,r′⟩
+� T
+0
+dt �n(r, t) − n(r′, t)�2 = a0
+JS 2
+2
+�
+d3x
+�
+(▽m(x, t))2 + 4l2(x, t)
+�
+(55)
+The massive ﬁeld l[x, t] represents ferromagnetic ﬂuctuations. Since this is a massive ﬁeld it can
+be integrated-out leading to an eﬀective ﬁeld theory for the antiferromagnetic ﬂuctuations m(x, t)
+whose Lagrangian is that of a non-linear sigma model
+L = 1
+2g
+� 1
+vs
+(∂tm(x, t) − vs(▽m(x, t))2
+�
+(56)
+where the coupling constant is g = 2/S and the spin-wave velocity is vs = 4a0JS . If we to allow
+for a weak next-nearest-neighbor interaction J′ > 0, the coupling constant g and the spin wave
+velocity vs become renormalized to g′ ≃ g/ √1 − 2J′/J and v′
+s ≃ vs
+√1 − 2J′/J.
+We conclude that that the quantum ﬂuctuations about a N´eel state are well described by a non-
+linear sigma model. Provided the frustration eﬀects of the next-nearest-neighbor interactions are
+weak enough, the long-range antiferromagnetic N´eel order should extend up to a critical value
+of the coupling constant gc where the RG beta function has a non-trivial zero, which signals a
+quantum phase transition to a strong coupling phase without long-range antiferromagnetic order.
+Motivated by the discovery of high temperature superconductivity in the strongly correlated
+quantum antiferromagnet La2CuO4 (at ﬁnite hole doping) in 1988 Chakravarty, Halperin and
+Nelson [81] utilized a quantum non-linear sigma model to analyze this system and its quantum
+phase transition. La2CuO4 is a quasi-two-dimensional material and so it exhibits strong quantum
+and thermal ﬂuctuations. The upshot of this analysis is that while at T = 0 the non-linear sigma
+model has a quantum phase transition, at T > 0 the long range order is absent in a strictly 2D
+system but present in the actual material due to the weak-three-dimensional interaction. So, in
+the strict 2D case there is no phase transition but two diﬀerent crossover regimes: a renormal-
+ized classical regime (without long range order), a quantum disordered regime and a quantum
+critical regime. La2CuO4 has long range N´eel (antiferromagnetic) order at T = 0 and is in the
+renormalized classical regime (with long range order due to the weak 3D interaction).
+The non-linear sigma model does not describe the nature of the ground state for g > gc
+beyond saying that there is no long range order. The problem is that, unlike the Ising model in a
+transverse ﬁeld, the microscopic tuning parameter is the next nearest neighbor antiferromagnetic
+coupling J′, and to reach the regime g ≃ gc one has to make J′ ≃ J. This is the regime in which
+frustration eﬀects become strong. In this regime the assumption that the important conﬁgurations
+are smooth and close to the classical N´eel state is incorrect. The nature of the ground state turns
+out to depend on the value of S .
+5. Topological Excitations
+Topology has come to play a crucial role both in Condensed Matter Physics and in Quan-
+tum Field Theory. Topological concepts have been used to classify topological excitations such
+as vortices and dislocations and to provide a mechanism for phase transitions, quantum num-
+ber fractionalization, tunneling processes in ﬁeld theories, and nonperturbative construction of
+vacuum states. Here we will discuss a few representative cases of what has become a very vast
+subject.
+17
+
+5.1. Topological Excitations: Vortices and Magnetic Monopoles
+In Condensed Matter Physics topological excitations play a central role in the description of
+topological defects and on their role in phase transitions. Here topology integers in the classiﬁ-
+cation of the conﬁguration space into equivalence classes characterized by topological invariants
+[82]. The most studied example are vortices. Vortices play a key role in the mixed phase of type
+II superconductors in a uniform magnetic ﬁeld [83]. Vortices also play a key role in the Statistical
+Mechanics of 2D superﬂuids and the the 2D classical XY model [84, 85] [86, 87] [88]. Disloca-
+tions and disclinations play an analogous role in the theory of classical melting [84, 89, 90], and
+2D and 3D classical liquid crystals [91, 92, 72].
+A similar problem occurs in Quantum Field Theory. Theories with global symmetries, such
+as the two-dimensional O(3) non-linear sigma model discussed above, when formulated in Eu-
+clidean space-time have instantons. Typically instantons are ﬁnite Euclidean action conﬁgu-
+rations, which are also classiﬁed into equivalence classes (associated with homotopy groups)
+labeled by topological invariants [93, 94]. Instantons play a central role in understanding the
+non-perturbative structure of gauge theories. Gauge theories with a compact gauge group cou-
+pled to matter ﬁelds have non-trivial vortex [95] and monopole [96, 97, 98] conﬁgurations, as
+do non-abelian Yang-Mills gauge theories [99]. Instantons have also played a central role in
+Condensed Matter Physics as well, notably in Haldane’s work on 1D quantum antiferromag-
+nets (discussed below), and in the problem of macroscopic quantum tunneling and coherence
+[100, 101].
+5.1.1. Vortices in two dimensions
+In this section I will focus on the the problem of the superﬂuid transition in 2D and the
+closely related problem of the phase transition of a magnet with an easy-plane anisotropy, the
+classical XY model. A superﬂuid is described by an order parameter that is a one-component
+complex ﬁeld φ(x). If electromagnetic ﬂuctuations are ignored, this description also applies
+to a superconductor. The complex ﬁeld can be written in terms of an amplitude |φ(x)|, whose
+square represents the local superﬂuid density, and a phase θ(x) = arg(φ(x)).
+Deep in the
+super���uid phase the amplitude is essentially constant, that we will set to be a real positive
+number φ0, while the phase ﬁeld θ(x) is periodic with period 2π and can ﬂuctuate.
+Simi-
+larly, an easy-plane ferromagnet is described by a two-component real order parameter ﬁeld
+M(x) = (M1(x), M2(x)) = |M(x)|(cosθ(x), sin θ(x)). Deep in the ferromagnetic phase the ampli-
+tude |M| is essentially constant but the phase ﬁeld θ(x) can ﬂuctuate.
+We will assume that we are in a regime where the local superﬂuid density |φ0|2 is well formed
+(or, equivalently that |M| is locally well formed) but that the phase ﬁeld is ﬂuctuating. In this
+regime the problem at hand is an O(2) ≃ U(1) non-linear sigma model, and its partition function
+takes the form
+Z =
+�
+Dθ exp
+�
+−
+�
+d2x 1
+2g
+�
+∂µθ(x)
+�2�
+(57)
+where we deﬁned the coupling constant g = T/J|φ0|2, where T is the temperature, J is an in-
+teraction strength, and |φ0|2 is the magnitude (squared) of the amplitude of the order parameter,
+which we will take to be constant; κ = J|φ0|2 is the phase stiﬀness.
+Except for the requirement that the phase ﬁeld be locally periodic, θ ≃ θ + 2π, superﬁcially
+this seems to be a trivial free (Gaussian) ﬁeld theory. We will see that the periodicity (or, com-
+pactiﬁcation) condition makes this theory non-trivial. Indeed, conﬁgurations of the phase ﬁeld
+that are weak enough that that do not see the periodicity condition, for all practical purposes, can
+regarded as being non-compact and ranging from −∞ to +∞. However there are many conﬁgu-
+rations for which the periodicity condition is essential. Such conﬁgurations are called vortices.
+Even in the absence of vortices, the periodic (compact) nature of the phase ﬁeld is essential
+to the physics of this problem. In fact the only allowed observables must be invariant under
+local periodic shifts of the phase ﬁeld. This implies that the phase ﬁeld θ itself is not a physical
+observable but that exponentials of the phase of the form exp(inθ(x)) are physical. This operator
+is just the order parameter ﬁeld of the XY model. In Conformal Field theory operators of this
+type are called vertex operators [102, 103]. We will see below that this theory has a dual ﬁeld ϑ,
+associated with vortices, and that there are vertex operators of the dual ﬁeld. In String Theory the
+model of a compactiﬁed scalar is known as the compactiﬁed boson and represents the coordinate
+of a string on a compactiﬁed space, in this case a circle S 1 [104].
+To picture a vortex consider a large closed curve C on the 2D plane. Hence, topologically
+a closed curve is isomorphic to a circle, C ≃ S 1. The phase ﬁeld θ(x) is equivalent to a unit
+18
+
+circle S 1. Therefore the conﬁguration space are maps of S 1 (the large circle) onto S 1 (the unit
+circle of the order parameter space). The conﬁgurations can be classiﬁed by the number of times
+the phase winds on the large circle C. The winding number is an integer called the topological
+charge of the conﬁguration, the vorticity. Thus, a vortex is a conﬁguration of the phase ﬁeld θ(x)
+that winds by 2πm (where m is an integer):
+(∆θ)C
+2π
+= 1
+2π
+�
+C
+dx · ▽θ(x) ≡ i
+� 2π
+0
+dϕ
+2π eiθ(ϕ)∂ϕe−iθ(ϕ) = m
+(58)
+where ϕ ∈ (0, 2π] is the azimuthal angle for a vector at the center of the large circle C. Here
+n is the vorticity or winding number of the conﬁguration; m > 0 is a vortex and m < 0 is an
+anti-vortex. The vorticity is a topological invariant of the ﬁeld the conﬁguration θ(x) which does
+not change under smooth changes.
+The winding number of a vortex is a topological invariant that classiﬁes the conﬁgurations
+of the phase ﬁeld as continuous maps of a large circle S 1 onto the unit circle S 1 deﬁned by the
+phase ﬁeld. In Topology such continuous maps are called homotopies. The winding number
+classiﬁes these maps into a discrete set of equivalence classes, which form a homotopy group
+under the composition of two conﬁgurations. In this case the homotopy group is called Π1(S 1).
+Since the equivalence classes are classiﬁed by a topological invariant that takes integer values,
+the homotopy group Π1(S 1) is isomorphic to the group of integers, Z [82].
+The ﬁeld jµ(x) = ∂µθ(x) is the superﬂuid current, and the vorticity ω(x) is the curl of the
+current, i.e.
+ω(x) = ǫµν∂µ jν(x) = ǫµν∂µ∂νθ(x)
+(59)
+which vanishes unless θ(x) has a branch-cut singularity across which the phase ﬁeld jumps by
+2πn. Let ω(x) be the vorticity ﬁeld with singularities at the locations {x j} of vortices with topo-
+logical charge m j
+ω(x) = 2π
+�
+j
+m jδ2(x − x j)
+(60)
+which is satisﬁed by the phase ﬁeld conﬁguration
+θ(x) =
+�
+j
+2πm jIm ln(z − zj)
+(61)
+where we have used the complex coordinates z = x1 + ix2. Away from the singularities {x j},
+this conﬁguration obeys the Laplace equation. Hence, it has a Cauchy-Riemann dual ﬁeld ϑ(x)
+which satisﬁes the Cauchy-Riemnann equation
+∂µϑ = ǫµν∂νθ
+(62)
+which satisﬁes the Poisson equation
+− ▽2ϑ(x) = ω(x)
+(63)
+whose solution is
+ϑ(x) =
+�
+d2y G(|x − y|) ω(y)
+(64)
+where G(|x − y|) is the Green function of the 2D Laplacian
+− ▽2G(|x − y|) = δ2(x − y)
+(65)
+In 2D this Green function is
+G(|x − y|) = 1
+2π ln
+�
+a
+|x − y|
+�
+(66)
+where a is a short distance cutoﬀ (a lattice spacing). In what follows we will assume that the
+Green function of Eq.(66) has been cutoﬀ so that G(|x − y|) = 0 for |x − y| ≤ a.
+The energy of a conﬁguration of vortices {n j} with vanishing total vorticity, �
+j m j = 0, is
+E[θ] = Jφ2
+0
+2
+�
+d2x
+�
+∂µθ
+�2
+= Jφ2
+0
+2
+�
+d2x
+�
+d2y ω(x) G(|x − y|) ω(y) = 2πJφ2
+0
+�
+j>k
+m jmk ln
+�
+a
+|x j − xk
+�
+(67)
+19
+
+where we used that conﬁgurations with non-vanishing vorticity do not contribute to the partition
+function since they have inﬁnite energy in the thermodynamic limit. We conclude that, up to an
+unimportant prefactor, that the partition function of Eq.(57) is the same as the partition function
+a gas of charges {m j} (the vortices) with total vanishing vorticity, �
+j m j = 0,
+Z2DCG =
+�
+[{mj}]
+exp
+−2π Jφ2
+0
+T
+�
+j<k
+m jmk ln
+�|x j − xk|
+a
+�
+(68)
+which is known as the two-dimensional (neutral) Coulomb gas [84].
+Kosterlitz and Thouless argued that the neutral two dimensional Coulomb gas has a phase
+transition a critical temperature TKT between a low temperature phase dielectric phase in which
+vortices and anti-vortices are bound into neutral pairs and a high temperature phase in which
+pairs unbind, the vortex charges are screened and the vortex gas is in a plasma phase. A simple
+estimate of the critical temperature is found by computing the free energy of a single vortex
+Fvortex = Evortex − TS vortex, where Evortex of a single vortex and S vortex is the vortex entropy. In
+2D the energy of a single vortex of unit topological charge, n = 1, is πJφ2
+0 ln(L/a), where L is the
+linear size of the system, and the vortex entropy is S vortex = ln(L/a)2, where (L/a)2 is the number
+of places where a vortex can be located, i.e. the area of the system in units of the spacing a.
+Vortices unbind (proliferate) when the entropy beats the energy. Hence, the critical point occurs
+at a temperature TKT where the vortex free energy vanishes, Fvortex[TKT] = 0. This yields the
+estimate
+TKT = π
+2 Jφ2
+0
+(69)
+which is the Kosterlitz-Thouless critical temperature.
+There is an alternative, equivalent way to see this physics. Let is consider the theory if
+Eq.(57) in the background of a U(1) gauge ﬁeld Aµ(x)
+Z[A] =
+�
+Dθ exp
+�
+− 1
+2g
+�
+d2x
+�
+∂µθ − Aµ
+�2�
+(70)
+where the curl of the background gauge ﬁeld Aµ is chosen to be equal to the local vorticity of
+Eq.(59),
+ǫµν∂µAν(x) = ω(x)
+(71)
+In other words, the gauge Aµ ﬁeld forces the phase ﬁeld to have vortices at the locations {x j}.
+Next we will use a Hubbard-Stratonovichtransformation (a Gaussian identity) to write Eq.(70)
+in the equivalent form
+Z[A] =
+�
+Dθ
+�
+Daµ exp
+�
+−g
+2
+�
+d2x a2
+µ + i
+�
+d2x aµ(x)
+�
+∂µθ(x) − Aµ(x)
+��
+(72)
+where we neglected an unimportant prefactor. In Eq.(72) the phase ﬁeld can be integrated-out
+(after an integration by parts) to yield the constraint ∂µaµ = 0. This constraint is solved by
+introducing the dual ﬁeld ϑ(x) such that
+aµ(x) = ǫµν∂νϑ(x)
+(73)
+Hence, the partition function Z[A] becomes
+Z[A] =
+�
+Dϑ exp
+�
+−g
+2
+�
+d2x (∂µϑ)2 + i
+�
+d2x ϑ(x) ω(x)
+�
+(74)
+where we have assumed that the total vorticity is zero,
+�
+d2x ω(x) = 2π �
+j m j = 0. This identity
+show that the quantization of the vortex topological charge requires that the dual ﬁeld ϑ should
+also be periodic (compact) with period 1.
+It is elementary to see that after performing the gaussian integral over the ﬁeld ϑ in Eq.(74)
+and summing over all vortex conﬁgurations (with total vanishing vorticity) we reproduce the
+partition function of the 2D neutral Coulomb gas [105, 106]. Thus, the theory of Eq.(70) deﬁned
+in terms of the phase ﬁeld θ and the theory of Eq.(74) are equivalent. This is an example of a
+duality transformation which amounts to the exchange of the ﬁelds
+θ ↔ ϑ,
+and
+g ↔ 1/g
+(75)
+20
+
+A more careful analysis done on the lattice shows that the dual ﬁeld ϑ is deﬁned on the dual of
+the square lattice (which is also a square lattice) [88, 105]. In addition, by diﬀerentiating both
+the partition function of the phase ﬁeld of Eq.(70) and the partition function of the dual ﬁeld
+ϑ of Eq.(74) with respect to the background ﬁeld Aµ we obtain the duality identiﬁcation of the
+currents
+1
+g∂µθ ←→ gǫµν∂νϑ
+(76)
+as an operator identity. This identity has the same form as the duality of forms in geometry.
+In this case, the dual of the current ∂µθ, which is a 1-form ﬁeld, is another 1-form ﬁeld, ∂µϑ.
+In both cases the ﬁeld θ and its dual ϑ are associated with global symmetries. Later we will
+encounter similar duality identiﬁcations but their content will be diﬀerent in diﬀerent dimensions.
+Moreover, running these identities backwards it is easy to see that the insertion of the order
+parameter operator exp(ipθ(x)) (with p and integer), which acts as a source on the phase ﬁeld θ,
+in the partition function is is equivalent to minimally couple the dual ﬁeld ϑ to a singular gauge
+ﬁeld Bµ(x) whose curl is p at the location x. In other words, charges are dual to vortices.
+We will now use the identity we derived in Eq.(74) to compute the full partition function
+which is a sum over all vortex conﬁgurations. Now we will assume that the vortices have a large
+core energy (i.e. the energy associated with the short distance behavior of the Green function of
+Eq.(66)). For a vortex with unit topological charge we will call this energy u, and for a vortex
+with charge m the core energy will be um2. The eﬀects of the core energy can be accounted for
+in terms of a vortex fugacity z = exp(−un2/T). The total partition function now is
+Z =
+�
+{mj}
+�
+Dϑ exp
+−g
+2
+�
+d2x (∂µϑ)2 + i
+�
+j
+2πm jϑ(x j) − u
+T
+�
+j
+m2
+j
+
+(77)
+where, once again, we assumed global charge neutrality.
+In the limit in which z = exp(−u/T) ≪ 1, only the vortices with the lowest topological
+charges m = ±1 contribute and, furthermore, they are dilute. In this limit we can perform over
+the conﬁgurations with the lowest topological charge and derive the eﬀective ﬁeld theory of the
+dual ﬁeld ϑ:
+ZSG =
+�
+Dϑ exp
+�
+−
+�
+d2x
+�g
+2(∂µϑ)2 − � cos(2πϑ)
+��
+(78)
+which is known as the sine-Gordon ﬁeld theory. The coupling constant of the sine-Gordon theory
+is � = 2 exp(−u/T)/a2, where a is the core size (the lattice spacing). Hence, the cosine operator
+of the sine-Gordon theory represents the eﬀects of the vortices.
+We can now analyze the role of vortices by looking at the renormalization group of the sine-
+Gordon theory. The ﬁrst question is what is the scaling dimension of the cosine operator. This
+can be computed by looking at the vortex operator correlator in the theory with � = 0:
+⟨exp(i2πϑ(x)) exp(−i2πϑ(y))⟩ =
+const.
+|x − y|2π/g
+(79)
+which implies that the vortex scaling dimension is
+∆vortex = π
+g
+(80)
+Eq.(12) implies that in D = 2 dimensions the operator is irrelevant if ∆ > 2, relevant for ∆ < 2
+and marginal for ∆ = 2. Hence, vortices are marginal if ∆vortex = 2 which happens if g = π/2.
+This is the same as to say that the system is at the Kosterlitz-Thouless critical temperature T =
+TKT. Hence, vortices are irrelevant if T < TKT and relevant for T > TKT.
+This RG analysis tells us that T > TKT, when vortices proliferate, the coupling constant ν
+ﬂows to strong coupling to a regime where ϑ is pinned to an integer value and the theory is in a
+massive phase. In this phase the connected vortex correlator decays exponentially with distance
+which is the same as to say that the vortex charge is screened. This is why in this phase the
+vortices proliferate. It can be shown that in this phase the correlator of the spins of the XY
+model, i.e. the correlator of the vertex operator exp(iθ(x)), decays exponentially with distance
+and the systems is in its disordered phase.
+There is still the question of the nature of the phase with T < TKT. The Mermin-Wagner The-
+orem [107] (and its generalizations by Hohenberg [108] and Coleman [109]) states that classical
+21
+
+statistical mechanical systems with a global continuous symmetry group cannot undergo spon-
+taneous symmetry breaking in space dimensions D ≤ 2 (and quantum systems with space-time
+dimensions D ≤ 2). Does this theory violate this the Mermin-Wagner Theorem? The answer is
+no. It is easy to see that in the phase in which the vortices are irrelevant, i.e. for T < TKT, the
+correlator of the order parameter operator is always a power law of the distance |x − y|,
+⟨exp(iθ(x)) exp(−iθ(y))⟩ =
+const.
+|x − y|g/2π
+(81)
+with an exponent that depends on temperature and satisﬁes
+g
+2π =
+T
+2πJφ2
+0
+≤ 1
+4
+(82)
+In other words, the entire low temperature phase is not an ordered phase of matter since the cor-
+relator is not constant at long distance. On the other hand, the correlators of the order parameter
+exp(iθ) and of the vortices exp(iϑ) have a power la behavior for T < TKT, we conclude that in
+this temperature range the system is scale invariant and that it has a line of critical points.
+In summary, we succeeded in expressing the partition function as a sum over conﬁgurations
+of the topological excitations, the vortices. We succeeded in doing that because vortices are
+labeled by their coordinates on the plane. In addition, we found a non-trivial phase transition
+since the entropy and the energy both scale logarithmically with the linear size of the system
+or, equivalently, that vortices became marginally relevant at a critical temperature. This is the
+mechanism behind the Kosterlitz-Thouless transition.
+One may ask if this construction is generic and the answer is no. As an example consider the
+Abelian Higgs model in D = 2. This model has a complex scalar ﬁeld minimally coupled to a
+Maxwell gauge ﬁeld and, hence, its gauge group is U(1). In the classical spontaneously broken
+phase, the gauge ﬁeld becomes massive. In this phase the long range coherence of the phase ﬁeld
+of the vortices of the scalar ﬁeld is screened at the scale of the penetration depth. As a result
+the Euclidean action of the vortices is now ﬁnite. Furthermore, on longer scales the interaction
+energy between vortices becomes short ranged. Hence, instead of a 2D Coulomb gas now one
+has a gas of particles (vortices and anti-vortices) with short range interactions. In this case the
+entropy always dominates and the vortices proliferate [110]. This behavior is quite analogous to
+the restoration of symmetry by proliferation of domain walls in one-dimensional classical spin
+chains [19] and to the analogous problem of tunneling in the path integral formulation of quantum
+mechanical double-well potentials [93]. As a caveat, we should note that, in spite of the obvious
+similarities, the 2D Abelian Higgs model does not describe correctly a 2D superconductor (i.e. a
+superconducting ﬁlm) since the electromagnetic ﬁeld is not conﬁned to the ﬁlm and this renders
+the electromagnetic action nonlocal.
+5.1.2. Magnetic monopoles in compact electrodynamics
+Instantons are of great interest in Quantum Field Theory since they provide for a mechanism
+to understand the non-perturbative structure of these theories. For this reasons they have been
+used to understand the mechanisms of quark conﬁnement and the role of quantum anomalies
+in non-Abelian gauge theories [99, 97, 98, 96, 111]. Instantons in non-Abelian gauge theories
+are magnetic monopoles and the condensation of monopoles have long been argued to be the
+mechanism behind quark conﬁnement.
+The simplest non-Abelian gauge theory that has magnetic monopoles is the Georgi-Glashow
+[112]. This model has a three-component real ﬁeld φ and an SU(2) Yang-Mills gauge ﬁeld Aµ. In
+its Higgs phase the scalar ﬁeld φ acquires an expectation value which breaks the gauge symmetry
+group SU(2) down to its diagonal U(1) subgroup. Since U(1) ⊂ SU(2), this Abelian gauge group
+is compact, meaning that its magnetic ﬂuxes are quantized. Polyakov [113] and ’t Hooft [111]
+showed that in 2+1 Euclidean dimensions have non-singular instanton solutions which at long
+distances resemble the magnetic monopole originally proposed by Dirac in 1931 [114]
+Bi(x) = q
+2
+xi
+|x|2 − 2πqδi,3δ(x1)δ(x2)θ(−x3)
+(83)
+The ﬁrst term in Eq.(83) is the magnetic radial ﬁeld of a monopole of magnetic charge q. The
+second term represents an inﬁnitely long inﬁnitesimally thin solenoid ending at the location of
+the monopole, x = 0, that supplies the quantized magnetic ﬂux 2πq. This singular term is
+known as the Dirac string. The string itself (and its orientation) is physically unobservable to any
+22
+
+electrically charged particle that obeys the Dirac quantization condition, qe = 2π (in units where
+ℏ = c = 1).
+In the language of a lattice gauge theory [32], a theory with a compact (i.e. periodic) U(1)
+gauge ﬁelds on a D = 3 cubic lattice, describing a compact gauge ﬁeld in 2 + 1 dimensions.
+This theory should have instantons that resemble magnetic monopoles much in the same way as
+a theory with a compact global U(1) symmetry has vortices. The simplest example is Polyakov’s
+compact electrodynamics [97] whose partition function is
+Z =
+�
+x,µ
+� 2π
+0
+dAµ(x)
+2π
+exp
+
+1
+4e2
+�
+x,µ,ν
+cos(Fµν(x))
+
+(84)
+where Fµν(x) = ∆µAν(x)−∆νAµ(x) ≡ �
+µ Aµ is the magnetic ﬂux through the elementary plaquette
+labeled by a site x and a pair of directions, µ and ν, with µ = 1, 2, 3. This theory is invariant under
+local gauge transformations Aµ(x) → Aµ(x) + ∆µΦ(x) and it is also invariant under local periodic
+shifts of the gauge ﬁelds Aµ(x) → Aµ(x) + 2πℓµ(x), where ℓµ(x) ∈ Z. The plaquette ﬂux operator
+satisﬁes the lattice version of the Bianchi identity that the product of exponentials of the ﬂux on
+the faces of every elementary cube of the lattice is
+�
+cubefaces
+eiFµν(x) = 1
+(85)
+which says that the theory can have magnetic monopoles of integer magnetic charge.
+We will analyze this theory following an approach analogous to what we used for vortices in
+section 5.1.1.To this end we will consider the partition function
+Z[Bµν] =
+�
+DAµ exp
+�
+− 1
+4e2
+�
+d3x
+�
+Fµν(x) − Bµν(x)
+�2�
+(86)
+where Fµν(x) = ∂µAν − ∂νAµ is the ﬁeld strength of the abelian U(1) Maxwell gauge ﬁeld Aµ.
+Here Bµν(x) is an (anti-symmetric) two-form background gauge ﬁeld. The coupling constant of
+this theory is e2. Since Aµ is a connection it has units of length−1, and F2
+µν is a dimension 4 ﬁeld.
+Then, in D = 3 dimensions, e2 has units of length−1.
+The theory is invariant under two local transformations, namely the usual invariance under
+gauge transformations
+Aµ(x) → Aµ(x) + ∂µΦ(x),
+Bµν(x) → Bµν(x)
+(87)
+where Φ(x) is an arbitrary smooth function of x. The presence of the background two-form ﬁeld
+Bµν now requires invariance under one-form gauge transformations
+Aµ(x) → Aµ(x) + aµ(x),
+Bµν(x) → Bµν(x) + ∂µaν − ∂νaµ
+(88)
+The two-form gauge ﬁeld Bµν essentially represents the magnetic monopoles. Let {m j} be a
+conﬁguration of monopoles of charges m j with coordinates {x j}, with total vanishing monopole
+charge, �
+j m j = 0. Let M(x) be the magnetic monopole density at x,
+M(x) = 2π
+�
+j
+m j δ3(x − x j)
+(89)
+which can be expressed as the curl of the two-form gauge ﬁeld Bµν,
+M(x) = 1
+2ǫµνλ∂µBνλ(x)
+(90)
+We will proceed next much in the same way as in Eq.(72) and rewrite the partition function
+of Eq.(86) in terms of a two-form Hubbard-Stratonovich ﬁeld bµν(x) such that
+Z[B] =
+�
+DAµ
+�
+Dbµν exp
+�
+−e2
+4
+�
+d3x b2
+µν(x) + i
+�
+d3x 1
+2bµν(x)
+�
+Fµν(x) − Bµν(x)
+��
+=
+�
+DAµ
+�
+Dbµν exp
+�
+−e2
+4
+�
+d3x b2
+µν(x) + i
+�
+d3x
+�
+Aµ(x)∂νbµν(x) − 1
+2bµν(x)Bµν(x)
+��
+(91)
+23
+
+Thus, the gauge ﬁeld Aµ plays the role of a Lagrange multiplier ﬁeld the enforces the constraint
+∂νbµν(x) = 0
+(92)
+which is solved in terms of a compact scalar ﬁeld ϑ(x)
+bµν(x) = ǫµνλ∂λϑ(x)
+(93)
+Using this identity and the deﬁnition of the monopole density M(x) we ﬁnd that the partition
+function Z[Bµν] of Eq.(86) becomes
+Z[B] =
+�
+Dϑ exp
+�
+−
+�
+d3x
+�e2
+2 (∂µϑ(x))2 + iM(x)ϑ(x)
+��
+=
+�
+Dϑ exp
+−e2
+2
+�
+d3x(∂µϑ(x))2 + 2πi
+�
+j
+m jϑ(x j)
+
+(94)
+which requires that the ﬁeld ϑ obeys the compactiﬁcation condition ϑ → ϑ + n, where n is an
+arbitrary integer. Eq.(94) says that the magnetic monopole instantons of the compact U(1) gauge
+theory are dual to charges of the dual phase ﬁeld ϑ, which has a compact U(1) global symmetry.
+The full partition function is obtained by summing over all monopole conﬁgurations satis-
+fying the total neutrality condition, �
+j m j = 0. As in section section 5.1.1, we will weigh the
+conﬁgurations with a coupling u and ﬁnd
+Z =
+�
+{mj}
+Z{m j}
+�
+Dϑ exp
+−e2
+2
+�
+d3x (∂µϑ)2 +
+�
+j
+2πim jϑ(x j) − u
+�
+j
+m2
+j
+
+(95)
+which is the same theory we found in Eq.(77) except that now we are in 3D. Moreover, summing
+only over dilute conﬁgurations of monopoles and anti-monopoles we ﬁnd, once again the sine-
+Gordon theory but now in D = 3 dimensions:
+Z =
+�
+Dϑ exp
+�
+−
+�
+d3x
+�e2
+2 (∂µϑ)2 − � cos(2πϑ)
+��
+(96)
+with � = 2 exp(−u)/a3.
+In spite of the similarities between Eq.(96) and the sine-Gordon theory in 2D, Eq.(78), the
+physics is very diﬀerent. It is straightforward to see that, in the limit v = 0, the monopole operator
+correlator is
+⟨exp �2πiϑ(x)) exp(−2πiϑ(y)� = exp
+�4π2
+e2 [G(|x − y|) − G(0)]
+�
+≃ exp
+� π
+2e2
+� 1
+R − 1
+a
+��
+(97)
+where a is the short-distance cutoﬀ. Unlike the behavior of the correlator of the vortex operators
+in 2D found in Eq.(79), Eq.(97) does not show a power-law behavior. The reason is that at � = 0
+the compactiﬁed ﬁeld ϑ should be regarded as the Goldstone boson of a spontaneously broken
+U(1) symmetry. However, the cosine operator is now always relevant and the ﬁeld ϑ is pinned
+and its ﬂuctuations are actually massive.
+Looking back at the partition function of Eq.(95), we could integrate out the ﬁeld ϑ and
+obtain an expression with the same form as the Coulomb gas of Eq.(68) except that now this is
+the three-dimensional neutral Coulomb gas [97]
+Z3DCG =
+�
+[{mj}]
+exp
+− π
+2e2
+�
+j<k
+m jmk
+�
+1
+|x j − xk| − 1
+a
+�
+(98)
+Now the energy of an isolated monopole is ﬁnite (in the infrared) while the entropy is still log-
+arithmically divergent. The conclusion is that the monopoles proliferate and the 3D neutral
+Coulomb gas is always in a plasma phase withe screened interactions. These results also imply
+that the expectation value of the Wilson loop operator in the U(1) gauge theory decays exponen-
+tially with the area of the loop. The conclusion is that in 2+1 dimensions compact electrody-
+namics is in a conﬁning phase for all values of the coupling constant e2 [97, 98].
+24
+
+5.2. Non-Linear Sigma Models and Antiferromagnetic Quantum Spin Chains
+We will now discuss the one-dimensional case in which topology plays a crucial role. In one
+space dimension, the sites of the chain are labelled by an integer j = 1, . . ., N (with N even).
+After some simple manipulations, the continuum limit of the staggered Wess-Zumino terms of
+Eq.(52) is
+lim
+a0→0 S
+N
+�
+j=1
+(−1) jSWZ[n(j, t)] =S
+2
+�
+d2x m(x, t) · ∂tm(x, t) × ∂xm(x, t)
++ S
+�
+d2x l(x, t) · m(x, t) × ∂tm(x, t)
+(99)
+The continuum limit of the second term of Eq.(52) is essentially the same as in Eq.(55).
+By putting together the results of Eq.(99) and Eq.(55), we ﬁnd that in the case of a 1D chain
+the continuum ﬁeld theory has the Lagrangian density
+L[m, l] = −2a0JS 2l2 + S l · m × ∂tm − a0JS 2
+2
+(∂xm)2 + S
+2 m · ∂tm × ∂xm
+(100)
+Finally, we integrate-out the ferromagnetic ﬂuctuations represented by the massive ﬁeld l(x, t) to
+ﬁnd that the eﬀective low-energy Lagrangian has the form
+L = 1
+2g
+� 1
+vs
+(∂tm)2 − vs (∂xm)2
+�
++ θ
+4π m · ∂xm × ∂tm
+(101)
+which is a non-linear sigma model with N = 3 components (and hence has a global O(3) sym-
+metry) with a topological term whose coupling constant is θ. This result was ﬁrst derived by
+Haldane [115].
+In Eq.(101) the eﬀective coupling constant g, the spin-wave velocity vs, and the θ-angle are
+given by g = 2
+S , vs = 2a0JS , and θ = 2π. As in the 2D case, since g = 2/S , large S implies that
+the non-linear sigma model is in the weak coupling regime. Provided that vs > 0, one can always
+rescale the time and space coordinates without aﬀecting the form of the Lagrangian, including
+the coupling constant or, as we will see, the value of the θ angle. In what follows we will assume
+that we have done the rescaling in such a way that we set vs = 1 and, that time and space scale as
+lengths. Thus we will use a relativistic notation and, after an analytic continuation to imaginary
+time, we label the time coordinate by x2 and the space direction by x1. The partition function
+now takes the form
+Z =
+�
+Dmexp
+�
+− 1
+2g
+�
+Ω
+d2x
+�
+∂µn
+�2 + iθ Q[m]
+�
+(102)
+where Ω is the spacetime manifold. In this notation the ﬁrst term of the exponent is called the
+Euclidean action of the non-linear sigma model. In Eq.(102) we denoted by Q[m] the quantity
+Q[m] = 1
+8π
+�
+Ω
+d2x ǫµνm · ∂µm × ∂νm
+(103)
+Here we will consider the case in which the spacetime manifold Ω is closed. In particular
+we will assume that it is a two-sphere S 2. The quantity Q[m] is the integral of a total derivative
+which counts the number of times the ﬁeld conﬁguration m(x1, x2) wraps around the sphere S 2.
+In other words, it yields a non-vanishing result only for “large” conﬁgurations which wind (or
+wrap) around the sphere S 2. Since Q[n] is an integer, it has the same for all smooth the ﬁeld
+conﬁguration m that can be smoothly deformed into each other and are homotopically equiva-
+lent. If we demand that the ﬁeld conﬁgurations m have ﬁnite Euclidean action, which requires
+that at inﬁnity the conﬁgurations take the same (but arbitrary) value of m, we have eﬀectively
+compactiﬁed the x1 − x2 plane into a two sphere S 2. On the other hand the ﬁeld m is restricted
+by the constraint m2 = 1 to take values on a two-sphere S 2. Therefore, the ﬁeld conﬁgurations
+m(x1, x2) are smooth maps of the S 2 of the coordinate space to the S 2 of the target space of the
+ﬁeld m. Hence, the integer Q[m] classiﬁes the smooth ﬁeld conﬁgurations into a set of equiva-
+lence classes each labeled by the integer Q. Under composition homotopies form groups, and the
+equivalence classes themselves also form a group which, in this case, is isomorphic to the group
+of integers, Z. In Topology, these statements are summarized by the notation Π2(S 2) ≃ Z. The
+25
+
+conﬁgurations with non-zero values of Q are called instantons which, in the quantum problem,
+represent tunneling processes of the non-linear sigma model.
+Another consequence of Q being an integer is that the contribution of its term to the weight of
+the path integral of Eq.(103) is a periodic function of θ angle. On the other hand, since the only
+allowed values of the θ are θ = 0 (mod 2π) for S integer, or θ = π (mod 2π) for S a half-integer,
+the contribution of the topological invariant Q to the weight of the path integral is
+exp(iθ Q[m]) = (−1)2S Q
+(104)
+Therefore, for spin chains with S integer, the weight is 1 and the topological invariant does not
+contribute to the path integral. But, if S is a half-integer, the weight is (−1)Q[m], and it does
+contribute. Moreover, its contribution is the same for all half-integer values of S .
+These results have important consequences for the physics of spin chains which led Haldane
+to some startling conclusions [115]. In the weak coupling regime, g ≪ 1 (equivalently, for
+large S ), we can use the perturbative renormalization group and derive the beta function for all
+these O(3) non-linear sigma models (with or without topological terms) and ﬁnd that their beta
+functions are the same as in Eq.(21) with N = 3. Hence, for all S , the eﬀective coupling ﬂows
+to large values. We can make this inference since the topological term yields no contribution for
+all conﬁgurations which are related by smooth deformations. Thus, we infer that all spin chains
+with S integer are in a massive phase with an exponentially small energy gap ∼ exp(−2πS ). This
+result is nowadays known as the Haldane gap.
+On the other hand, these results also imply that all spin chain with half-integer spin S are
+also the same and, in particular, th same as spin-1/2 chains. However, the Hamiltonian of the
+quantum spin chain with spin-1/2 degrees of freedom is an example of an integrable system and
+its spectrum is known to be gapless from its Bethe Ansatz solution [116, 117]. However, the
+spin-1/2 chain is not only gapless but its low energy states are gapless solitons with a relativistic
+spectrum. The low energy description of a theory of this type must be described by a conformal
+ﬁeld theory. Haldane concluded that the RG ﬂow for spin-1/2 chains must have an IR stable ﬁxed
+point at some ﬁnite (and large) value of the coupling constant. This conjecture was conﬁrmed by
+Aﬄeck and Haldane [118] who showed that the spin-1/2 Heisenberg chains are in the universality
+class of the SU(2)1 Wess-Zumino-Witten model [119] whose CFT was solved by Knizhnik and
+Zamolodchikov [120].
+5.3. Topology and open integer-spin chains
+In the preceding section we showed that integer spin chains have a Haldane gap. We did that
+by showing that in that case the topological term is absent. However, the derivation is correct
+provided the spacetime manifold is closed, e.g. a sphere, a torus, etc. What happens if the system
+has a boundary? Let us denote the boundary of Ω by Γ = ∂Ω. For example we will take Γ to be
+along the imaginary time direction and hence that it is a circle of circumference 1/T, where T
+is the temperature. The topological term has the same form as the Berry phase term of the path
+integral for spin, the “Wess-Zumino” term of Eq.(44), but its prefactor is 1/2 as big. Thus, for
+a system with an open boundary the topological term yields a net contribution equal to a Berry
+phase with a net prefactor of S/2.
+In other words, the boundary of the integer spin chain behaves as a localized degree of free-
+dom whose spin is 1/2 (mod an integer). Form the periodicity requirement we also see that if
+the spin chain is made of odd-integer degrees of freedom, there should be a spin 1/2 degree of
+freedom localized at the open boundary!. Conversely, if the chain is made of even-integer spins,
+there is no boundary degree of freedom! This line of argument implies that an antiferromagnetic
+chain with odd-integer spins must have a spin-1/2 degree of freedom at the boundary whereas
+a chain of even-integer spins should not. Notice that the “bulk” behavior is the same for both
+odd and even integer spin chains. The diﬀerence is whether or not they have a non-trivial “zero-
+mode” state at the boundary. In particular, the existence of this state is robust, i,.e. it cannot
+be removed by making smooth changes to the quantum Hamiltonian or, what is the same, the
+boundary state is topologically protected.
+A simple system that displays a protected spin-1/2 zero mode at the boundary is a generalized
+S = 1 spin chain with Hamiltonian
+H = α
+N
+�
+j=1
+S(j) · S(j + 1) + β
+N
+�
+j=1
+(S(j) · S(j + 1))2
+(105)
+26
+
+where S(j) = (S x(j), S y(j), S z(j) are the spin 1 matrices at each lattice site j. This problem was
+examined in great detail by Aﬄeck, Kennedy, Lieb and Tasaki [121] who showed that at the
+special value of the parameters α = 1/2 and β = 1/6 this Hamiltonian takes the form of a sum of
+projection operators
+H =
+�
+j
+P2(S(j) + S(j + 1))
+(106)
+where P2 is an operator that projects out the spin 2 states. These authors constructed the exact
+ground state, known as the AKLT state, of this Hamiltonian by writing each spin 1 degree of
+freedom of two spin-1/2 degrees of freedom at each site. They showed that the ground state is a
+projected product state in which the “constituent’ spin-1/2 degrees of freedom (each labeled by
++ and − respectively) on nearby sites j and j + 1 are in a valence bond singlet state of the form
+1√
+2(| ↑j,+, ↓j+1,−⟩− ↓j,+, ↑j+1,−⟩ (and symmetrizing at each site to project onto a spin 1 state).
+This state of the spin-1 chain is translation invariant and gapped and hence agrees with Haldane’s
+result. Moreover, it has a spin-1/2 degree of freedom at each open boundary [122].
+The arguments discussed above show that the AKLT state is a gapped topological state. The
+gapless spin-1/2 boundary states are an example of spin fractionalization. The spin-1/2 boundary
+degrees of freedom are an example of an edge state which are present in many, thought not all,
+topological phases of matter. One may ask what symmetry protects the gaplessness of these edge
+degrees of freedom. The only perturbation that would give a ﬁnite energy gap to the spin-1/2
+edge states is an external magnetic ���eld. However, this perturbation would break the global
+SU(2) symmetry of the Hamiltonian as well as time reversal invariance.
+6. Duality in Ising Models
+Duality plays a signiﬁcant role of our understanding of statistical physics and of quantum
+ﬁeld theory. Many seemingly unrelated correspondences between diﬀerent theories have come
+to be called dualities.
+6.1. Duality in the 2D Ising Model
+One of the earliest versions of duality transformations was used to relate the high-temperature
+expansion of the 2D classical Ising model and its low-temperature expansion [123]. In all dimen-
+sions, for concreteness we will think of a hypercubic lattice, the high temperature expansion is a
+representation of the partition function as a sum of contributions of closed loops on the lattice. At
+temperature T, a conﬁguration of loops γ contributes with a weight which in the Ising model has
+the form C(γ) tanhL(γ)(β), where β = 1/T and L(γ) is the length of the loop γ, i.e. the number
+of links on the loop, and C(γ) is an entropic factor that counts the number of allowed loops with
+ﬁxed perimeter L(γ). However not all loop conﬁgurations are allowed as in the Ising model they
+satisfy constraints such as being non-overlapping, etc.
+In section 4.2 we noted that the partition function of the classical Ising model (in any dimen-
+sion) can be interpreted as the path-integral of a quantum spin model in one dimension less on a
+lattice with a discretized imaginary time. In this picture, the loops γ can be regarded as processes
+in which pairs of particles are created at some initial (imaginary) time, evolve and eventually are
+annihilated at a later (imaginary) time. In other words, the high temperature phase is a theory of
+a massive scalar ﬁeld. The restrictions on the allowed loop conﬁgurations represents interactions
+among these particles. In the temperature range in which the expansion in loops is convergent
+the particles are massive as the loops are small. As the radius of convergence of the expansion is
+approached, longer and increasingly fractal-like loops begin to dominate the partition function,
+and concomitantly the mass of the particles decreases. This process is the signal of the approach
+to a continuous phase transition where the particles become massless. In fact, right at the critical
+point the particle interpretation is lost as the associated ﬁelds acquire anomalous dimensions.
+Returning to the 2D classical Ising model, Kramers and Wannier also considered the low
+temperature expansion. This is an expansion around one of the broken symmetry state, e.g.
+the state with all spins up. In this low temperature regime the partition function is a sum of
+conﬁgurations of ﬂipped spins. A typical conﬁguration is a set of clusters of ﬂipped spins. In the
+absence of a uniform ﬁeld, a conﬁguration of ﬂipped spins has a energy cost only on links of the
+lattice with oppositely aligned spins (“broken bonds”). Thus a cluster of ﬂipped spins costs an
+energy equal to 2 (I assumed that I set J = 1) for each broken bond at the boundary of the cluster.
+This boundary is domain wall which is a closed loop on the dual lattice. In 2D the dual of the
+square lattice is the square lattice of the dual sites (the centers of the elementary plaquettes of the
+27
+
+2D lattice). In 2D the links of the direct lattice pierce the links of the dual lattice. We can see
+that this analogous to the the geometric duality of forms: sites (“0-forms”) are dual to plaquettes
+(“2-forms”) and links (“1-forms”) are dual to links (also “1-forms”).
+Thus, in 2D the low temperature expansion is an expansion in the loops γ∗ of the dual lattice
+that represent the domain walls. We can also regard the domain walls (the dual loops) as the
+histories of of pairs of particles on he dual lattice. However, the weight of each dual loop is
+exp(−2β) per link of γ∗. Except for that, the counting and restrictions on the dual loops γ∗
+are the same as those of γ. This means that there is a one-to-one correspondence between the
+two expansions with the replacement tanh β ↔ exp(−2β∗). This mapping means that the dual
+of the 2D classical Ising model (with global Z2 symmetry) at inverse temperature β is a dual
+Ising model (also with global Z2 symmetry) on the dual lattice at inverse temperature β∗. This
+correspondence is in close analogy with what we discussed in section 5.1.1.
+In particular if one assumes that there is a transition at βc = 1/Tc then there should also be a
+transition at β∗
+c = −1/2 lntanh βc. Moreover, if one further assumes (as Kramers and Wannier did
+[123]) that there is a unique transition (correct in the Ising model but not in other cases), then the
+critical point must be such that exp(−2βc) = tanh βc, which yields the value Tc = 2/ ln(
+√
+2 + 1),
+which agrees with the Onsager result [20].
+In section 4.2 we showed that the classical 2D Ising model is equivalent to the one-dimensional
+Ising model in a transverse ﬁeld whose Hamiltonian is given in Eq.(23). The 1D Ising model on
+a transverse ﬁeld has spin degrees of freedom deﬁned on the sites of a one-dimensional chain
+labeled by an integer-valued variable n. The 1D Hamiltonian is expressed in terms of local op-
+erators, the Pauli matrices σ3(n) and σ1(n). The 1D chain has a dual lattice whose sites are the
+midpoints of the chain. Thus in 1D sites (“0-forms”) are dual to links (“1-forms”) and viceversa.
+We will now see that there is a Hamiltonian version of the Kramers-Wannier duality [70].
+In Eq.(26) we introduced the kink creation operator which ﬂips are σ3 operators to the left and
+including site j. For clarity we will denote the kink creation operator operator as τ3(˜n), where
+˜n is the site of the dual lattice between the sites n and n + 1 of the original lattice. We will now
+deﬁne an operator τ1(˜n) = σ3(n)σ3(n+1). The operators τ1(˜n) and τ3(˜n) satisfy the same algebra
+os the Pauli operators σ1(n) and σ3(n). Furthermore, we readily ﬁnd the Hamiltonian of the dual
+theory is the same (up to boundary conditions) as the original Hamiltonian of Eq.(23) except
+that the dual coupling constant is λ∗ = 1/λ. So, once again, if we assume that there is a single
+(quantum) phase transition we require λ∗
+c = λc which is only satisﬁed by λc = 1. In this language
+the kink creation operator plays the role of a disorder operator [70].
+6.2. The 3D duality: Z2 gauge theory
+We will now discuss the role of duality in the 3D Ising model on a cubic lattice. This case,
+and its generalizations to higher dimensions, was considered ﬁrst by Franz Wegner [124].
+In section 6.1 we discussed the loop representation of the high temperature expansion and
+showed that it has the form form in all dimensions. Hence, in 3D the high temperature expansion
+is an expansion in closed loops γ with weight of tanh β per unit link of loop. Hence, in 3D as
+well, the high temperature phase can be regarded as ﬁeld theory of a massive scalar ﬁeld. As in
+the 2D case, the weight of a loop conﬁguration is tanh β for each link of the loop γ.
+However, the low temperature expansion has a radically diﬀerent form and physical inter-
+pretation. Much as in 2D, the low temperature expansion is an expansion in clusters of ﬂipped
+spins. Here too, in the absence of an external ﬁeld, the only energy cost resides at the boundary
+of the clusters of overturned spins, the domain walls. But in 3D the clusters occupy volumes
+whose boundaries are closed surfaces Σ∗. In 3D a link (bond) is dual to a plaquette (expressing
+the fact that in 3D a 1-form is dual to a 2-form). Hence the closed domain walls of overturned
+spins is dual to a closed surface Σ∗ on the dual lattice. The weight of each conﬁguration of closed
+surfaces is exp(−2β) for each plaquette of the surface Σ∗.
+These facts mean that the dual of the 3D Ising model is a theory on the dual lattice with
+coupling constant β∗, with exp(−2β∗) = tanh β, such that its expansion for small β∗ is a sum over
+conﬁgurations of closed surfaces σ∗, and for large β∗ is a sum of conﬁgurations of closed loops
+γ∗ on the dual lattice. The dual theory is the naturally deﬁned on (dual) plaquettes, not on links.
+To this end, let us deﬁne a set of Ising-like degrees of freedom σµ(x) = ±1 located on the links
+(x, µ) of the dual lattice. These degrees of freedom are coupled on each plaquette of the lattice.
+Since each plaquette has four links, the coupling involves the degrees of freedom on all four links
+28
+
+of each plaquette. The partition function of the dual theory is
+Z =
+�
+{σµ(x)}
+exp
+β∗
+�
+plaquettes
+σµ(x)σν(x + eµ)σµ(x + eµ + eν)σν(x)
+
+(107)
+where the sum in th exponent (the negative of the “action”) runs on all the plaquettes of the dual
+lattice.
+The action of the theory of Eq.(107) is invariant under the reversal of all six Ising degrees of
+freedom on links sharing a given site x. This is a local symmetry. Unlike the 3D Ising model,
+which has a global Z2 symmetry of ﬂipping all spins simultaneously, this theory has local (or
+gauge) Z2 symmetry. This is the simplest example of a lattice gauge theory in which the degrees
+of freedom are gauge ﬁelds that take values on the Z2 gauge group [124, 32, 71].
+We saw that in the spin model the high temperature expansion (i.e. the expansion in powers of
+tanh β) is a sum over loop conﬁgurations which can be interpreted in terms of processes in which
+pairs of particles are created, evolve (in imaginary time) and then are destroyed (also in pairs).
+The analogous interpretation of the expansion of the Z2 gauge theory in powers of tanh β∗ =
+exp(−2β) as a sum over the conﬁgurations of closed surfaces is not in terms of the histories of
+particles but in terms of the histories of closed strings which, as they evolve, sweep the closed
+surfaces. The physics is, however, more complex as the sum over surfaces runs over all surfaces
+of arbitrary topology with arbitrary number of handles (or genus). Thus, over (imaginary) time a
+closed strong is created, evolves, splits into two closed strings, etc.
+Therefore, the 3D Ising model, which has a Z2 global symmetry is dual to a Z2 gauge theory
+which has a Z2 local symmetry. The duality maps the high temperature (disordered) phase of
+the Ising model to the strong coupling (small β∗) phase of the gauge theory. In the gauge theory
+language this is the conﬁning phase. This can be seen by computing the expectation value of the
+Wilson loop operator on the closed loop Γ, which here reads ⟨�
+(x,µ)∈Γ σµ(x)⟩. In the small β∗
+phase this expectation value decays exponentially with the size of the minimal surface bounded
+by the loop Γ. This behavior of the area law of the Wilson loop. Under duality the insertion of
+this operator is equivalent to an Ising model with a domain wall terminating on the loop Γ, and
+the area law of the Wilson loop is the consequence of the fact that if the Ising model has long
+range order, the free energy cost of the domain wall scales with its area. Moreover, in this phase
+of the gauge theory the closed strings are small meaning that the string tension (the energy per
+unit length of string) is ﬁnite. In the Ising model language, the string tension becomes surface
+tension of the domain wall.
+In the preceding subsection we discussed a Hamiltonian version of the duality. We will brieﬂy
+do the same in the case of the 2+1 dimensional Ising model. The hamiltonian of the Ising model
+in a transverse ﬁeld on a square lattice is
+H2D−TFIM = −
+�
+r
+σ1(r) − λ
+�
+r, j=1,2
+σ3(r)σ3(r + e j)
+(108)
+where r labels the sites of the square lattice and e j (with j = 1, 2) are the two (orthonormal)
+primitive unit vectors of the lattice. Here too, at each site we have a two-level system (the spins),
+σ1 and σ3 are Pauli matrices acting on these states at each site and λ is the coupling constant.
+Just as in the 1D case, this system has two phases: an ordered phase for λ > λc and a disordered
+phase for λ < λc, where λc is a critical coupling. This Hamiltonian is invariant under the global
+Z2 symmetry generated by the global spin ﬂip operator Q = �
+r σ1(r) which commutes with the
+Hamiltonian, [Q, H2D−TFIM] = 0.
+Let is consider now a Z2 gauge theory on the dual of the square lattice. We will now deﬁne
+a two-dimensional Hilbert space on each link, denoted by (˜r, j) (with j = 1, 2) of the dual lattice
+with sites labelled by ˜r. We will further deﬁne at each link (˜r, j) the Pauli operators σ1
+j(˜r) and
+σ3
+j(˜r). The Hamiltonian of the Z2 gauge theory is
+HZ2gauge = −
+�
+˜r, j
+σ1
+j(˜r) − g
+�
+˜r
+σ3
+1(˜r)σ3
+2(˜r + ˜e1)σ3
+1(˜r + ˜e1 + ˜e2)σ3
+2(˜r)
+(109)
+where ˜e j = ǫjkek (with j, k = 1, 2).
+Unlike the Hamiltonian of Eq.(108), the Hamiltonian of Eq.(109) has a local symmetry. Let
+˜Q(˜r) be the operator that ﬂips the Z2 gauge degrees of freedom on the four links that share the
+site ˜r,
+˜Q(˜r) =
+�
+j=1,2
+σ1
+j(˜r)σ1
+j(˜r − ˜e j)
+(110)
+29
+
+These operators commute with each other, [ ˜Q(˜r), ˜Q(˜r′)] = 0, and commute with the Hamiltonian,
+[HZ2gauge, ˜Q(˜r)] = 0. The Hilbert space of gauge-invariant states are eigenstates of ˜Q(r) with unit
+eigenvalue,
+˜Q(r)|Phys⟩ = |Phys⟩
+(111)
+(for all r). This constraint is the the Z2 Gauss Law. The Hamiltonian HZ2gauge has two phases: a
+conﬁning phase for g < gc, and a deconﬁned phase for g > gc (where gc is a critical coupling).
+The duality transformation is deﬁned so that
+σ1(r) =
+�
+plaquette(r)
+σ3
+j(˜r)
+(112)
+is the product of the σ3
+j operators of the gauge theory on a dual plaquette centered at the site r,
+and
+σ1
+1(˜r) = σ3(r)σ3(r + e2),
+σ1
+2(˜r) = σ3(r)σ3(r + e1)
+(113)
+These identities imply that the gauge theory constraint of the Z2 Gauss Law, Eq.(111), is satisﬁed.
+This also means that duality is a mapping of the gauge-invariant sector of the Z2 gauge theory
+to the Hilbert space of the Ising model in a transverse ﬁeld.
+Eq.(112) and Eq.(113) imply that the Hamiltonians H2D−TFIM and HZ2gauge are equal to each
+other with the identiﬁcation of the coupling constants g = 1/λ. Hence, the ordered phase of the
+Ising model, λ > λc, maps onto the conﬁning phase of the Z2 gauge theory and, conversely, the
+disordered phase of the Ising model maps onto the deconﬁned phase of the gauge theory.
+Eq.(113) also implies that the operator σ3(r) of the Ising model can be identiﬁed with the
+operator
+σ3(r) =
+�
+γ(r)
+σ1
+j(˜r)
+(114)
+where the product is on the links of the dual lattice pierced by the path γ(r) on the direct lattice
+ending at the site r.
+While σ3(r) is of course just the order parameter of the Ising model, the dual operator deﬁned
+by Eq.(114) anti-commutes with the plaquette term of the Hamiltonian of Eq.(109) and creates an
+Z2 ﬂux excitation at the plaquette. With some abuse of language, this operator can be regarded as
+creating a Z2 “magnetic monopole”. Since in the conﬁning phase this operator has an expectation
+value, we can regard this phase as a magnetic condensate.
+We will now discuss the role of boundary conditions which is diﬀerent in both theories. Let
+us examine the behavior of the Z2 gauge theory with periodic boundary conditions. Periodic
+boundary conditions means that the 2D space is topologically a 2-torus. It is straightforward to
+show that the ground state in the conﬁning phase is unique and insensitive to boundary condi-
+tions. The physics of the deconﬁned phase is more subtle. We will now see that on a 2-torus
+it has a four-fold degenerate ground state and that the degeneracy is not due to the spontaneous
+breaking of a global symmetry.
+Let γ1 and γ2 be two non-contractible loops on the torus along the directions 1 and 2 re-
+spectively. Let us consider the (“electric”) Wilson loop operators along γ1 and γ2, W[γ1] =
+�
+(r, j)∈γ1 σ3
+j(r) and similarly for γ2. Similarly let us consider the (“magnetic”) ’t Hooft operators
+˜W[˜γ1] and ˜W[γ2] deﬁned on the non-contractible closed paths of the dual lattice ˜γ1 and ˜γ2, such
+that ˜W[˜γ1] = �
+˜γ1 σ1
+2(˜r) and ˜W[˜γ2] = �
+˜γ2 σ1
+1(˜r). It is easy to see that the Wilson and’t Hooft
+operators satisfy that W[γi]2 = ˜W[˜γ j]2 = I, and that
+[W[γi], W[γ j] = [ ˜W[˜γi], ˜W[˜γ j]] = 0,
+{W[γ1], ˜W[˜γ2]} = {W[γ2], ˜W[˜γ1]} = 0
+(115)
+Let us consider the special limit of g → ∞. In this limit the Wilson loop operators W[γi] on
+the two non-contractibleloops of the 2-torus commute with the Hamiltonian HZ2gauge of Eq.(109).
+Therefore the eigenstates of the Hamiltonian can be chosen to be the eigenstates of these two Wil-
+son loops. Since the Wilson loop operators are hermitian and obey W[γi]2 = I, the spectrum of
+each loop is two-dimensional | ± 1⟩i (i = 1, 2) with eigenvalues ±1, respectively. At g → ∞ these
+states are also eigenstates of HZ2gauge. Thus, the ground state of HZ2gauge is four dimensional. The
+’t Hooft loops ˜W[˜γi] act as ladder operators in this restricted Hilbert space.
+The conclusion is that, on a 2-torus, at least in the limit g → ∞, the ground state of HZ2gauge
+is degenerate. However, this degeneracy is not due to the spontaneous breaking of a global
+symmetry. Rather, it reﬂects the topological character of the theory. To see this, one can extend
+this analysis to a theory to be on a more general two-dimensional and instead of a 2-torus we can
+30
+
+consider a surface with g handles (not to be confused with the coupling constant!), e.g. g = 0 for
+a sphere (or disk), g = 1 for a 2-torus, g = 2 for a pretzel, etc. For each of these two-dimensional
+surfaces the number diﬀerent number of non-contractible loops is g, and the Wilson loops deﬁned
+on them commute with each other (and with HZ2gauge). Hence, in the limit g → ∞ HZ2gauge has
+a ground state degeneracy of 2g which, clearly, depends only on the topology of the surface. We
+conclude that, at least as g → ∞, the Hamiltonian HZ2gauge is in a topological phase.
+However, is the exact degeneracy found in the limit g → ∞ a property of the entire deconﬁned
+phase? For a ﬁnite system the expansion in powers of 1/g is convergent. On the other hand, in
+the thermodynamic limit, L → ∞, the expansion has a ﬁnite radius of convergence with gc being
+an upper bound. Let us consider the ground state at g = ∞ with eigenvalue +1 for the Wilson
+loop W[γ1]. The degenerate state with eigenvalue −1 is created by the ’t Hooft loop ˜W[˜γ2], i.e.
+| − 1⟩1 = ˜W[˜γ2] | + 1⟩1. The ’t Hooft operator is a product of σ1 operators on links along the
+direction 1 crossed by the path ˜γ2 of the dual lattice, which involves (at least) L links. Then, it
+takes L orders in the expansion in powers of 1/g to mix the state | + 1⟩1 with the state | − 1⟩1,
+and this amplitude is of order 1/gL, which is exponentially small. The same argument applies to
+the mixing between all four states. For a ﬁnite but large system of linear size L, the degeneracy
+is lifted but the energy splitting is exponentially small in the system size. Hence, the topological
+protection is a feature of the deconﬁned phase in the thermodynamic limit L → ∞.
+We conclude that the deconﬁned phase of the Z2 lattice gauge theory is a topological phase.
+This result extends to the case of a gauge theory with discrete gauge group Z, the cyclic group
+of k elements. In general spacetime dimension D > 2 the Zk gauge theory also has a conﬁned
+and a deconﬁned phase and if k ≳ 4 it also has an intermediate “Coulomb phase”[125, 126]. In
+section 9.4 we will see that the low-energy (IR) regime of the deconﬁned phase is described by
+a topological quantum ﬁeld theory known as the level k BF theory.
+7. Bosonization
+The behavior of one-dimensional electronic systems is of great interest in Condensed Matter
+Physics for many reasons. One is that, even for inﬁnitesimally weak interactions, these one-
+dimensional metals violate the basic principles of the Landau Theory of the Fermi Liquid [13].
+A central assumption of Femi Liquid theory is that at low energies the excitation energy of the
+fermion (electron) quasiparticle is always much larger than its width. Hence, at asymptotically
+low energies the quasiparticle excitations become increasingly sharp.
+A manifestation of this feature is that the quasiparticle propagator (the “Green function”) has
+a pole on the real frequency axis with a ﬁnite residue Z. In one-dimensional metals this assump-
+tion always fails since the residue Z vanishes, the Fermi ﬁeld acquires a non-trivial anomalous
+dimension, and the pole is replaced by a branch cut. To this date this is the best example of what
+is called a ”non-Fermi liquid”. In one-dimensional metals this non-Fermi liquid is often called a
+“Luttinger Liquid” [127]. A detailed analysis can be found in chapter 6 of Ref. [9], and in other
+books.
+Bosonization provides for a powerful tool to understand the physics of these non-trivial sys-
+tems. Bosonization is a duality between a massless Dirac ﬁeld in 1+1 dimensions and a (also
+massless) relativistic Bose (scalar) ﬁeld [128, 129, 130, 131]. In this context, bosonization is a set
+of operator identities relating observables between two diﬀerent (dual) continuum ﬁeld theories.
+These identities have a close resemblance to the Jordan-Wigner transformation which relates op-
+erators of a theory of spinless (Dirac) fermions on a one-dimensional lattice to a theory of bosons
+with hard cores (i.e. spins) on the same lattice [22].
+7.1. Dirac fermions in one space dimensions
+To understand how these operator identities come about and what these ﬁelds mean in the
+Condensed Matter context we will consider the simple problem of a system of non-interacting
+spinless fermions c(n) on a one-dimensional chain of length L (with L → ∞), for simplicity with
+periodic boundary conditions. The Hamiltonian is
+H0 = −t
+L
+�
+n=1
+c(n)†c(n + 1) + h.c.
+(116)
+In momentum space, and in the thermodynamic limit L → ∞, the Hamiltonian becomes
+H0 =
+� π
+−π
+dp
+2π (ε0(p) − µ) c†(p)c(p)
+(117)
+31
+
+where −π ≤ p ≤ π, µ is the chemical potential (which ﬁxes the fermion number) and ε0(p) is the
+(free) quasiparticle energy which, in this case, is
+ε0(p) = −2t cos p
+(118)
+This simple system has a global internal symmetry
+c′(n) = eiαc(n),
+c′(n)† = e−iαc(c)
+(119)
+where α is a constant phase (with period 2π). This global symmetry reﬂects the conservation of
+fermion number
+NF =
+�
+n
+c†(n)c(n)
+(120)
+The free fermion Hamiltonian of Eq.(116) is also invariant under lattice translations.
+The ground state of this system is obtained by occupying all single particle states with energy
+below the chemical potential, E ≤ µ ≡ EF, which deﬁnes the Fermi energy EF. In what follows
+we will redeﬁne the zero of the energy at the Fermi energy. In the thermodynamic limit the
+one-particle states are labeled by momenta deﬁned in the ﬁrst Brillouin zone [−π, π). The ground
+state |G⟩ of this system is obtained by ﬁlling up all single-particle states with momentum p in
+the range [−pF, pF), where pF is the Fermi momentum. Hence, the occupied states have energy
+ε0(p) ≤ EF. The ground state is
+|G⟩ =
+�
+|p|≤pF
+c†(p)|0⟩
+(121)
+and is called the ﬁlled Fermi sea. We will further assume that the fermionic system is dense in
+the sense that the occupied states are a ﬁnite fraction of the available states, i.e. we assume that
+|EF| is a ﬁnite fraction of the bandwidth W = 4t.
+The single-particle excitations have low energy if |ε0(p) − EF| ≪ |EF|. This range can only
+be accessed by quasiparticles with momenta p close to ±pF. For states wit p close to pF we can
+a linearized approximation and write
+ε0(p) ≃ �F(p − pF) + . . .
+(122)
+and similarly for the states near −pF. Here �F = ∂ǫ0
+∂p
+���pF = 2t sin pF is the Fermi velocity. Let q
+being the momentum measured from pF (or −pF in the other case), This approximation is correct
+in a range of momenta −Λ ≤ q ≤ Λ, where 2π
+L ≪ Λ ≪ π. In this regime we can approximate
+the lattice ﬁelds c(n) with two continuum right moving ψR(x) and left moving ψL(x) Fermi ﬁelds,
+such that (with x = na0)
+c(n) ∼ eipF xψR(x) + e−ipF xψL(x)
+(123)
+in terms of which the Hamiltonian takes the continuum form
+Hcontinuum =
+�
+dx
+�
+ψ†
+R(x)(−i�F)∂xψR(x) − ψ†
+L(x)(−i�F)∂xψL(x)
+�
+=
+� dq
+2πq�F
+�
+ψ†
+R(q)ψR(q) + ψ†
+R(q)ψR(q)
+�
+(124)
+In what follows I will rescale time and space in such a way as to set �F = 1.
+Eq.(124) is the Hamiltonian of a massless Dirac ﬁeld in 1+1 dimensions. It describes the
+eﬀective low energy behavior of the excitations of the fermionic system deﬁned on a lattice by
+the Hamiltonian of Eq.(116). Here low energy means asymptotically close to the Fermi energy
+(which we have set to zero) and momenta close to ±pF. We will see that this eﬀective relativistic
+ﬁeld theory gives a complete description of the universal low energy physics encoded in the
+microscopic model of Eq.(116) up to some subtle issues associated with what are called quantum
+anomalies.
+Let us denote by ψ(x) the bi-spinor ﬁeld ψ(x) = (ψR(x), ψL(x)) and the 2 × 2 Dirac gamma-
+matrices in terms of the three Pauli matrices are
+γ0 = σ1,
+γ1 = −iσ2,
+γ5 = σ3
+(125)
+which obey the Dirac algebra
+{γµ, γν} = 2gµν
+(126)
+32
+
+where gµν is the metric tensor in 1+1-dimensional Minkowski spacetime
+gµν =
+�1
+0
+0
+−1
+�
+(127)
+Using the standard notation ¯ψ(x) = ψ†(x)γ0 (where I left the spinor index α = 1, 2 implicit)
+the Lagrangian density of the free massless Dirac fermion is
+L = ¯ψi/∂ψ
+(128)
+where we have used the Feynman slash notation which denotes the contraction of a vector ﬁeld,
+say Aµ with the Dirac gamma matrices with a slash, Aµγµ ≡ /A.
+Formally, the Dirac lagrangian of Eq.(128) (and the Dirac Hamiltonian of Eq.(124)) are in-
+variant under two separate global transformations. It has a global U(1) (gauge) symmetry trans-
+formation
+ψ′(x) = eiθψ(x)
+(129)
+(where θ is constant) under which the two components of the bi-spinor ψ transform in the same
+way
+ψ′
+R(x) = eiθψR(x),
+ψ′
+L(x) = eiθψL(x)
+(130)
+The massless Dirac theory is also invariant under a global gauge U(1) chiral transformation
+ψ′(x) = eiθγ5ψ(x)
+(131)
+which in components it reads
+ψ′
+R(x) = eiθψR(x),
+ψ′
+L(x) = e−iθψL(x)
+(132)
+In general, if a system has a global continuous symmetry it should have a locally conserved
+current and a globally conserved charge. This is the content of Noether’s Theorem. Since the
+massless Dirac theory has these two global symmetries one would expect that it should have two
+separately conserved currents.
+The global U(1) symmetry has an associated current jµ = (j0, j1) given by
+jµ = ¯ψγµψ
+(133)
+which is invariant under the global U(1) symmetry. In terms of the right and left moving Dirac
+ﬁelds, ψR and ψL, the components of the current are
+j0 = ψ†
+RψR + ψ†
+LψL,
+j1 = ψ†
+RψR − ψ†
+LψL
+(134)
+This current is locally conserved and satisﬁes the continuity equation
+∂µ jµ = 0
+(135)
+It has an associated conserved global charge, which we will call fermion number
+Q =
+� ∞
+−∞
+dxj0(x) =
+� ∞
+−∞
+dx
+�
+ψ†
+RψR + ψ†
+LψL
+�
+(136)
+This is the continuum version of the conservation of fermion number NF of the free fermion
+lattice model discussed above.
+Since this current is conserved it can be coupled to an electromagnetic ﬁeld through a term
+in the Lagrangian
+Lint = −ejµAµ
+(137)
+where Aµ is the electromagnetic vector potential, which in 1+1 dimensions has only two com-
+ponents, Aµ = (A0, A1). Here e is a coupling constant which is interpreted as the electric charge.
+This coupling amounts to making the U(1) symmetry a local gauge symmetry under which the
+ﬁelds transform as follows
+ψ′(x) = eiθ(x)ψ(x),
+A′
+µ(x) = Aµ(x) + 1
+e∂µθ(x)
+(138)
+33
+
+Now the conservation of fermion number becomes the conservation of the total electric charge
+Qe = −eQ
+(139)
+In a continuum non-relativistic model the Fermi ﬁeld is ψ(x) (ignoring spin), e.g. an electron
+gas in one dimension such as a quantum wire, the analog of decomposition shown in Eq.(123) of
+the electron ﬁeld ψ(x) becomes
+ψ(x) = eipF xψR(x) + e−ipF xψL(x)
+(140)
+The electron ﬁeld ψ(x) transforms as ψ′(x) = eiαψ(x) under the global U(1) gauge symmetry of
+Eq.(130). In particular, the local density operator ρ(x) = ψ†(x)ψ(x) is invariant under this sym-
+metry. Under both decompositions of Eqs. (123) and (140), the local density operator becomes
+ρ(x) = ψ†
+R(x)ψR(x) + ψ†
+L(x)ψL(x) + ei2pF xψ†
+R(x)ψL(x) + e−i2pF xψ†
+L(x)ψR(x)
+(141)
+Clearly the non-relativistic density operator ρ(x) is invariant under the global U(1) gauge sym-
+metry. The same considerations applies to the lattice version of the density operator (the local
+occupation number).
+Thus, the density operator ρ(x) can be decomposed into a slowly varying part (the ﬁrst two
+terms in Eq.(141)) and the last two terms which oscillate with wave vectors Q = ±2pF. These
+observations imply that we can express ρ(x) in the form
+ρ(x) = ¯ρ + j0(x) + eiQxψQ(x) + e−iQxψ−Q(x)
+(142)
+This decomposition can be interpreted as a Fourier expansion of the density operator in term of
+newly deﬁned slowly-varying ﬁelds. In Eq.(142) Q = 2pF and ¯ρ is the average density, where we
+assumed that the Dirac density operator j0(x) has vanishing expectation value (i.e. it is normal-
+ordered). In Eq.(142) we deﬁned the (bosonic) operators
+ψQ(x) = ψ†
+R(x)ψL(x),
+ψ−Q(x) = ψ†
+L(x)ψR(x)
+(143)
+which characterize the oscillatory component of the density. The operators ψ±Q(x) are the order
+parameters of a charge density wave state in one dimension and Q is the ordering wavevector.
+Repulsive interactions between the electrons can cause scattering process between the right
+and left moving components to become relevant (in the Renormalization Group sense) leading
+to the spontaneous breaking of translation invariance. The resulting state is known as a charge-
+density-wave (CDW). In this state the operators ψ±Q(x) (or a linear combination of them) acquire
+a non-vanishing expectation value, and the expectation value of the density operator ρ(x) has a
+(static) modulated component, and the Dirac Hamiltonian density becomes
+H = ψ†
+R(x)(−i)∂xψR(x) − ψ†
+L(x)(−i)∂xψL(x) + m
+�
+ψ†
+R(x)ψL(x) + ψ†
+L(x)ψR(x)
+�
+(144)
+where m is the Dirac mass. The operator in the second term of Eq.(144) mixes the right and left
+moving components of the Dirac spinor. This hermitian operator is known as the Dirac mass
+term. In relativistic notation this operator is written
+¯ψψ = ψ†
+R(x)ψL(x) + ψ†
+L(x)ψR(x)
+(145)
+When m � 0, i.e. when ⟨ ¯ψψ⟩ � 0, the fermionic spectrum has a mass gap, and the electronic
+states with momenta ±pF have an energy gap. Alternatively we could have considered a state
+with the in which the (hermitian) operator that has an expectation value is
+i ¯ψγ5ψ = i
+�
+ψ†
+R(x)ψL(x) − ψ†
+L(x)ψR(x)
+�
+(146)
+which is known as the γ5 mass term. In a more general CDW state both mass terms can be
+present.
+7.2. Chiral symmetry and chiral symmetry breaking
+The CDW states we introduced have diﬀerent symmetries. To see this let us observe that the
+operators ψ±Q(x) transform non-trivially under a global U(1) chiral transformation
+ψ′
+±Q(x) = e∓2iθψ±Q(x)
+(147)
+34
+
+Upon substituting this transformation in the expansion of the density ρ(x) of Eq.(142), we see
+that a chiral transformation is equivalent to a uniform displacement of the density operator by
+θ/pF,
+ρ(x) → ρ (x + θ/pF)
+(148)
+Under a chiral transformation with arbitrary angle θ, the Dirac and γ5 mass term operators trans-
+form as an orthogonal transformation of a two-component vector. In particular for a chiral trans-
+formation with θ = π/4,
+¯ψψ �→ i ¯ψγ5ψ,
+i ¯ψγ5ψ �→ − ¯ψψ
+(149)
+which is equivalent to a displacement of the density by 1/4 of the period of the CDW. In this
+sense these two states are equivalent, as is the state with a more general linear combination.
+The microscopic lattice model (and the continuum non-relativistic model) are invariant under
+the spatial inversion symmetry x ↔ −x. This also implies a symmetry under the exchange of
+right and left moving components of the Dirac spinor, ψR ↔ ψL. In the massless Dirac theory
+this operation is equivalent to the multiplication of the spinor by the Pauli matrix σ1. In the
+massless theory the multiplication by σ2 (followed by a chiral transformation with θ = π/2) has
+the same eﬀect.
+These symmetries have an important eﬀect on the fermionic spectrum. To understand what
+they do let us consider the one-particle Dirac Hamiltonian with a Dirac mass m (jn momentum
+space)
+h =
+�p
+m
+m
+−p
+�
+(150)
+This operator anti-commutes with the Pauli matrix σ2, {h, σ2} = 0. Let |E⟩ be an eigenstate of
+the Hamiltonian h with energy eigenvalue E =
+�
+p2 + m2. Let us consider the state σ2|E⟩. It is
+also an eigenstate of h but with energy −E, i.e.
+hσ2|E⟩ = −σ2h|E⟩ = −E|E⟩,
+⇒ σ2|E⟩ = | − E⟩
+(151)
+Hence if |E⟩ is an eigenstate of energy E, then σ2|E⟩ is an eigenstate of energy −E. This means
+that the spectrum is invariant under charge conjugation symmetry. Notice that under this opera-
+tion the spinor transforms as
+σ2
+�ψR
+ψL
+�
+=
+�−iψL
+iψR
+�
+= e−iγ5π/2
+�ψL
+ψR
+�
+(152)
+In other words, this theory is invariant under charge conjugation C and parity P which, combined,
+it implies that it is invariant under time-reversal T .
+The same consideration applies in the case of a γ5 mass term in which case the one-particle
+Dirac Hamiltonian now is
+h =
+� p
+−im5
+im5
+−p
+�
+(153)
+This Hamiltonian now anti-commutes with the Pauli matrix σ1 which also implies that the same
+charge conjugation symmetry C, |E⟩ ↔ |−E⟩, is present in the spectrum of the case of a γ5 mass.
+We can also repeat the argument on parity invariance P, which is now multiplication by σ1, Thus
+the theory is invariant under time-reversal T .
+However, if the theory has both a Dirac mass m and a γ5 mass m5 these symmetries are
+broken. Indeed, the one-particle Dirac Hamiltonian now is
+h =
+�
+p
+m − im5
+m + im5
+−p
+�
+= pσ3 + mσ1 + m5σ2
+(154)
+which no longer has a spectral symmetry. In this case CP is broken and, hence, so it T since
+CPT remains unbroken (as it should).
+In a lattice system this is a symmetry transformation only if θ = πn/pF is a lattice displace-
+ment, which restricts the allowed values of the chiral angle to be discrete. Although this is true
+interactions play a signiﬁcant role in the actual behavior. In fact, there are physical situations
+in which an eﬀective continuous symmetry actually emerges in this he infrared (long-distance)
+limit. This is what happens when the CDW is incommensurate and, as we will see in the next
+subsection, it slides under the action of an electric ﬁeld.
+In the case of a half-ﬁlled system (with only nearest-neighbor hopping matrix elements) the
+Fermi wave vectors are ±π/2. In this case the allowed discrete chiral transformation has a chiral
+35
+
+angle π/2 under which ¯ψψ �→ − ¯ψψ (and similarly for i¯γ5ψ) corresponding by a translation by one
+lattice spacing. The allowed four fermion operator is ( ¯ψψ)2. If the lattice fermions are spinless
+this operator reduces to
+( ¯ψψ)2 = −2 jR(x)jL(x) + lim
+y→x ψ†
+R(x)ψ†
+L(x)ψL(y)ψR(y) + h.c.
+(155)
+where introduced the right and left moving (chiral) components of the current operator
+jR(x) = 1
+2(j0(x) + j1(x)) = ψ†
+R(x)ψR(x),
+jL(x) = 1
+2(j0(x) − j1(x)) = ψ†
+L(x)ψL(x)
+(156)
+The ﬁrst term in Eq.(155) is known as the backscattering interaction and has scaling dimension
+2. Hence, it is a marginal operator. The theory with only the ﬁrst term is known in Condensed
+Matter Physics as the Luttinger model and in High-Energy Physics at the (massless) Thirring
+model. The second operator in Eq.(155) formally violates momentum conservation as its total
+momentum is 4pF = 2π which is a reciprocal lattice vector and, as such, it is equivalent to
+zero (mod 2π). Such an operator is not formally allowed in a (naive) continuum theory. This
+operator is due to a lattice Umklapp process and breaks the formal continuous chiral symmetry
+to a discrete Z2 subgroup. Although the naive scaling dimension of this operator is 2 (and hence
+it is formally marginal). If the fermions are spinless, the leading operator actually vanishes and
+the leading non-vanishing operator actually has dimension 4, which is irrelevant. However, as
+shown above, backscattering processes of the form jR(x) jL(x) are part of this operator and are
+exactly marginal.
+If the interaction is strong enough the backscattering interaction can make the Umklapp op-
+erator relevant. When this happens the fermionic system has a quantum phase transition to an
+insulating state with a period 2 (commensurate) CDW state. On the other hand, for spin 1/2
+fermions the operator ( ¯ψψ)2 is allowed and is marginally relevant. If the interactions are re-
+pulsive the resulting state is an antiferromagnetic N´eel state at quantum criticality, while for
+attractive interactions it is a period 2 CDW. This is what happens in the 1D Hubbard model (see,
+e.g. Ref. [9] for a detailed analysis).
+There are two theories in relativistic systems which are closely related to this problem. One
+if the Gross-Neveu model [132] which is a theory of N species of massless Dirac spinors with
+Lagrangian density
+LGN = ¯ψai/∂ψa + g( ¯ψaψa)2
+(157)
+with a = 1, . . ., N (summation of repeated induces is implies). The spin-1/2 Hubbard model
+corresponds to the case N = 2. This Lagrangian is invariant only under the discrete chiral
+symmetry ¯ψaψa �→ − ¯ψaψa. This is a discrete, Z2, symmetry and as such it can be spontaneously
+broken in 1+1 dimensions. For N ≥ 2, the resulting state has a (dynamically) broken Z2 chiral
+symmetry and that there is a chiral condensate ⟨ ¯ψaψa⟩ � 0 corresponding to a period 2 CDW.
+The other theory is known as the chiral Gross-Neveu model whose Lagrangian density is
+LcGN = ¯ψai/∂ψa + g
+�
+( ¯ψaψa)2 − ( ¯ψaγ5ψa)2�
+(158)
+which has the full continuous chiral symmetry. For N = 1 this theory is equivalent to the Lut-
+tinger model (and to the Gross-Neveu model if we ignore the umklapp term). For N ≥ 2 the chi-
+ral symmetry is formally broken. If this were true this theory would violate the Mermin-Wagner
+theorem. However a detailed study (most easily done using bosonization methods) shows that
+instead of long range order the correlator of both mass terms decay as a power law as a function
+of distance, consistent with the requirements by this theorem.
+We close this subsection by noting that if the lattice model is not half ﬁlled but its density
+is either incommensurate or has a higher degree of commensurability, say p/q, the chiral sym-
+metry is actually continuous (in the incommensurate case) or eﬀectively continuous since the
+requisite umklapp terms are strongly irrelevant. However, if the lattice model is not at half ﬁlling
+charge conjugation symmetry C is broken at the lattice scale (in the UV), where it is equivalent
+to particle-hole symmetry, but it is recovered in the low-energy, IR, regime (up to irrelevant op-
+erators). In this sense, both the continuous chiral symmetry and charge conjugation symmetry
+can be regarded as emergent IR symmetries
+7.3. The chiral anomaly
+In section 7.2 we showed that the theory of massless Dirac fermions, in addition to a global
+U(1) gauge gauge symmetry, has a second conservation law which we called a global U(1) chiral
+36
+
+symmetry, shown in Eq.(131). This symmetry implies that there is a locally associated chiral
+current j5
+µ, given by
+j5
+µ(x) = ¯ψ(x)γµγ5ψ(x)
+(159)
+which also satisﬁes a continuity equation
+∂µ j5
+µ = 0
+(160)
+and there is a globally conserved chiral charge Q5
+Q5 =
+� ∞
+−∞
+dx j5
+0(x) =
+� ∞
+−∞
+dx
+�
+ψ†
+RψR − ψ†
+LψL
+�
+(161)
+It is easy to check that the Dirac current jµ and the chiral current j5
+µ are related by
+j5
+µ = ǫµν jµ
+(162)
+where ǫµν is the second rank Levi-Civita tensor.
+The simultaneous conservation of both currents jµ and j5
+µ in the massless Dirac theory implies
+that the right and left moving densities jR and jL, deﬁned in Eq.(156), should be separately
+conserved. In fact, if the Dirac theory has a mass term
+L = ¯ψi/∂ψ − m ¯ψψ
+(163)
+it is straightforward to show that
+∂µ j5
+µ = 2mi ¯ψγ5ψ
+(164)
+which means that in the massive theory the axial current is not conserved. This is easy to under-
+stand since the mass term mixes the right and left moving components of the Dirac spinor and,
+hence, the right and left moving densities are not conserved.
+What happens in the massless limit, m → 0, is more subtle. This problem was investigated
+in the late 1960’s in 3+1 dimensions by S. Adler [133] and by J. S. Bell and R. Jackiw [134]
+who were interested in the anomalous decay of a neutral pion into two photons, π0 → 2γ.
+This process appears at third order of perturbation theory and it involves the computation of a
+triangle diagram (a fermion loop). In 3+1 dimensions this process has a UV divergence which
+needed to be regulated. These authors showed that it is not possible to ﬁnd a regularization
+in which both the Dirac (gauge) current jµ = ¯ψγµψ and the axial current j5
+µ = ¯ψγµγ5ψ are
+conserved. In other words, if gauge invariance is preserved then the axial current is not and has
+an anomaly and results in a non-conservation of the axial current, ∂µ j5
+µ � 0. On the other hand,
+at least in the case of the physical gauge-invariant regularization, the obtained expression for
+the anomaly in the axial current is universal, independent of the value of the UV regulator (the
+cutoﬀ). Sometime later G. ’t Hooft showed that in non-abelian gauge theories instant processes
+also lead to anomalies and, furthermore, the result was also universal [96, 111]. Since the result
+is universal and, hence, independent of the UV scale, this led to the concept of anomaly matching
+conditions.
+We will examine this problem in 1+1 dimensions (although it plays a key role in the theory of
+topological insulators in three space dimensions [135]). As we noted above, the lattice model is
+gauge-invariant and has only one conserved current. The conserved axial current appeared only
+in the low-energy regime in which the lattice model is described by a theory of massless Dirac
+fermions. To understand this problem we will consider the theory of fermions in 1+1 dimen-
+sions coupled to a U(1) gauge ﬁeld ﬁeld. In the presence of a background (i.e. not quantized)
+electromagnetic ﬁeld in the A0 = 0 gauge the free fermion Hamiltonian of Eq.(116) becomes
+H[A] = −t
+L
+�
+n=1
+c†(n)eieA(n,t)/ℏc(n) + h.c.
+(165)
+An uniform and constant electric ﬁeld is E is represented by a vector potential A = −cEt (with
+c being the speed of light). The net eﬀect of this gauge ﬁeld is to shift of the momentum of the
+fermion quasiparticles p → p + ecEt/ℏ or, what is the same, to displace the fermion dispersion
+relation in momentum by the ecEt/ℏ. This means that the Fermi points are also shifted by that
+amount, pF → pF + ecEt/ℏ and −pF → −pF + ecEt/ℏ. This means that the single particle states
+between pF and pF +ecEt/ℏ that were empty for E = 0 are now occupied and the states between
+37
+
+−pF and −pF + ecEt/ℏ that were occupied for E = 0 are now empty. This means that number
+of right-moving fermions is increasing at a rate of ecE/ℏ and that the number of left-moving
+fermions decreases at the same rate. This results in a net current. Throughout this process the
+total number of fermions is not changed, gauge invariance is satisﬁed, but the number of right
+and left moving fermions are not separately concerned. Notice that this is an eﬀect that involved
+the entire Femi sea but the net eﬀect is at low energies.
+Let us see now how this plays out in the eﬀective Dirac theory. The massless Dirac La-
+grangian density in the background of an unquantized electromagnetic ﬁeld Aµ is
+L = ¯ψ(i/∂ − e /A)ψ
+(166)
+Since there is no mass term the Dirac equation still decouples into two equations, for the right
+and left moving components of the Dirac spinor. In the A0 = 0 gauge they are
+i∂0ψR = (−i∂1 − A1)ψR,
+i∂0ψL = (i∂1 − A1)ψL
+(167)
+In the temporal gauge, A0 = 0, a uniform electric ﬁeld E = ∂0A1, and A1 increases linearly with
+time. As A1 increases, the Fermi momentum pF (which is equal to the Fermi energy EF) also
+increases at the rate eE. The density of states of a system of length L is L/(2π). So, the rate of
+change of the number of right-moving fermions is
+dNR
+dx0
+= e
+2πE
+(168)
+where we deﬁned NR =
+� L
+0 dx jR and similarly for NL. If the UV regulator of the theory is
+compatible with gauge invariance, then the total fermion number must be conserved and the total
+vacuum charge must remain equal to zero,
+Q =
+� L
+0
+dx j0(x) = NR + NL = 0
+(169)
+Thus, if NR increases, then NL must decrease by the same amount. Or, equivalently, the electric
+ﬁeld E creates an equal number of particles NR and of antiparticles ¯NL = −NL.
+On the other hand, the chiral charge Q5 = NR − NL must increase at the rate
+dQ5
+dx0
+= dNR
+dx0
++ d ¯NL
+dx0
+= e
+π E
+(170)
+Again, the details of the UV regularization do not matter, only that it is gauge-invariant. We can
+also interpret Eq.(170) as the rate of particle-antiparticle pair creation by an electric ﬁeld.
+In a relativistic notation these results are expressed as
+∂µ j5
+µ = e
+2πǫµνFµν
+(171)
+Hence, the formally conserved current j5
+µ has an anomaly and is not conserved due to quantum
+eﬀects. Since it is not conserved, we cannot gauge the chiral symmetry. In the next subsection
+we will see that the the anomaly is closely related with bosonization.
+7.4. Bosonization, anomalies and duality
+We will reexamine the problem at hand from the point of view of the fermionic currents of
+the Dirac theory jµ as operators. Since the currents obey the continuity equation, ∂µ jµ = 0 one
+expects that this would imply that it may be possible to write them as a curl of a scalar ﬁeld, i.e.
+jµ(x) = ǫµν∂νφ(x) where φ(x) should be a scalar ﬁeld. Since this should be an operator identity
+we will need to understand how the currents act on the physical Hilbert space.
+The Fermi-Bose mapping in 1D systems is closely related to the chiral anomaly we just
+discussed. Bosonization of a system of 1+1 dimensional massless Dirac fermions is a set of
+operator identities understood as matrix elements of the observables in the physical Hilbert space.
+These identities were ﬁrst derived by Daniel Mattis and Elliott Lieb [128], based on earlier work
+by Julian Schwinger [136]. These identities were rediscovered (and their scope greatly expanded)
+by Alan Luther and Victor Emery [129], by Sidney Coleman [131], by Stanley Mandelstam
+[130], and by Edward Witten [137]. A non-abelian version of bosonization was subsequently
+derived by Witten [119].
+38
+
+The physical Hilbert space is deﬁned as follows. Let |FS⟩ ≡ |0⟩ denote the ﬁlled Fermi
+sea. In what follows we will assume that the physical system is macroscopically large abd that
+local operators create the physical states by acting ﬁnitely on the ﬁlled Fermi sea. Physical
+observables, such as the right and left moving densities jR(x) and jL(x), need to be normal-
+ordered with respect to the physical vacuum state, the ﬁlled Fermi (Dirac) sea |0⟩. The normal
+ordered densities are : jR(x) : ≡ jR(x) − ⟨0|jR(x)|0⟩ and : jL(x) : ≡ jL(x) − ⟨0|jR(x)|0⟩. Since
+the densities are products of fermion operators they need to be deﬁned as a limit in which the
+operators are separated by a short distance η. Crucial to this construction is that the computation
+the expectation values be regularized in such a way that the charge (gauge) current jµ(x) is locally
+conserved and satisﬁes the continuity equation ∂µ jµ = 0 as an operator identity. In what follows
+all expectation values will refer to the ﬁlled Fermi sea state |0⟩.
+The propagators of the right and left moving Fermi ﬁelds are given by
+⟨ψ†
+R(x0, x1)ψR(0, 0)⟩ =
+−i
+2π(x0 − x1 + iǫ),
+⟨ψ†
+L(x0, x1)ψL(0, 0)⟩ =
+i
+2π(x0 + x1 + iǫ)
+(172)
+The expectation value of the currents at a space location x1 are
+⟨jR(x1)⟩ = lim
+η→0⟨ψ†
+R(x1 + η)ψR(x1 − η)⟩ =
+i
+4πη,
+⟨jL(x1)⟩ = lim
+η→0⟨ψ†
+L(x1 + η)ψL(x1 − η)⟩ = −i
+4πη
+(173)
+which are divergent at short distances. It follows that the normal-ordered right and left moving
+current densities satisfy the equal-time commutation relations
+[jR(x1), jR(x′
+1)] = − i
+2π∂1δ(x1 − x′
+1),
+[jL(x1), jL(x′
+1)] = + i
+2π∂1δ(x1 − x′
+1)
+(174)
+These identities imply that the normal-ordered space-time components of the current jµ = (j0, j1)
+satisfy the equal-time commutation relations
+[j0(x1), j1(x′
+1)] = − i
+π∂1δ(x1 − x′
+1),
+[j0(x1), j0(x′
+1)] = [ji(x1), j1(x′
+1)] = 0
+(175)
+The non-vanishing right-hand sides of these commutators are known as Schwinger terms. These
+identities deﬁne the the U(1) (Kac-Moody) current algebra.
+We should note that in a theory of non-relativistic Fermi ﬁelds ψ(x, t) in all dimensions, the
+charge density ρ(x) and the current operators j(x) =
+1
+2i
+�
+ψ†(x)▽ψ(x) − ▽ψ†(x)▽ψ(x)
+�
+satisfy a
+similar expression (also at equal times)
+[ρ(x), jk(x′)] = −i e2
+mc2 ∂k
+�δ(x − x′)ρ(x)� ,
+[ρ(x), ρ(x′)] = 0,
+[jk(x), jl(x′)] = 0
+(176)
+In one dimension, and in the regime in which the fermions have a macroscopic density so that
+ρ(x) ≃ ⟨ρ(x)⟩, after a multiplicative rescaling of the operators, the non-relativistic identities of
+Eq.(176) are equivalent to the U(1) current algebra of Eq.(175).
+The U(1) current algebra of Eq,(175) is reminiscent of the equal-time canonical commutation
+relations of a scalar ﬁeld. Indeed, if φ(x) is a scalar ﬁeld and Π(x) = ∂0φ(x) is its canonically
+conjugate momentum, then they obey the equal-time canonical commutation relations
+[φ(x1), Π(x′
+1)] = iδ(x1 − x′
+1)
+(177)
+We can then identify the charge density j0 and the current density j1 with the scalar ﬁeld operators
+j0(x) =
+1√π∂1φ(x),
+j1(x) = − 1√πΠ(x) = − 1√π∂0φ(x)
+(178)
+which obey the U(1) current algebra of Eq.(175) as a consequence of the canonical commutation
+relations, Eq. (177). Furthermore, we can rewrite Eq.(177) in the Lorentz covariant form
+jµ(x) =
+1√πǫµν∂νφ(x)
+(179)
+which is clearly consistent with the local conservation of the current jµ,
+∂µ jµ(x) = 0
+(180)
+39
+
+Let us examine now the question of the conservation of the chiral current j5
+µ. In Eq.(162)
+we showed that the gauge current and the chiral current are related by j5
+µ = ǫµν jν. Therefore the
+divergence of the chiral current is
+∂µ j5
+µ = ǫµν∂µ jν =
+1√πǫµνǫνλ∂µ∂λφ = − 1√π∂2φ
+(181)
+where we used the identiﬁcation of the gauge current in terms of the scalar ﬁeld φ, Eq.(179).
+Therefore we conclude that
+∂µ j5
+µ = 0 ⇔ ∂2φ = 0
+(182)
+This equation states that the chiral current as an operator identity is conserved if and only if the
+ﬁeld φ is a free massless scalar ﬁeld, whose Lagrangian density is
+LB = 1
+2(∂µφ)2
+(183)
+In Eq.(166) we considered the free massless Dirac Lagrangian coupled to a background (not
+quantized) gauge ﬁeld Aµ through the usual minimal coupling which here is Lint = −ejµAµ. using
+the bosonization identity for the gauge current, Eq.(179) we see that the Lagrangian density of
+the bosonized theory now becomes
+LB[A] =1
+2(∂µφ)2 −
+e√πǫµν∂νφ(x) Aµ(x)
+≡1
+2(∂µφ)2 + J(x)φ(x)
+(184)
+where the source J(x) is
+J(x) =
+e√πǫµν∂νAµ(x) =
+e
+√
+4π
+F∗(x)
+(185)
+where F∗(x) = ǫµνFνµ(x) is the (Hodge) dual of the ﬁeld strength Fµν. Hence, F∗ is (essentially)
+the source for the scalar ﬁeld φ(x). This implies that the equation of motion of the scalar ﬁeld
+must be
+− ∂2φ(x) = J(x) =
+e√πǫµν∂νAµ
+(186)
+Retracing our steps we ﬁnd that the chiral current j5
+µ obeys
+∂µ j5
+µ = − 1√π∂2φ = e
+2πǫµνFµν
+(187)
+which reproduces the the chiral anomaly given in Eq.(170).
+These results suggest that the theory of a free massless Dirac spinor must be equivalent to
+the theory of the free massless scalar ﬁeld. This statement is known as bosonization. However
+for this identiﬁcation to be correct there must be an identiﬁcation of the Hilbert spaces and of all
+the operators of each theory. We will not do this detailed analysis here but we will highlight the
+most signiﬁcant statements.
+Let us begin with the fermion number of the Dirac theory. Consider a system of fermions
+of total length L. Using the bosonization identity of Eq.(179) we ﬁnd that the fermion number
+NF ≡ Q is given by
+NF =
+� L
+0
+dx1 j0(x0, x1) =
+1√π
+� L
+0
+dx1 ∂1φ(x0, x1) =
+1√π(φ(x0, L) − φ(x0, 0))
+(188)
+Thus, the vacuum sector of the Dirac theory, with NF = 0, corresponds to the theory of the scalar
+ﬁeld with periodic boundary conditions, φ(x1 = 0) = φ(x1 = L). Furthermore, since the fermion
+number is quantized, NF ∈ Z, changing the fermion number is the same as twisting the boundary
+conditions of the scalar so that
+φ(x1 + L) = φ(x1) + √πNF
+(189)
+In String Theory [104] the scalar ﬁeld is interpreted as the coordinate of a string. Compactifying
+the space where the string lives to be a circle of radius R means that the string coordinate is
+deﬁned modulo 2πRn, where n is an integer. We see that the condition imposed by Eq.(189)
+40
+
+is equivalent to say that the scalar ﬁeld is compactiﬁed and that the compactiﬁcation radius is
+R = 1/
+√
+4π. This identiﬁcation also imposes the restriction that the allowed operators of the
+scalar ﬁeld must obey the identiﬁcation
+φ(x) ∼ φ(x) + 2πRn
+(190)
+as an equivalency condition.
+The simplest bosonic operators that obey the compactiﬁcation condition are the vertex oper-
+ators Vα(x),
+Vα(x) = exp(iαφ(x))
+(191)
+The compactiﬁcation condition then requires that the allowed vertex operators should have α =
+n/R =
+√
+4π n, where n is an integer. Since the propagator of the scalar ﬁeld in 1+1-dimensional
+(Euclidean) spacetime is
+G(x − x′) = − 1
+2π ln
+�|x − x′|
+a
+�
+(192)
+where a is a short-distance cutoﬀ, we ﬁnd that the scaling dimension of the vertex operator is
+∆α = α2/(4π) = n2. We will see shortly that the vertex operator with α =
+√
+4π is essentially the
+Dirac mass operator (which has scaling dimension 1).
+The free massless scalar ﬁeld can be decomposed into right and left moving components, φR
+and φL respectively,
+φ = φR + φL,
+ϑ = −φR + φL
+(193)
+where
+ϑ(x0, x1) =
+� x1
+−∞
+dx′
+1Π(x0, x′
+1)
+(194)
+is the Cauchy-Riemann dual of the ﬁeld φ(x) since they satisfy the Cauchy-Riemann equation
+∂µφ = ǫµν∂νϑ
+(195)
+The right and left moving component of the Dirac spinor are found to have the bosonized expres-
+sion [130]
+ψR(x) =
+1
+√
+2πa
+: exp(i
+√
+4πφR(x)) :,
+ψL(x) =
+1
+√
+2πa
+: exp(−i
+√
+4πφL(x)) :
+(196)
+It is easy to check that the propagators of these operators agree with the expressions given in
+Eq.(172), and that they have scaling dimension 1/2 and spin 1/2.
+How does a chiral transformation act on the scalar ﬁeld? A chiral transformation by an angle
+θ, c.f. Eq.(132), acts on the right and moving fermions as ψ′
+R = exp(iθ)ψR and ψ′
+L = exp(−iθ)ψL.
+From Eq.(196) we see that the right and left moving components of the scalar ﬁeld transform as
+φ′
+R = φR + θ/
+√
+4π and φ′
+L = φL + θ/
+√
+4π. This means that a chiral transformation by an angle
+θ of the Dirac fermion by an angle θ is equivalent to a translation (a shift) of the scalar ﬁeld
+φ′ = φ + 2θ/
+√
+4π.
+We can use the Operator Product Expansion discussed in section 3.3.2 to show that the
+fermion mass terms ¯ψψ and i ¯ψγ5ψ are given by the following identiﬁcations
+¯ψψ =
+1
+2πa : cos(
+√
+4πφ) :,
+i ¯ψγ5ψ =: sin(
+√
+4πφ) :
+(197)
+These operators have scaling dimension 1 and transform properly under chiral transformations.
+These identiﬁcations imply that a theory of free massive Dirac fermions
+LD = ¯ψi/∂ ψ − m ¯ψψ
+(198)
+is equivalent to the sine-Gordon ﬁeld theory [131] whose Lagrangian is
+LSG = 1
+2(∂µφ)2 − g : cos(
+√
+4πφ) :
+(199)
+where g = m/(2πa).
+Given the central role played by the current algebra identities of Eqs. (175) one may wonder
+if a similar approach might apply in higher dimensions. Schwinger terms in current algebra play
+an important role in relativistic ﬁeld theories. However in higher dimensions their structure is
+41
+
+more complex and does not lead to identities of the type we have discussed. The reason at the
+root of this problem is largely kinematical. The Bose (scalar) ﬁeld φ is qualitatively a bound
+state, a collective mode in the language of Condensed Matter Physics. In 1+1 dimensions this
+collective mode exhausts the spectrum at low energies due to the strong kinematical restriction
+on one spatial dimension.
+The equivalency between the theory of free massive Dirac fermions and the sine-Gordon
+theory is an example of the power of bosonization. On the Dirac side the mapping the theory is
+free and its spectrum is well understood. But on the sine-Gordon side the theory is non-linear.
+In fact in the sine-Gordon theory the fermions are essentially solitons, domain walls of the scalar
+ﬁeld. For these and many other reasons that we do not have space here bosonization plays a
+huge role in understanding the non-perturbative behavior of systems both in Condensed Matter
+Physics and in Quantum Field Theory in 1+1 dimensions. We will see in section 11.3.1 that to
+an extent some of these ideas can and have been extended to relativistic systems and classical
+statistical mechanical systems in 2+1 dimensions.
+8. Fractional Charge
+8.1. Solitons in one dimensions
+We begin by returning to the equivalency between the theory of free massive Dirac fermions
+and sine-Gordon theory, in 1+1 dimensions. In Eq.(188) we showed that the boundary conditions
+of the compactiﬁed scalar ﬁeld φ(x) are determined by the fermion number NF of the dual Dirac
+theory, and that the vacuum sector of the Dirac theory maps onto the sine-Gordon theory with
+periodic boundary conditions. We will now examine the sector with one fermion, NF = 1. This
+sector of the Dirac theory maps onto the sine-Gordon theory with twisted boundary conditions,
+φ(L) − φ(0) = √π
+(200)
+The Hamiltonian of the sine-Gordon theory is
+HSG =
+� ∞
+−∞
+dx
+�1
+2Π2(x) + 1
+2 (∂xφ(x)) + g cos
+� √
+4πφ(x)
+��
+(201)
+In the sector with periodic boundary conditions the classical ground states are static and uniform
+conﬁgurations that minimize the potential energy. Since the potential energy is a periodic func-
+tional of φ(x) the classical minima are at φn(x) = (n + 1/2) √π, where n is an arbitrary integer.
+The classical energy of these ground states is extensive and is given by Egnd = −gL where L is
+the linear size of the system.
+The classical ground state in the twisted sector is a domain wall (or soliton) which interpo-
+lates between the static and uniform ground states φ(x) = ± √π/2. The classical ground state in
+this sector is the static solution of the Euler-Lagrange equation
+d2φ
+dx2 = −2g √π sin
+�
+2 √πφ(x)
+�
+(202)
+such that asymptotically satisﬁes limx→±∞ = ± √π/2. The solution is the classical soliton con-
+ﬁguration
+φ(x) =
+2√π tan−1 �
+exp(2 √πg(x − x0))
+�
+−
+√π
+2
+(203)
+The soliton solution represents a domain wall between two symmetry-related classical ground
+states with φ = ± √π/2.
+The energy of the soliton (measured from the energy of the ground state in the trivial sector)
+is ﬁnite and is given by
+Esoliton = 4
+�
+g
+π
+(204)
+where x0 is a zero mode of the soliton solution and represents its coordinate. By coupling the
+bosonized theory to a weak electromagnetic ﬁeld Aµ, as given in Eq.(184), it is easy to check that
+it has electric charge −e and, in this sense represents the electron. Then the identities of Eq.(196)
+can be used that as a quantum state it is indeed a fermion.
+42
+
+8.2. Polyacetylene
+In section 7.4 we saw that solitons of a scalar ﬁeld can be regarded as being equivalent to
+electrons, fermions with charge −e. We will now see that in a theory of fermions coupled to
+a domain wall of a scalar ﬁeld, the soliton carries fractional charge. This problem has been
+extensively studied in one-dimensional conductors such as polyacetylene, in particular by the
+work of Wu-Pei Su, J. Robert Schrieﬀer and Alan Heeger [138] and by Roman Jackiw and J.
+Robert Schrieﬀer [139]. In quantum ﬁeld theory this problem was ﬁrst discussed by Roman
+Jackiw and Claudio Rebbi [140] and by Jeﬀrey Goldstone and Frank Wilczek [141].
+In section 7.1 we showed that the physics of lattice fermions in one dimension at low energies
+is well described by a theory of massless Dirac fermions. In a one-dimensional conductor, such as
+polyacetylene, the fermions couple to the lattice vibrations (phonons). Su, Schrieﬀer and Heeger
+(SSH) [138] proposed a simple model in which the electrons couple to the lattice vibrations
+through a modulation of the hopping amplitude between two consecutive sites n and n + 1,
+instead of being a constant t, becomes tn,n+1 = t − g(un+1 − un), where un is the displacement of
+the ion (a CH group in polyacetylene) at site n from its classical equilibrium position and g is
+the electron-phonon coupling constant. In polyacetylene there number of electrons (which are
+spin-1/2 fermions) is equal to the number of sites of the lettuce and the electronic band is half-
+ﬁlled. At half ﬁlling this simple band structure is invariant under a particle-hole transformation.
+If the coupling to the lattice vibrations is included this symmetry remains respected provided the
+displacements change sign un → −un for all lattice sites. In polyacetylene the lattice dimerizes
+(a process known as a Peierls distortion) and the discrete translation symmetry by one lattice
+spacing is spontaneously broken: the system becomes a period 2 CDW on the bonds of the
+lattice. The broken symmetry state is still invariant under a particle-hole transformation.
+The eﬀective ﬁeld theory of this system is a theory of two Dirac spinors ψα,σ(x), where
+α = 1, 2 denotes right and left-moving fermions, and σ =↑, ↓ are the two spin polarizations. The
+Lagrangian density of this system is
+L = ¯ψσi/∂ψσ − gφ(x) ¯ψσ(x)ψσ(x) − 1
+2φ(x)2
+(205)
+The real scalar ﬁeld φ(x) represents the distortion ﬁeld of the polyacetylene chain. Here we will
+assume that the chains has spontaneously distorted and we will regard the scalar ﬁeld as static
+and classical. This is a good approximation since the masses of the CH complexes is much
+bigger than the electron mass. This continuum model is due to Takayama, Lin-Liu and Maki
+[142] and further developed by Campbell and Bishop [143, 144]. Many of the results fund in
+this (adiabatic) approximation remain qualitatively correct upon taking into account the quantum
+dynamics of the chain, even in the limit in which the ions are treated as being “light” (provided
+the spin of the fermions is taken into account) [145, 146].
+In the ﬁeld theory the discrete symmetry of displacements by one lattice spacing becomes
+the Z2 symmetry φ → −φ. This is a symmetry of the electron phonon system once combined
+with the discrete chiral transformation ψ → γ5ψ under which ¯ψψ → − ¯ψψ. The ground state is
+two fold degenerate ±φ0 with
+φ0 = 2Λ�F
+g
+exp
+�
+−π�F
+g2
+�
+(206)
+and the Dirac fermion (the electron) has a exponentially small mass, m = gφ0.
+8.3. Fractionally charged solitons
+Jackiw and Rebbi showed that the 1+1-dimensional classical φ4 theory has the following
+soliton solution which interpolates between the two classically ordered states at ±φ0 [140]
+φ(x) = φ0 tanh
+� x − x0
+ξ
+�
+(207)
+where ξ is the correlation length of φ4 theory, and x0 is the (arbitrary) location of the soliton.
+They further showed that, when coupled to a theory of massless relativistic fermions through a
+Yukawa coupling, as in Eq.(205), this soliton carries fractional charge. The argument goes as
+follows. The one-particle Dirac Hamiltonian for a Dirac fermion with a position-dependent mass
+m(x) is
+H = −iσ1∂x + m(x)σ3 =
+�m(x)
+−i∂x
+−i∂x
+−m(x)
+�
+(208)
+43
+
+where m(x) = gφ(x), with φ(x) being the soliton solution of Eq.(207). This Hamiltonian is her-
+mitian and real. Furthermore, this Hamiltonian anti-commutes with the Pauli matrix σ2. This
+implies that for every positive-energy state |E⟩ with energy +E there is a negative-energy eigen-
+state with energy −E given by σ2|E⟩. Hence, the spectrum is particle-hole symmetric (or, what
+is the same, charge-conjugation invariant). In addition, and consistent with charge-conjugation
+symmetry, the Hamiltonian of Eq.(208) has state with E = 0, a zero-mode, with a normalizable
+spinor wave function
+ψ0(x) =
+1√
+2
+�−i
+1
+�
+exp
+�
+−sgn(m)
+� x
+0
+dx′ m(x′)
+�
+(209)
+which exists for an arbitrary function m(x) which changes sign once at some location (which
+we took to be x0 = 0). Jackiw and Rebbi further showed that the soliton (anti-soliton) carries
+fractional charge
+Q = ∓e
+2
+(210)
+This result follows from the spectral asymmetry identity of the density of states ρS (E) in the
+presence of the soliton
+Q = −e
+2
+� ∞
+0
+(ρS (E) − ρS (−E)) = −e
+2η
+(211)
+where η is the spectral asymmetry of the Dirac operator in the soliton background, and it is known
+as the APS η-invariant of Atiyah-Patodi-Singer [147]. Given the one-to-one correspondence
+that exists between positive and negative energy states in the spectrum, the spectral asymmetry
+follows from the condition that the zero mode be half-ﬁlled, which is required by normal-ordering
+or, what is the same, by charge neutrality. Another way to understand this result is that adding (or
+removing) a fermion of charge −e results in the creation of a soliton-antisoliton pair, with each
+topological excitation carrying half of the charge of the electron. In other words, in the dimerized
+phase the electron is fractionalized. In this analysis we ignored the spin of the electron. If we
+take it into account the spin degree of freedom the soliton is instead a boson with charge ∓e.
+There is an alternative, complementary, way to think about the charge of the soliton. Gold-
+stone and Wilczek [141] considered a theory in which the (massless) Dirac fermion is coupled to
+two real scalar ﬁelds, ϕ1 and ϕ2, with Lagrangian
+L = ¯ψi/∂ψ − gϕ1 ¯ψψ − igϕ2 ¯ψγ5ψ ≡ ¯ψi/∂ψ − g|ϕ| ¯ψ exp(iθγ5)ψ
+(212)
+where |ϕ|2 = ϕ2
+1 + ϕ2
+2 and θ = tan−1(ϕ2/ϕ1). They considered a soliton in which gϕ1 = m is the
+constant (in space) Dirac mass and ϕ2 winds slowly between two values ±ϕ0 for x → ±∞. In
+this theory the one-particle Dirac Hamiltonian is
+H = −iσ1∂x + gϕ1σ3 + gϕ2σ2
+(213)
+which is hermitian and complex and, hence, it violates CP invariance.
+A perturbative calculation of the induced (gauge-invariant) current jµ, which is given by the
+triangle diagram of a fermion loop with two gauge ﬁeld insertions and a coupling of the scalar
+ﬁelds, yields the result (with a = 1, 2)
+⟨jµ(x)⟩ = 1
+2πǫµνǫab
+ϕa∂νϕb
+|ϕ|2
+= 1
+2πǫµν∂νθ
+(214)
+which is locally conserved. Notice, however, that the induced axial current j5
+µ is not conserved,
+∂µ⟨j5
+µ⟩ = − 1
+2π∂2θ � 0 and, hence, this current is anomalous. We can now compute the total
+charge accumulated as the soliton is created adiabatically to be given by the Goldstone-Wilczek
+formula
+Q = −e∆θ
+2π
+(215)
+where ∆θ = θ(+∞) − θ(−∞). Since limx→±∞ ϕ2(x) = ±ϕ0, we obtain the result
+Q = − e
+π tan−1 �gϕ0
+m
+�
+(216)
+In the limit m = gϕ → 0, where CP (or T) invariance is recovered, we get
+lim
+m→0 Q = −e
+2
+(217)
+44
+
+which is the Jackiw-Rebbi result for the fractional charge of a soliton of Eq.(210).
+The results for the fractional charge of the soliton can also be derived using the bosonization
+identities of section 7.4. Indeed, the bosonized expression for the Lagrangian of Eq.(212) is
+L = 1
+2
+�
+∂µφ
+�2 − g|ϕ|
+2πa cos
+� √
+4πφ − θ
+�
+(218)
+Deep in the phase in which the Bose ﬁeld φ is massive, the non-linear term in Eq.(218) locks this
+ﬁeld to the chiral angle θ, i.e.
+φ =
+1
+√
+4π
+θ
+(219)
+However, the bosonization identities also tell us that the gauge current jµ is given by the curl of
+the scalar ﬁeld φ. Therefore, in this state the current jµ is
+jµ =
+1√πǫµν∂νφ = 1
+2πǫµν∂νθ
+(220)
+which is the same as the Goldstone-Wilczek result of Eq.(214). That these two seemingly diﬀer-
+ent approaches yield the same result is not accidental as they both follow from the axial anomaly.
+9. Fractional Statistics
+A fundamental axiom of Quantum Mechanics is that identical particles are indistinguish-
+able [148, 149]. In non-relativistic Quantum Mechanics this leads to the requirement that the
+quantum states of a system of identical particles must be eigenstates of the pairwise particle ex-
+change operator. Since two exchanges are equivalent to the identity operation this implies that the
+states must be even or odd under pairwise exchanges. This result, in turn, implies that particles
+can be classiﬁed as either being bosons (whose states are invariant under pairwise exchanges)
+or fermions (whose states change sign under pairwise particle exchanges). A consequence is
+that bosons obey the the Bose-Einstein (and can condense into a single particle state) whereas
+fermions obey the Fermi-Dirac distribution and must obey the Pauli exclusion principle. It is
+an implicit assumption of this line of reasoning that all relevant states of a system of identical
+particles can be eﬃciently represented by a (suitably symmetrized or antisymmetrized) product
+state.
+This classiﬁcation is present at an even deeper level in (relativistic) Quantum Field Theory
+where locality, unitarity and Lorentz invariance require that the ﬁelds be classiﬁed as represen-
+tations of the Lorentz group and obey the Spin-Statistics Theorem [150]. The Spin-Statistics
+Theorem is actually an axiom of local relativistic Quantum Field theory which requires that
+ﬁelds that transform with an integer spin representation of the Poincar´e group (i.e. scalars, gauge
+ﬁelds, gravitational ﬁelds, etc) must be bosons while ﬁelds that transform with a half-integer
+spin representation (i.e. Dirac spinors, etc) must be fermions. This spin-statistics connection is
+intrinsic to the construction of String Theory [104].
+Given these considerations there was a general consensus that fermions and bosons were
+the only possible types of statistics. Nevertheless several exceptions to this rule were known
+to exist. One is the construction of the magnetic monopole in 3+1 dimensional gauge theory
+by Tai-Tsun Wu and Chen-Ning Yang who showed that a scalar coupled to a Dirac magnetic
+monopole behaves as a Dirac spinor [151]. This was an early example of statistical transmutation
+by coupling a matter ﬁeld to a non-trivial conﬁguration of a gauge ﬁeld. As we will note below,
+the construction of anyons (particles with fractional statistics) in 2+1 dimensions has a close
+parentage to the Wu-Yang example. Examples of statistical transmutation were known to exist
+in 1+1 dimensional theories where a system of hard-core bosons was shown to be equivalent to a
+theory of free fermions using the Jordan-Wigner transformation [22] which represents a fermion
+as a composite operator of a hard-core boson and an operator that creates a kink (or soliton).
+This construction also underlies the fermion-boson mapping in 1+1 dimensional ﬁeld theories
+[128, 129, 131, 130] that we discussed in section 7.4. Finally, in the late 1970s it was found that
+1+1 dimensional ZN spin systems harbor operators known as parafermions which obey the same
+algebra shortly afterwards found to be obeyed by anyons in 2+1 dimensions [152].
+45
+
+9.1. Basics of fractional statistics
+Jon Magne Leinaas and Jan Myrheim wrote an insightful paper in 1977 in which they exam-
+ined the structure of the conﬁguration space of the histories of a system of N identical particles
+[153]. Using the Feynamn path-integral approach, they showed that if the worldlines of the iden-
+tical particles are not allowed to cross, then the conﬁguration space is topologically non-trivial.
+Through a detailed analysis they showed that the three and higher dimensions under a pairwise
+particle exchange the states must be either even or odd and hence the particles are either bosons
+or fermions. Leinaas and Myrheim also showed that in one and two space dimensions the wave
+functions can change by a phase, nowadays known as the statistical angle. In retrospect this re-
+sult could have been anticipated (but was not) in an earlier paper by Michael G. G. Laidlaw and
+C´ecile Morette DeWitt [154] who did a similar analysis of the conﬁguration space of identical
+particles in the Feynman path integral.
+Frank Wilczek generalized the Aharonov-Bohm eﬀect [155] to describe the quantum me-
+chanics of composite objects made of electric charge and magnetic ﬂux in two space dimensions
+[156]. Wilczek showed that composite objects made of a non-relativistic particle of charge q
+bound to a magnetic ﬂux of (magnetic) charge Φ behaves as an object with fractional angu-
+lar momentum qΦ/2π. Here Φ is measured in units of the ﬂux quantum 2π (in units in which
+ℏ = c = e = 1). Furthermore, in a subsequent paper Wilczek [157] showed that, upon an
+adiabatic process in which the two composites exchange positions without their worldlines co-
+inciding, the wave function of two identical ﬂux-charge composites changes by a phase factor
+exp(±iqΦ). For instance if Φ = π (i.e. a half-ﬂux quantum) the acquired phase is equal to
+exp(iπ) = −1. thus if the particle was a boson, the composite becomes a fermion and viceversa.
+Wilczek coined the term anyon to describe the behavior of an arbitrary charge-ﬂux composite.
+Furthermore, this construction also implies that not only fractional statistics but also fractional
+spin, consistent with a generalization of the Spin-Statistics Theorem. In other terms, “ﬂux-
+attachment” implies fractional statistics (and fractional spin). Clearly Wilczek’s construction
+gave an explicit physical grounding to the general arguments of the 1977 paper by Leinaas and
+Myrheim. It is worth note the close analogy between this construction in 2+1 dimensions and the
+Wu-Yang construction in 3+1 dimensions (whose consistency with the Spin-Statistics Theorem
+had been shown earlier on by Alfred Goldhaber [158]).
+In this description the statistics of the composites (the anyons) enters in the form of complex
+weights (phases) given in terms of the linking numbers of the worldlines. Hence, the concept of
+fractional statistics is intimately related to the theory of knots and of the representations of the
+braid group. These concepts were originally introduced in physics to describe statistical trans-
+mutation in the theory of solitons [159] in the context of the Skyrme model [160]. Yong-Shi Wu
+[161] developed an explicit connection between the work of Leinaas and Myrheim and Wilczek’s
+work on anyons in terms of operations acting on the worldlines of the anyons and described by
+the Braid Group. Wu’s work and the somewhat early paper by Wilczek and Zee on the statistics
+of solitons [162] marked the deﬁnite entry of the theory of knots (and of the braid group) in
+physics in general and in condensed matter physics in particular. The classiﬁcation of anyons
+in terms of representations of the braid group labeled by the fractional statistics (determined by
+linking numbers) as well as of fractional (or topological) spin (determined by the writhing num-
+ber of the worldlines) leads to a rich set of physical consequences. As it will turn out, Wilczek’s
+anyons are described one-dimensional representations of the braid group. As we will see below,
+additional and intriguing (non-abelian) representations will also play a role.
+9.2. What is a topological ﬁeld theory
+We will now consider a special class of gauge theories known as topological ﬁeld theories.
+These theories often (but not always) arise as the low energy limit of more complex gauge theo-
+ries. In general, one expects that at low energies the phase of a gauge theory be either conﬁning
+or deconﬁned. While conﬁning phases have (from really good reasons!) attracted much atten-
+tion, deconﬁned phases are often regarded as trivial, in the sense that the general expectation
+is that their vacuum states be unique and the spectrum of low lying states is either massive or
+massless.
+Let us consider a gauge theory whose action on a manifold M with metric tensor tensor
+gµν(x) is
+S =
+�
+M
+dDx √g L(g, Aµ)
+(221)
+46
+
+At the classical level, the the energy-momentum tensor T µν(x) is the linear response of the action
+to an inﬁnitesimal change of the local metric,
+T µν(x) ≡
+δS
+δgµν(x)
+(222)
+That a theory is topological means that depends only on the topology of the space in which is
+deﬁned and, consequently, it is independent of the local properties that depend on the metric, e.g.
+distances, angles, etc. Therefore, at least at the classical level, the energy-momentum tensor of a
+topological ﬁeld theory must vanish identically,
+T µν = 0
+(223)
+In particular, if the theory is topological, the energy (or Hamiltonian) is also zero. Furthermore, if
+the theory is independent of the metric, it is invariant under arbitrary coordinate transformations.
+Thus, if the theory is a gauge theory, the expectation values of Wilson loops will be independent
+of the size and shape of the loops. Whether or not a theory of this type can be consistently deﬁned
+at the quantum level is a subtle problem which we will brieﬂy touch on below.
+It turns out that, due to the non-local nature of the observables of a gauge theory, the low
+energy regime of a theory in its deconﬁned phase can have non-trivial global properties. In what
+follows, we will say that a gauge theory is topological if all local excitations are massive (and in
+fact we will send their mass gaps to inﬁnity). The remaining Hilbert space of states is determined
+by global properties of the theory, including the topology of the manifold of their space-time. In
+several cases, the eﬀective action of a topological ﬁeld theory does not depend on the metric of
+the space-time, at least at the classical level. In all cases, the observables are non-local objects,
+Wilson loops and their generalization.
+9.3. Chern-Simons Gauge Theory
+Gauge theories play a key role in physics. In 2+1 dimensions it is possible to deﬁne a special
+gauge theory which is odd under time reversal invariance and parity: Chern-Simons gauge theory.
+Originally introduced in Quantum Field Theory in1982 by Stanley Deser, Roman Jackiw and
+Stephen Templeton [163, 164], Chern-Simons gauge theory can be deﬁned for any compact Lie
+group, as well as an extension of Einstein’s gravity in 2+1 dimensions. In 1989 Edward Witten
+[165] showed that Chern-Simons gauge theory computes the expectation values of conﬁgurations
+of Wilson loops, regarded as the worldlines of heavy particles, in terms of a set of topological
+invariants known as the Jones polynomial that classify knots in three dimensions.
+We will consider the simplest case, the U(1) Chern-Simons gauge theory. The Chern-Simons
+action for a U(1) gauge ﬁeld Aµ in 2+1 dimension is
+S [A] = k
+4π
+�
+Ω
+d3x ǫµνλAµ∂νAλ +
+�
+Ω
+d3x JµAµ
+(224)
+where Jµ is a set of conserved currents (representing the worldlines of a set of heavy particles).
+On a closed 3-manifold Ω (e.g. a sphere, a torus, etc) the Chern-Simons action is gauge invariant
+provided the parameter k (known as the level) is an integer. If the manifold Ω has a boundary,
+the action is not gauge invariant at the boundary. Gauge invariance is restored by additional
+boundary degrees of freedom. This structure is general, and not just a feature of the U(1) theory.
+Since the Chern-Simons action is ﬁrst order in derivatives it is not invariant under time re-
+versal and under parity (which in 2+1 dimensions is a reﬂection). These symmetries make this
+theory relevant to the description of the fractional quantum Hall eﬀect. In the absence of exter-
+nal sources, Jµ = 0, at the classical level the Chern-Simons action is invariant under arbitrary
+changes of coordinates. This means that the theory is, at least classically, a topological ﬁeld
+theory.
+For a general non-abelian gauge group G the Chern-Simons action becomes
+S = k
+4π
+�
+M
+d3x tr
+�
+AdA + 2
+3A ∧ A ∧ A
+�
+(225)
+Here, the cubic term is shorthand for
+tr
+�
+A ∧ A ∧ A
+�
+≡ tr
+�
+ǫµνλAµAνAλ�
+(226)
+for a gauge ﬁeld Aµ that takes values on the algebra of the gauge group G.
+47
+
+9.4. BF gauge theory
+A closely related (abelian) gauge theory is the so-called BF theory [166] which, in a general
+spacetime dimension D (even or odd), is a theory of a vector ﬁeld Aµ (a one-form) and an an-
+tisymmetric tensor ﬁeld B with D − 2 Lorentz indices (a D2 form), known as a Kalb-Ramond
+ﬁeld. In 2+1 dimensions the action of the BF theory is
+S = k
+2π
+�
+M
+d3x ǫµνλBλ∂µAν
+(227)
+where, once again, k is an integer.
+The BF gauge theory has the same content as the topological sector of a discrete Zk gauge
+theory. To see this we will consider the theory of compact electrodynamics which is a U(1)
+gauge theory minimally coupled to charged scalar ﬁeld φ. The will assume that the gauge theory
+is deﬁned for a compact U(1) gauge group (meaning that the gauge ﬂux is quantized) and that
+the complex scalar ﬁeld has charge integer k ∈ Z. As usual minimal coupling is implemented by
+introducing the covariant derivative Dµ − ∂µ + ikAµ, where Aµ is the U(1) gauge ﬁeld. Deep in
+the phase in which the global U(1) symmetry is spontaneously broken, usually called the Higgs
+regime, the amplitude of the scalar ﬁeld is frozen at its (real) vacuum expectation value φ0 but its
+phase ω, representing the Goldstone mode, is unconstrained. In this limit the Lagrangian of this
+theory becomes
+L = |φ0|2 �
+∂µω − kAµ
+�2 − 1
+4e2 F2
+µν
+(228)
+where e is the coupling constant of the gauge ﬁeld (the electric charge) and Fµν is the ﬁeld
+strength of the gauge ﬁeld Aµ. In general spacetime dimension D the charge e has units of
+length(D−4)/2. In this limit the gauge ﬁeld becomes massive (this is the Higgs mechanism). This
+theory has ﬂuxes quantized in units of 2π/k and has only k distinct ﬂuxes. This is the Zk gauge
+theory.
+Here we will consider this theory in 2+1 dimensions. We will use a gaussian (Hubbard-
+Stratonovich) decoupling of the ﬁrst term of Eq.(228) in terms of a gauge ﬁeld Cµ to write the
+Lagrangian in the equivalent form
+L = −
+1
+4|φ0|2C2
+µ + Cµ(∂µω − kAµ) − 1
+4e2 F2
+µν
+(229)
+Up to an integration by parts, we see that the phase ﬁeld ω plays the role of a Lagrange multiplier
+ﬁeld which forces the vector ﬁeld Bµ to obey the constraint ∂µCµ = 0. This constraint is solved
+by writing Cµ as
+Cµ = 1
+2πǫµνλ∂νBλ
+(230)
+of a 1-form gauge ﬁeld Bµ. Upon solving the constraint the Lagrangian of Eq.(229) becomes
+L = k
+2πǫµνλAµ∂νBλ − 1
+4e2 Fµν(A)2 −
+1
+32π2|φ0|2 Fµν(B)2
+(231)
+For spacetime dimensions D < 4 the IR the Maxwell terms for the ﬁelds Aµ and Bµ are irrelevant
+in the IR and, in this limit, this theory reduces to the BF theory at level k of Eq.(227). Therefore,
+Zk gauge theory is equivalent to the BF theory at level k. In fact, this result is essentially valid
+in all dimensions with the main diﬀerence being that the ﬁeld Bµ is, in general, a rank D − 2
+Kalb-Ramond antisymmetric ﬁeld.
+9.5. Quantization of Abelian Chern-Simons Gauge Theory
+By expanding the action of Eq.(224) the Lagrangian density becomes
+L = k
+4πǫi jAi∂0A j + A0
+� k
+2π B − J0
+�
+− JiAi
+(232)
+where B = ǫi j∂iA j is the local ﬂux, J0 is a local classical density and Ji a local classical current.
+Then, the ﬁrst term of Eq.(232) implies that the spatial components of the gauge ﬁeld obey
+equal-time canonical commutation relations
+[A1(x), A2(x′)] = i2π
+k δ(x − x′)
+(233)
+48
+
+The second term of the Lagrangian enforces the Gauss Law which for this theory simply
+implies that the states in the physical Hilbert space obey the constraint
+B(x) = 2π
+k J0(x)
+(234)
+Thus the Gauss Law requires that a charge density necessarily has a magnetic ﬂux attached to it.
+In other terms, the physical states are charge-ﬂux composites as postulated in Wilczek’s theory
+[157]. This is the theoretical basis to the concept of ﬂux attachment which, as we will see in
+section 10.3, is widely used in the theory of the fractional quantum Hall eﬀect.
+The third term in Eq.(232) simply states that the Hamiltonian density is just
+H = JiAi
+(235)
+Hence, in the absence of sources the Hamiltonian vanishes, H = 0.
+The Chern-Simons action is locally gauge-invariant, up to boundary terms. To see this let us
+perform a gauge transformation, Aµ → Aµ + ∂µΦ, where Φ(x) is a smooth, twice diﬀerentiable
+function. Then,
+S [Aµ + ∂µΦ] =
+�
+M
+(Aµ + ∂µΦ)ǫµνλ∂ν(Aλ + ∂λΦ)
+=
+�
+M
+d3x ǫµνλAµ∂νAλ +
+�
+M
+d3x ǫµνλ∂µΦ∂νAλ
+(236)
+Therefore, the change is
+S [Aµ + ∂µΦ] − S [Aµ] =
+�
+M
+d3x ∂µΦF∗
+µ =
+�
+M
+d3x ∂µ(ΦF∗
+µ) −
+�
+M
+d3x Φ∂µF∗
+µ
+(237)
+where F∗
+µ = ǫµνλ∂νAλ, is the dual ﬁeld strength. However, in the absence of magnetic monopoles,
+this ﬁeld satisﬁes the Bianchi identity, ∂µF∗µ = ∂µ(ǫµνλ∂νAλ) = 0. Therefore, using the Gauss
+Theorem, we ﬁnd that the change of the action is a total derivative and integrates to the boundary
+δS =
+�
+M
+d3x ∂µ(ΦF∗
+µ) =
+�
+Σ
+dS µΦF∗
+µ
+(238)
+where Σ = ∂M is the boundary of M. In particular, if Φ is a non-zero constant function on M,
+then the change of the action under such a gauge transformation is
+δS = Φ × ﬂux(Σ)
+(239)
+Hence, the action is not invariant if the manifold has a boundary, and the theory must be supplied
+with additional degrees of freedom at the boundary.
+Indeed, the ﬂat connections, i.e. the solution of the equations of motion, Fµν = 0, are pure
+gauge transformations, Aµ = ∂µφ, and have an action that integrates to the boundary. Let the
+M = D × R where D is a disk in space and R is time. The boundary manifold is Σ = S 1 × R,
+where S 1 is a circle. Thus, in this case, the boundary manifold Σ is isomorphic to a cylinder. The
+action of the ﬂat conﬁgurations reduces to
+S =
+�
+S 1×R
+d2x k
+4π∂0φ∂1φ
+(240)
+This implies that the dynamics on the boundary is that of a scalar ﬁeld on a circle S 1, and obeys
+periodic boundary conditions.
+Although classically the theory does not depend on the metric, it is invariant under arbitrary
+transformations of the coordinates. However, any gauge ﬁxing condition will automatically break
+this large symmetry. For instance, we can specify a gauge condition at the boundary in the form
+of a boundary term of the form Lgauge ﬁxing = A2
+1. In this case, the boundary action of the ﬁeld ϕ
+becomes
+S [ϕ] =
+�
+S 1×R
+d2x k
+4π
+�
+∂0φ∂1φ − (∂1φ2)
+�
+(241)
+The solutions of the equations of motion of this compactiﬁed scalar ﬁeld have the form φ(x1∓x0),
+and are right (left) moving chiral ﬁelds depending of the sign of k. This boundary theory is not
+topological but is is conformally invariant.
+49
+
+A similar result is found in non-abelian Chern-Simons gauge theory. In the case of the
+S U(N)k Chern-Simons theory on a manifold D × R, where D is a disk whose boundary is Γ, and
+R is time, the action is
+S CS[A] =
+�
+D×R
+d3x
+� k
+8πtr
+�
+ǫµνλAµ∂νAλ + 2
+3ǫµνλAµAνAλ
+��
+(242)
+This theory integrates to the boundary, Γ×R where it becomes the chiral (right-moving) S U(N)k
+Wess-Zumino-Witten model (at level k) at its IR ﬁxed point, λ2
+c = 4π/k
+S WZW[g] =
+1
+4λ2c
+�
+Γ×R
+d2x tr
+�
+∂µg∂µg−1�
++
+k
+12π
+�
+B
+ǫµνλtr
+�
+g−1∂µg g−1∂νg g−1∂λg
+�
+(243)
+Here, g ∈ S U(N) parametrizes the ﬂat conﬁgurations of the Chern-Simons gauge theory. The
+boundary theory is a non-trivial CFT, the chiral Wess-Zumino-Witten CFT [165].
+9.6. Vacuum degeneracy a torus
+We will now construct the quantum version of the U(1) Chern-Simons gauge theory on a
+manifold M = T 2 × R, where T 2 is a spatial torus, of linear size L1 and L2. Since this manifold
+does not have boundaries, the ﬂat connections, ǫi j∂iA j = 0 do not reduce to local gauge transfor-
+mations of the form Ai = ∂iΦ. Indeed, the holonomies of the torus T 2, i.e. the Wilson loops on
+the two non-contractible cycles of the torus Γ1 and Γ2 are gauge-invariant observables:
+� L1
+0
+dx1A1 ≡ ¯a1,
+� L2
+0
+dx1A2 ≡ ¯a2
+(244)
+where ¯a1 and ¯a2 are time-dependent. Thus, the ﬂat connections now are
+A1 = ∂1Φ + ¯a1
+L1
+,
+A2 = ∂2Φ + ¯a2
+L2
+(245)
+whose action is
+S = k
+4π
+�
+dx0ǫi j¯ai∂0¯a j
+(246)
+Therefore, the global degrees of freedom ¯a1 and ¯a2 at the quantum level become operators that
+satisfy the commutation relations
+[¯a1, ¯a2] = i2π
+k
+(247)
+We ﬁnd that the ﬂat connections are described by the quantum mechanics of ¯a1 and ¯a2. A
+representation of this algebra is
+¯a2 ≡ −i2π
+k
+∂
+∂¯a1
+(248)
+Furthermore, the Wilson loops on the two cycles become
+W[Γ1] = exp
+�
+i
+� L1
+0
+A1
+�
+≡ ei¯a1,
+W[Γ2] = exp
+�
+i
+� L2
+0
+A2
+�
+≡ ei¯a2
+(249)
+and satisfy the algebra
+W[Γ1]W[Γ2] = exp(−i2π/k)W[Γ2]W[Γ1]
+(250)
+Under large gauge transformations
+¯a1 → ¯a1 + 2π,
+¯a2 → ¯a2 + 2π
+(251)
+Therefore, invariance under large gauge transformations on the torus implies that ¯a1 and ¯a2 deﬁne
+a two-torus target space.
+Let us deﬁne the unitary operators
+U1 = exp(ik¯a2),
+U2 = exp(−ik¯a1)
+(252)
+which satisfy the algebra
+U1U2 = exp(i2πk)U2U1
+(253)
+50
+
+The unitary transformations U1 and U2 act as shift operators on ¯a1 and ¯a2 by 2π, and hence
+generate the large gauge transformations. Moreover, the unitary operators U1 and U2 leave the
+Wilson loop operators on non-contractible cycles invariant,
+U−1
+1 W[Γ1]U1 = W[Γ1],
+U−1
+2 W[Γ2]U2 = W[Γ2]
+(254)
+Let |0⟩ be the eigenstate of W[Γ1] with eigenvalue 1, i.e. W[Γ1]∥0⟩ = |0⟩. The state W[Γ2]|0⟩ is
+also an eigenstate of W[Γ1] with eigenvalue exp(−i2π/k), since
+W[Γ1]W[Γ2]|0⟩ = ei2π/kW[Γ2]W[Γ1]|0⟩ = e−i2π/kW[γ2|0⟩
+(255)
+More generally, since
+W[Γ1]W p[Γ2]∥0⟩ = e−i2πp/kW p[Γ2]|0⟩
+(256)
+we ﬁnd that, provided k ∈ Z, there are k linearly independent vacuum states |p⟩ = W p[Γ2]|0⟩, for
+the U(1) Chern-Simons gauge theory at level k. It is denoted as the U(1)k Chern-Simons theory.
+Therefore the ﬁnite-dimensional topological space on a two-torus is k-dimensional. It is trivial
+to show that, on a surface of genus g, the degeneracy is kg.
+We see that in the abelian U(1)k Chern-Simons theory the Wilson loops must carry k possible
+values of the unit charge. This property generalizes to the non-abelian theories, which are tech-
+nically more subtle. We will only state some important results. For example, if the gauge group
+is SU(2) we expect that the Wilson loops will carry the representation labels of the group SU(2),
+i.e. they will be labelled by (j, m), where j = 0, 1
+2, 1, . . . and the 2 j+1 values of m satisfy |m| ≤ j.
+However, it turns out SU(2)k Chern-Simons theory has fewer states, and that the values of j are
+restricted to the range j = 0, 1
+2, . . . , k
+2.
+9.7. Fractional Statistics and Braids
+Another aspect of the topological nature of Chern-Simons theory is the behavior of expec-
+tation values of products of Wilson loop operators. Let us compute the expectation value of
+a product of two Wilson loop operators on two positively oriented closed contours γ1 and γ2.
+We will do this computation in the abelian Chern-Simons theory U(1)k in 2+1-dimensional Eu-
+clidean space. Note that the Euclidean Chern-Simons action is pure imaginary since the action
+is ﬁrst-order in derivatives. The expectation value to be computed is
+W[γ1 ∪ γ2] =
+�
+exp
+�
+i
+�
+γ1∪γ2
+dxµAµ
+��
+CS
+(257)
+The result changes depending on whether the loops γ1 and γ2 are linked or unlinked. In this
+section we will compute the contribution to this expression for a pair of contours γ1 and γ2. Here
+we will not include the contribution to this expectation value for each contours. We will return
+to this problem in section 11.3.1 where we discuss the problem of fractional spin.
+The expectation value of a Wilson loop on the union of two contours, as in the present case,
+γ can be written as
+�
+exp
+�
+i
+�
+γ
+dxµAµ
+��
+CS =
+�
+exp
+�
+i
+�
+d3xJµAµ
+��
+CS
+(258)
+where the current Jµ is
+Jµ(x) = δ(xµ − zµ(t))dzµ
+dt
+(259)
+Here zµ(t) is a parametrization of the contour γ. Therefore, the expectation value of the Wilson
+loop is [165]
+�
+exp
+�
+i
+�
+γ
+dxµAµ
+��
+CS ≡ exp(iI[γ]CS) = exp
+�
+− i
+2
+�
+d3x
+�
+d3y Jµ(x)Gµν(x − y)Jν(y)
+�
+(260)
+where Gµν(x − y) = ⟨Aµ(x)Aν(y)⟩CS is the propagator of the Chern-Simons gauge ﬁeld. Since the
+loops are closed, the current Jµ is conserved, i.e. ∂µJµ = 0, and the eﬀective action I[γ]CS of the
+loop γ is gauge-invariant.
+The Euclidean propagator of Chern-Simons gauge theory (in the Feynman gauge) is
+Gµν(x − y) = 2π
+k G0(x − y)ǫµνλ∂λδ(x − y)
+(261)
+51
+
+where G0(x − y) is the propagator of the massless Euclidean scalar ﬁeld, which satisﬁes
+− ∂2G0(x − y) = δ3(x − y)
+(262)
+Using these results, we ﬁnd the following expression for the eﬀective action
+I[γ]CS =π
+k
+�
+d3x
+�
+d3y Jµ(x)Jν(y)G0(x − y)ǫµνλ∂λδ(x − y)
+=π
+k
+�
+γ
+dxµ
+�
+γ
+dyνǫµνλ∂λG0(x − y)
+(263)
+Since the current Jµ is conserved, it can be written as the curl of a vector ﬁeld, Bµ, as
+Jµ = ǫµνλ∂νBλ
+(264)
+In the Lorentz gauge, ∂µBµ = 0, we can write
+Bµ = ǫµνλ∂νφλ
+(265)
+Hence,
+Jµ = −∂2φµ
+(266)
+where
+φµ(x) =
+�
+d3y G0(x − y)Jµ(y)
+(267)
+Upon substituting this result into the expression for Bµ, we ﬁnd
+Bµ =
+�
+d3yǫµνλ∂νG0(x − y)Jλ(y) =
+�
+γ
+ǫµνλ∂νG0(x − y)dyλ
+(268)
+Therefore, the eﬀective action I[γ]CS becomes
+I[γ]CS = π
+k
+�
+γ
+dxµ
+�
+γ
+dyνǫµνλ∂λG0(x − y) = π
+k
+�
+γ
+dxµBµ(x)
+(269)
+Let Σ be an oriented open surface of the Euclidean three-dimensional space whose boundary is
+the oriented loop (or union of loops) γ, i.e. ∂Σ = γ. Then, using Stokes Theorem we write in the
+last line of Eq.(269) as
+I[γ]CS = π
+k
+�
+Σ
+dS µǫµνλ∂νBλ = π
+k
+�
+Σ
+dS µJµ
+(270)
+The integral in the last line of this equation is the ﬂux of the current Jµ through the surface Σ.
+Therefore, this integral counts the number of times nγ the Wilson loop on γ pierces the surface Σ
+(whose boundary is γ), and therefore it is an integer, nγ ∈ Z. We will call this integer the linking
+number (or Gauss invariant) of the conﬁguration of loops. In other words, the expectation value
+of the Wilson loop operator is
+W[γ]CS = eiπnγ/k
+(271)
+The linking number is a topological invariant since, being an integer, its value cannot be changed
+by smooth deformations of the loops, provided they are not allowed to cross.
+We will now see that this property of Wilson loops in Chern-Simons gauge theory leads to the
+concept of fractional statistics. Let us consider a scalar matter ﬁeld that is massive and charged
+under the Chern-Simons gauge ﬁeld. The excitations of this matter ﬁeld are particles that couple
+minimally to the gauge ﬁeld. Here we will be interested in the case in which these particles are
+very heavy. In that limit, we can focus on states that have a few of this particles which will be in
+their non-relativistic regime.
+Consider, for example, a state with two particles which in the remote past, at time t = −T →
+−∞, are located at two points A and B. This initial state will evolve to a ﬁnal state at time t = T →
+∞, in which the particles either go back to their initial locations (the direct process), or to another
+one in which they exchange places, A ↔ B. At intermediate times, the particles follow smooth
+worldlines. These two processes, direct and exchange. There we see that the direct process is
+equivalent to a history with two unlinked loops (the worldlines of the particles), whereas in the
+exchange process the two loops form a link. It follows from the preceding discussion that the
+52
+
+two amplitudes diﬀer by the result of the computation of the Wilson loop expectation value for
+the loops γ1 and γ2. Let us call the ﬁrst amplitude Wdirect and the second Wexchange. The result is
+Wexchange = Wdirecte±iπ/k
+(272)
+where the sign depends on how the two worldlines wind around each other.
+An equivalent interpretation of this result is that if Ψ[A, B] is the wave function with the two
+particles at locations A and B, the wavefunction where their locations are exchanged is
+Ψ[B, A] = e±iπ/kΨ[A, B]
+(273)
+where the sign depends on whether the exchange is done counterclockwise or clockwise. Clearly,
+for k = 1 the wave function is antisymmetric and the particles are fermions, while for k → ∞
+they are bosons. At other values of k the particles obey fractional statistics and are called anyons
+[156, 153]. The phase factor φ = ±π/k is called the statistical phase.
+Notice that, while for fermions and bosons the statistical phase ϕ = 0, π is uniquely deﬁned
+(mod 2π), for other values of k the statistical angle is speciﬁed up to a sign that speciﬁes how
+the worldlines wind around each other. Indeed, mathematically the exchange process is known
+as a braid. Processes in which the worldlines wind clock and counterclockwise are braids that
+are inverse of each other. Braids can also be stack sequentially yielding multiples of the phase
+ϕ. In addition to stacking braids, Wilson loops can be fused: seen from some distance, a pair of
+particles will behave as a new particle with a well deﬁned behavior under braiding. This process
+of fusion is closely related to the concept of fusion of primary ﬁelds in Conformal Field Theory.
+Furthermore, up to regularization subtleties [167], the self-linking terms (those with a = b)
+yield a topological spin 1/2k, consistent with the spin-statistics connection [168]. For k = 1 this
+means that the ﬂux-charge composites have spin 1/2.
+What we have just described is a mathematical structure called the Braid group. The example
+that we worked out using abelian Chern-Simons theory yields one-dimensional representations
+of the Braid group with the phase ϕ being the label of the representations. For U(1)k there are k
+types of particles (anyons). That these representations are abelian means that, in the general case
+of U(1)k, acting on a one-dimensional representation p (deﬁned mod k) with a one-dimensional
+representation q (also deﬁned mod k) yields the representation one-dimensional p+q (mod k). We
+will denote the operation of fusing these representations (particles!) as [q]mod k × [p]mod k = [q +
+p]mod k. These representations are in one-to-one correspondence with the inequivalent charges of
+the Wilson loops, and with the vacuum degeneracy of the U(1)k Chern-Simons theory on a torus.
+A richer structure arises in the case of the non-abelian Chern-Simons theory at level k [165],
+such as SU(2)k. For example, for SU(2)1 the theory has only two representations, both are one-
+dimensional, and have statistical angles ϕ = 0, π/2.
+However, for SU(2)k, the content is more complex. In the case of SU(2)2 the theory has
+a) a trivial representation [0] (the identity, (j, m) = (0, 0)), b) a (spinor) representation [1/2]
+((j, m) = (1/2, ±1/2)), and c) a the representation [1] ((j, m) = (1, m), with m = 0, ±1). These
+states will fuse obeying the following rules: [0] × [0] = [0], [0] × [1/2] = [1/2], [0] × [1] = [1],
+[1/2]×[1/2] = [0]+[1], [1/2]×[1] = [1/2], and [1]×[1] = [0] (note the truncation of the fusion
+process!).
+Of particular interest is the case [1/2] × [1/2] = [0] + [1]. In this case we have two fusion
+channels, labeled by [0] and [1]. The braiding operations now will act on a two-dimensional
+Hilbert space and are represented by 2 × 2 matrices. This is an example of a non-abelian repre-
+sentation of the braid group. These rather abstract concepts have found a physical manifestation
+in the physics of the fractional quantum Hall ﬂuids, whose excitations are vortices that carry
+fractional charge and anyon (braid) fractional statistics.
+Why this is interesting can be seen by considering a Chern-Simons gauge theory with four
+quasi-static Wilson loops. For instance in the case of the SU(2)2 Chern-Simons theory the Wilson
+loops carry the spinor representation, [1/2]. If we call the four particles A, B, C and D, we
+would expect that their quantum state would be completely determined by the coordinates of
+the particles. This, however, is not the case since, if we fuse A with B, the result is either a
+state [0] or a state [1]. Thus, if the particles were prepared originally in some state, braiding
+(and fusion) will lead to a linear superposition of the two states. This braiding process deﬁnes
+a unitary matrix, a representation of the Braid Group. The same is true with the other particles.
+However, it turns out that for four particles there are only two linearly independent states. This
+two-fold degenerate Hilbert space of topological origin is called a topological qubit.
+53
+
+Moreover, if we consider a system with N (even) number of such particles, the dimension of
+the topologically protected Hilbert space is 2
+N
+2 −1. Hence, for large N, the entropy per particle
+grows as 1
+2 ln 2 = ln
+√
+2. Therefore the qubit is not an “internal” degree of freedom of the
+particles but a collective state of topological origin. Interestingly, there are physical systems,
+known as non-abelian fractional quantum Hall ﬂuids that embody this physics and are accessible
+to experiments! For these reasons, the non-abelian case has been proposed as a realization of a
+topological qubit [169, 170].
+10. Topological Phases of Matter
+10.1. Topological Insulators
+We will now give a brief discussion of the physics of Topological Insulators from a ﬁeld-
+theoretic perspective. Topological insulators are systems whose electronic states (band struc-
+tures) have special topological properties which manifest in the existence of symmetry-protected
+edge states. For this reason these systems are known as symmetry-protected topological states
+(or SPTs). The simplest example is found in one space dimension where it is related to the fas-
+cinating problem of fractionally charged solitons and electron fractionalization. In several ways
+many of the concepts involved in these 1+1-dimensional systems can and have been extended to
+higher dimensions.
+10.1.1. Dirac fermions in 2+1 dimensions
+We will now see that, in spite of the formal similarities withe the 1+1 dimensional case, this
+theory has diﬀerent symmetries, particularly concerning parity and time reversal invariance. In
+addition, in 2+1 dimensions it is not possible to deﬁne a γ5 Dirac matrix which implies that there
+is no chiral symmetry and no chiral anomaly. We will consider ﬁrst a theory of a Dirac ﬁeld
+which in 2+1 dimensions is a bi-spinor (as is in 1+1 dimensions). The Lagrangian of the Dirac
+theory coupled to a background gauge ﬁeld Aµ is
+L = ¯ψiγµ∂µψ − m ¯ψψ − eAµ ¯ψγµψ ≡ ¯ψiγµDµ(A)ψ − m ¯ψψ
+(274)
+where ¯ψ = ψ†γ0, ¯ψγµψ ≡ jµ is the gauge-invariant (and conserved) Dirac current, and Dµ(A) =
+∂µ + ieAµ is the covariant derivative.
+In Eq.(274), γµ are the three 2 × 2 Dirac matrices which obey the algebra, {γµ, γν} = 2gµν,
+where gµν = diag(1, −1, −1) is the metric of 2+1 dimensional Minkowski spacetime. The Dirac
+matrices γµ can be written in terms of the three 2 × 2 Pauli matrices. For instance we can choose
+γ0 = σ3, γ1 = iσ2 and γ2 = −iσ1 which satisfy the Dirac algebra. Since the three gamma
+matrices involve all tree Pauli matrices, it is not possible to deﬁne a γ5 matrix which would
+anticommute with the gamma matrices. In this sense, it is not possible to deﬁne a chirality for
+bi-spinors in 2+1 dimensions
+We could have also chosen a diﬀerent set of gamma matrices. For instance, we could have
+chosen γ0 = σ3, γ1 = iσ2 and γ2 = +iσ1 which also satisfy the Dirac algebra. These two choices
+are equivalent to the 2D parity transformation x1 → x1 and x2 → −x2. In other words, we can
+choose the three gamma matrices to be deﬁned in terms of a right handed or a left handed frame
+(or triad). Thus, the choice of handedness of the frame used to deﬁne the gamma matrices can
+be regarded as a chirality.
+The parity transformation is also equivalent to a unitary transformation (a change of basis)
+U = γ1 = iσ1 which ﬂips the sign of both γ2 and γ0. Thus, a parity transformation is equivalent
+to a change of the sign of γ0 or, what is the same, as changing the sign of the Dirac mass
+m → −m. Since the Dirac theory is charge conjugation invariant, the parity transformation is
+equivalent to time reversal. It is easy to see that in 2+1 dimensions the massive Dirac theory
+is not invariant under time reversal since this the single particle Dirac Hamiltonian H(p) for
+momentum p involves all three Pauli matrices,
+H(p) = α · p + βm
+(275)
+where, as usual, α = γ0γ and β = γ0. In fact, time reversal is equivalent to the change of the
+sign of the Dirac mass. These considerations also apply to the massless Dirac theory coupled to
+a background gauge ﬁeld.
+54
+
+10.1.2. Dirac Fermions and Topological Insulators
+In addition systems in one space dimension, discussed in section 7.1, in which Dirac fermions
+naturally describe the low energy electronic degrees of freedom, Dirac fermions also play a role
+in many other system in Condensed Matter Physics. Such systems range from the nodal Bogoli-
+ubov fermionic excitations of d-wave superconductors in 2D and p-wave superﬂuids in 3D, to
+Chern and Z2 topological insulators in 2D and Z2 topological insulators and Weyl semimetals in
+3D. The also play a role in spin liquid phases of frustrated spin-1/2 quantum antiferromagnets,
+such as Dirac and chiral spin liquids. Dirac fermions play also a signiﬁcant role in our under-
+standing of the compressible limit of 2D electron gases (2DEG) in large magnetic ﬁelds (see
+section 10.3). We will not cover here all of these examples. Instead we will focus of the 2D
+Chern topological insulators (which exhibit the anomalous quantum Hell eﬀect) and on the 3D
+Z2 topological insulators.
+In Condensed matter Physics the important low energy electronic degrees of freedom belong
+to the electronic states close to the Fermi energy. In such systems the lattice periodic potential
+determines the properties of their band structures. The only signiﬁcant exception to this general
+rule is the case of the 2DEGs in GaAs-AlAs heterostructures (the most commonly used platform
+for the study of quantum hall eﬀects) whose electronic densities are low enough so that the lattice
+periodic potential can for practical purposes almost always be ignored.
+In general, Dirac (and Weyl) fermions arise when the conduction and valence bands cross
+at isolated points of the Brillouin zone. In such situations, the near the crossing points the low
+energy electronic states locally (in momentum space) look like cones and, ignoring possible
+anisotropies, look like the states of a Dirac-Weyl theory. For these reasons, Dirac fermions play
+a central role in the theory of graphene type materials [171] and, more signiﬁcantly, in the theory
+of topological insulators [172].
+Dirac fermions play a central role in Quantum Electrodynamics and in the Standard Model of
+Particle Physics. The problem of quark conﬁnement requires an understanding of these theories
+in a regime inaccessible to Feynman-diagrammatic perturbation theory and an intrinsically non-
+perturbative formulation, known as Lattice Gauge Theory [55, 56, 173], needed to be developed.
+For this reason in the 1970s fermionic Hamiltonians with crossings at points became of great
+interest in High Energy Physics as a way to describing the short-distance dynamics of quarks in
+Lattice Gauge Theory.
+This program run into diﬃculties when extended to the theory of Weak Interactions which
+have an odd number of species of chiral (Weyl) Dirac fermions. All local discretizations of the
+Dirac equation yield an even number of species. This fact came to be known as the fermion
+doubling problem. These results are special cases of a general theorem, due to Holger Nielsen
+and Masao Ninomiya [174, 175, 176] (and extended by Daniel Friedan [177]), which proves that
+for systems whose kinetic energy is local in space (as it must be) there must always be an even
+number of crossings. Therefore, it is impossible to write a local theory with an odd number of
+chiral fermionic species.
+In the context of Condensed Matter Physics, the simplest example of Dirac fermions is found
+in the electronic structure of graphene (a single layer of graphite) discussed by A. H. Castro Neto
+and coworkers [171]. Graphene is an allotrope of crystalline carbon in which the carbon atoms
+are arranged in a 2D hexagonal lattice, which is a lattice with two inequivalent sites in its unit
+cell, labeled by A and B. Let rA and rB be the sites of the two sublattices. Each site rA has three
+nearest neighbors on the B sublattice located at ri
+B = rA + di, where i = 1, 2, 3 and
+d1 =
+� 1
+2
+√
+3
+, 1
+2
+�
+,
+d2 =
+� 1
+2
+√
+3
+, −1
+2
+�
+,
+d3 =
+�
+− 1√
+3
+, 0
+�
+(276)
+(in units in which the lattice spacing is a = 1). The nearest neighbor A sites are related by the
+vectors
+a1 =
+
+√
+3
+2 , −1
+2
+ ,
+a2 = (0, 1),
+a3 =
+−
+√
+3
+2 , −1
+2
+
+(277)
+where a1 and a2 are the primitive lattice vector of the hexagonal lattice. The nearest neighbor
+sites of the B sublattice are also related by these same three vectors.
+The low energy electronic degrees of freedom of graphene can be described by a single
+fermionic state (ignoring spin) at each carbon atom . Let c†(rA) and c†(rB) be the fermion oper-
+ators that creates an electron at the site rA and at the site nearest neighbor sites rB, respectively.
+The Hamiltonian is
+H = t1
+�
+rA,i=1,2,3
+c†(rA)c(rA + di) + h.c.
+(278)
+55
+
+where t1 is the hopping matrix element between nearest neighbor A and B sites.
+In Fourier space
+c(rA) =
+�
+BZ
+d2k
+(2π)2ψA(k) exp(ik · rA),
+c(rB) =
+�
+BZ
+d2k
+(2π)2 ψB(k) exp(ik · rB)
+(279)
+where BZ is the ﬁrst Brillouin zone of the hexagonal lattice, a hexagonal region of reciprocal
+space with vertices at ±2π/
+√
+3(1, 1/
+√
+3) (which are denoted by K and K′, respectively) and their
+two images under 2π/3 rotations. In momentum space the Hamiltonian is [178]
+H = t1
+�
+BZ
+d2k
+(2π)2
+�
+ψA(k)
+ψB(k)
+� �
+0
+�
+j=1,2,3 exp(ik · d j)
+�
+j=1,2,3 exp(−ik · d j)
+0
+� �ψA(k)
+ψB(k)
+�
+(280)
+The one-particle states of this Hamiltonian have eigenvalues E(k) = ±
+����
+�
+j=1,2,3 exp(ik · d j
+����.
+Clearly E(k) vanishes at the K and K′ points, the corners of the Brillouin zone. Hence, there
+is a crossing of the two bands at the K and K′ points near which the energy eigenvalues are a two
+cones at K and K′, respectively. Thus, the low energy states of graphene consists of two massless
+(gapless) Dirac bi-spinors, with opposite chirality/parity.
+Other examples with similar fermion content are a theory of fermions on a 2D square lattice
+with a π magnetic ﬂux (1/2 of the ﬂux quantum) per plaquette which arises, for instance, in the
+theory of the integer quantum Hall eﬀect on lattices by David Thouless, Mahito Kohmoto, Peter
+Nightingale and Marcel den Nijs [179] (albeit for more general ﬂux per plaquette), and in the
+theory of ﬂux phases of 2D spin liquids of Ian Aﬄeck and J. Brad Marston [180]. It also arises
+in the theory of the chiral spin liquid of Xiao-Gang Wen, Frank Wilczek and Anthony Zee [181].
+We will discuss these theories in section 10.
+10.1.3. Chern invariants
+The single-particle quantum states of free fermions (electrons) in the periodic potential of a
+crystal are Bloch wave functions labeled by the lattice momentum k and band indices m. The
+Bloch states are periodic functions of the lattice momenta whose periods are the Brillouin zones.
+Topologically each Brillouin zone is a torus: in 1D is a 1-torus, in 2D a 2-torus, etc. The sym-
+metries (and shapes) of the Brillouin zone are dictated by the symmetries of the host crystal.
+In this section we will consider the quantum states of systems of fermions on a 2D lattice with
+M bands with eigenvalues {Em(k)} with m = 1, . . ., M. The eigenstates are Bloch states {|um(k)⟩}
+such that the wave functions ar ψm(x) = um(k) exp(ik · x), where k is a (quasi) momentum in the
+ﬁrst Brillouin zone [182]. We will assume that the band spectrum is such that all the bands are
+separated from each other by a ﬁnite gap for all momenta in the ﬁrst Brillouin zone. Hence, the
+eigenvalues obey the inequality |Em(k) − En(k)| > 0. We will assume that the system of interest
+has N < M ﬁlled bands and, consequently, that there is a gap separating the top-most occupied
+band N (the “valence band”) and the lowest unoccupied band N + 1 (the “conduction band”) that
+does not close everywhere in the ﬁrst Brillouin zone.
+Let us consider speciﬁcally a 2D system and a Bloch state |um(k)⟩ at a momentum k, and let
+∂k|um(k)⟩ be an inﬁnitesimally close Bloch state with momentum k′ = k+dk. The inner product
+of these two inﬁnitesimally close Bloch states is the vector ﬁeld (deﬁned on the ﬁrst Brillouin
+zone for each band m)
+A(m)
+j (k) = i⟨um(k)|∂ j|um(k)⟩
+(281)
+where j = 1, 2 are the orthogonal direction in momentum space and we have used the notation
+∂i = ∂/∂ki. This vector ﬁeld is known as the Berry Connection of the electronic states in the mth
+band.
+The theory of the quantum states of non-interacting electrons in periodic potentials as devel-
+oped by Felix Bloch [182] is the foundation of much of what we know about the physics of many
+semiconductors and simple metals. This theory is the core subject of many textbooks [183, 184].
+This theory is based on the classiﬁcation of the electronic states as representations of the group
+of lattice translations and point (and spatial) group symmetry transformations. A key unstated
+assumption in much of this body of work is that the Bloch states are globally well-deﬁned func-
+tions on the Brillouin zone of a given crystal. It is rather remarkable that this assumption was not
+stated explicitly since the original work by Bloch in 1929 until the work by Thouless, Kohmoto,
+Nightingale and den Nijs on the quantum states of electrons on 2D lattices in the presence of an
+uniform magnetic ﬁeld [179]. The realization that there is a topological obstruction to deﬁne the
+56
+
+electronic states globally in the Brillouin zone is the key to the development of the the theory of
+topological insulators and, more generally, of the modern theory of band structures [185, 186].
+In quantum mechanics states are deﬁned up to a phase. This means that Bloch states that
+diﬀer by a phase describe the same quantum state. In other words each quantum state |um(k)⟩ is
+a member of a ray (or ﬁber) of states each labeled by a phase,
+|um)(k)⟩ �→ exp(ifm(k)) |um(k)⟩
+(282)
+Changing the states by phase factors deﬁnes a U(1)M unitary transformation of physically equiva-
+lent states. Since we have a ﬁbre at each point k, this amount at deﬁning a ﬁbre bundle. However,
+under this transformation the Berry connection A(m)
+j (k) changes by a gradient of the phases
+A(m)
+j (k) �→ A(m)
+j (k) + ∂ j fm(k)
+(283)
+where we assumed that the phases fm(k) are continuous and diﬀerentiable functions on the entire
+ﬁrst Brillouin zone. Since the physical states cannot change by redeﬁnitions of the phases of
+the basis states we are led to the condition that only the data that is invariant under the gauge
+transformations deﬁned by Eqs. (282) and (283) is physically meaningful, which is encoded in
+the gauge-invariant pseudo-scalar quantity F (m)(k)
+F (m)(k) = ǫi j∂iA(m)
+j (k)
+(284)
+which is known as the Berry curvature.
+In what follows we will be interested in the quantity
+C(m)
+1
+= 1
+2π
+�
+Γ
+dk jA(m)
+j (k) = 1
+2π
+�
+BZ
+d2k F (m)(k)
+(285)
+where Γ is the boundary of the 1st Brillouin zone (which we denote by BZ). We will now show
+that the quantity C(m)
+1 , which measure the ﬂux of the Berry connection A(m)
+j (k) through the ﬁrst
+Brillouin zone (in units of 2π), is an integer independent of the particular connection A(m)
+j (k) that
+we have chosen. This integer-valued quantity is a topological invariant known as the ﬁrst Chern
+number.
+However, since the ﬁrst Brillouin zone is a 2-torus ,which is a closed manifold, a Berry
+connection with a net ﬂux cannot obey periodic boundary conditions. Instead, we must allow
+for generalized (large) gauge transformations that wrap around the 2-torus of the ﬁrst Brillouin
+zone. This problem is similar (and closely related to) the problem of the wave functions of a
+charged particle moving in the presence of a magnetic monopole [151]. We will consider a 2D
+system whose ﬁrst Brillouin zone is spanned by two reciprocal lattice vectors b1 and b2, related
+by the two primitive lattice vectors a1 and a2 by the relation bi · a j = 2πδi j (with i, j = 1, 2) The
+generalized gauge transformations now are
+|um(k + b1)⟩ = exp(if (1)
+m (k)) |um(k)⟩,
+|um(k + b2)⟩ = exp(if (2)
+m (k)) |um(k)⟩
+A(m)
+j (k + b1) =A(m)
+j (k) + ∂ j f (1)
+m (k),
+A(m)
+j (k + b2) = A(m)
+j (k) + ∂ j f (2)
+m (k)
+(286)
+where f (1)
+m (k) and f (2)
+m (k) are smooth functions of k.
+We will now show that the Bloch states cannot be deﬁned globally over the Brillouin zone if
+the ﬂux of the Berry connection does not vanish. More speciﬁcally let us assume that at some
+point k0 ∈ T we know the Bloch state |um(k0)⟩ which satisﬁes the generalized periodic boundary
+conditions of Eq.(286). The question is, can we determine the Bloch state at some other also
+arbitrary point k′
+0? We will now that there is a topological obstruction that does not allow it if
+the ﬂux of the Berry phase is not zero. The reason is that the phase of the Bloch state cannot be
+deﬁned since, in general, the Bloch state will vanish at some point of the Brillouin zone where
+the phase is undeﬁned.
+We will prove that this si true by following an elegant construction due to Mahito Kohmoto
+which goes as follows [187]. The Brillouin zone is a torus, that we will denote by T, can be
+regarded as the tensor product of two circles, T ≡ S 1 × S 1. Let us split the torus T into two
+disjoint subsets (or patches) HI and HII such that T = HI
+� HII. We will assume that in region
+HI. Let us assume that the Bloch state vanishes at some point k0 ∈ TI and that the Bloch state
+does not vanish for all points k ∈ HII. This means that we can choose the Bloch state |um(k)⟩
+to be real for all k ∈ HII. On the other hand we can always assign some arbitrary phase to the
+57
+
+Bloch state at k0 ∈ TI. Once we have done that we can extend the deﬁnition of the phase to a
+neighborhood of k0 wholly included in HI. If we the assume that there is only one zero, then
+the phase can be deﬁned over all of HI. The result is that we now have two diﬀerent deﬁnitions
+of the phase of the Bloch state on HI and on HII. Let |um(k)⟩I and |um(k)⟩II be the two resulting
+deﬁnitions of the Bloch state. Let the closed curve γ be the common boundary of regions HI
+and HII, γ = ∂HI = ∂HII. However on the common boundary γ these the two deﬁnitions of the
+Bloch state must be a gauge transformation,
+|um(k)⟩I = exp(ifm(k)) |um(k)⟩II
+(287)
+on all points k ∈ γ. The gauge transformation fm(k), known as the transition function, is a
+smooth periodic function of the points k on the closed curve γ. Likewise, the Berry connection
+A(m)
+j (k) has two deﬁnitions on the regions HI and HII which diﬀer by a gauge transformation on
+all the points k of the common boundary γ,
+A(m)
+j (k)
+����I − A(m)
+j (k)
+����II = ∂ j fm(k)
+(288)
+The transition function deﬁnes a mapping of the closed curve γ, which is topologically equivalent
+to a circle S 1, to the phase of the Bloch state which is deﬁned mod 2π and hence is also a circle
+S 1. Hence the transition functions fm(k) are homotopies which can be classiﬁed by the homotopy
+group Π1(S 1) ≃ Z. We recognize that the classiﬁcation of the the transition functions is the same
+that we used for the vortices of section 5.1.1. Hence, the change of the transition function on a
+full revolution of the closed curve γ must be an integer multiple of 2π.
+We will now compute the quantity C(m)
+1 , given in Eq.(285), which measures the ﬂux of the
+Berry connection through the 2-torus T that deﬁnes the ﬁrst Brillouin zone. Using the partition
+of the torus T = TI
+� TII we can write
+C(m)
+1
+= 1
+2π
+�
+T
+d2k F (m)(k) = 1
+2π
+��
+HI
+d2k F (m)(k) +
+�
+HII
+d2k F (m)(k)
+�
+(289)
+Using Stokes Theorem on the two regions HI and HII we get
+C(m)
+1
+= 1
+2π
+�
+γ
+dk j
+�
+A j(k)
+����I − A j(k)
+����II
+�
+(290)
+Since the two deﬁnitions of the Berry connection diﬀer by a gauge transformation, Eq.(288), we
+can express C(m)
+1
+in terms of the transition function fm(k) on the curve γ
+C(m)
+1
+= 1
+2π
+�
+γ
+dk · ∂fm(k) = 1
+2π (∆fm(k))γ = n
+(291)
+where we used that the transition functions are classiﬁed by the integer-valued quantity n ∈ Z.
+We conclude that the ﬂux of the Berry curvature C(m)
+1
+takes only integer values. It is known of
+the ﬁrst Chern number C(m)
+1
+which is a topological invariant since it cannot change by a smooth
+redeﬁnition of the curve γ. Since C(m)
+1
+� 0 implies that the Bloch states must vanish at least
+at some point (or points) k ∈ HI, then the integer-valued Chern number can only change if a
+redeﬁnition of the curve γ crosses at least one point k at which the Bloch state vanishes.
+The conclusion of this analysis is that whenever the ﬂux of the Berry curvature through the
+Brillouin zone does not vanish the Bloch states cannot be deﬁned globally on the Brillouin zone.
+A direct consequence is that in this case the band is characterized by a topological invariant
+called the ﬁrst Chern number. This number is a property of a given band and, in general, it is
+diﬀerent for each band. This is a particular case of a more general topological classiﬁcation of
+the states of free fermions on a lattice which depends on on the dimensionality of the system
+[188, 189, 190].
+10.1.4. The quantum Hall eﬀect on a lattice
+We will now show that 2D free fermion systems with Chern bands, i.e. bands characterized
+by a non-vanishing Chern number, are insulators that have a quantized Hall conductivity. We will
+do this for the theory of the integer quantum Hall eﬀect on a 2D lattice of Thouless, Kohmoto,
+Nightingale and den Nijs (TKNN) [179, 187] (see, also, Ref. [191]). Although this problem
+played a key role in the development of the theory of topological phases of matter, for many
+58
+
+years it was viewed as an academic problem since it would require gigantic magnetic ﬁelds to
+be in the regime of interest for typical solids. The situation changed with the development of
+twisted bilayer graphene and similar systems which have unit cells large enough for this physics
+to be observable. These new materials has allowed to study this problem experimentally.
+TKNN considered a system of free fermions on a planar (actually square) lattice in a uniform
+magnetic ﬁeld B with ﬂux 2πp/q per plaquette (in units in which the ﬂux quantum φ0 = eℏ/c = 1)
+with p and q two co-prime integers. The tight-binding Hamiltonian for this simple problem is
+H =
+�
+r, j=1,2
+t j c†(r) eiA j(r) c(r + e j) + h.c.
+(292)
+where c(r) and c†(r) are fermion creation and annihilation operators on the lattices sites {r =
+(m, n)}, e j are lattice unit vectors, and t j, with j = x, y, is a hopping matrix element between
+nearest neighbor sites of the square lattice along the x and y directions. On each link of the
+lattice we deﬁned a vector potential A j(r), where the oriented sum of the lattice vector potentials
+on each plaquette 2πφ, where φ = p
+q, is the ﬂux on each plaquette of the square lattice. In the
+axial (Landau) gauge A1(m, n) = 0, A2(m, n) = 2πmφ is a periodic function of m with period q.
+In this gauge the Hamiltonian becomes
+H =t
+�
+m,n
+�
+c†(m + 1, n)c(m, n) + c†(m, n)c(m + 1, n)
+�
++t
+�
+m,n
+�
+c†(m, n + 1) ei2πφm c(m, n) + c†(m, n) e−i2πφm c(m, n + 1)
+�
+(293)
+In this gauge the problem reduces to a system with a theory with a q × 1 unit cell with q inequiv-
+alent sites. In momentum space, the (magnetic) Brillouin zone of this system is − π
+q ≤ k1 ≤ π
+q and
+−π ≤ k2 ≤ π. The Fourier transform of the operators c†(r) and c(r) are, respectively, c†(k) and
+c(k) which obey the standard anticommutation relations, {c(k), c(q)} = {c†(k), c†(q)} = 0, and
+{c†(k), c(q)} = δ(k − q). In momentum space the Hamiltonian becomes
+H = q
+� π/q
+−π/q
+dkx
+2π
+� π
+−π
+dky
+2π
+�H(kx, ky)
+(294)
+where
+�H(kx, ky) =1
+q
+q−1
+�
+n=0
+�
+2tx cos(kx + 2πφn) c†(kx + 2πφn, ky) c(kx + 2πφn, ky)
++ ty
+�
+e−iky c†(kx + 2π(n + 1)φ, ky) c(kx + 2πnφ, ky) + eiky c†(kx + 2π(n − 1)φ, ky) c(kx + 2πnφ, ky)
+��
+(295)
+The spectrum of this system was ﬁrst investigated by Hofstadter [192] consists of q bands which
+for q an odd integer are separated by ﬁnite energy gaps.
+For ﬁxed values of kx and ky (in the magnetic Brillouin zone), the Hamiltonian �H(kx, ky) is the
+tight-binding model of a one-dimensional chain on the 1D lattice of q sites located at kx + 2πnφ
+with nearest neighbor hopping. The single-particle (Bloch) states un(k) of this 1D model obeys
+the Schr¨odinger Equation
+2tx cos(kx + 2πnφ)un(k) + ty
+�
+e−iky un−1(k) + eiky un+1(k)
+�
+= Enun
+(296)
+which is known as Harper’s equation. In general this equation does not admit an analytic solution.
+However, the nature qualitative features of the spectrum can be obtained by an expansion either
+on tx/ty or on ty/tx. TKNN used degenerate perturbation theory to show that the spectrum has q
+bands and that the r-th band is characterized by two integers sr and ℓr which are the solution of
+the Diophantine equation
+r = q sr + p ℓr
+(297)
+with ℓ0 = s0 = 0. It isl also obvious that for r = q sq = 1 and ℓq = 0 (for all p).
+Furthermore, TKNN showed that there was an, until then unsuspected, relation between the
+Hall conductivity σxy of a gapped system of electrons in periodic potentials in a uniform magnetic
+ﬁeld and the Berry connection of the ﬁlled bands. This relation implies the quantization of the
+59
+
+Hall conductivity and its computation in terms of a topological invariant, the Chern number of the
+occupied bands. They showed that in this context each band has a non-trivial Berry connection
+A(r)
+j = i⟨ur(k)|∂ j|ur(k⟩ (with ∂ j = ∂kj) of the form of Eq.(281) with a non-vanishing (ﬁrst) Chern
+number C(r)
+1 , i.e. the ﬂux through the magnetic Brillouin zone.
+The conductivity tensor characterizes the electrical properties of a physical system. Linear
+Response Theory provides a framework for computing the conductivity tensor by perturbing the
+system with a weak external electromagnetic ﬁeld Aµ and computing the currents that they induce
+[193, 10]. The expectation value of the gauge-invariant current operator Jµ(x) is
+⟨Jµ(x)⟩ = − i
+ℏ
+δ
+δAµ(x) ln Z[Aµ]
+(298)
+where Z[Aµ] is the partition function in the presence of a background (i.e. classical) electro-
+magnetic ﬁeld Aµ(x). In general spacetime dimension D = d + 1, the lowest order in the vector
+potential Aµ, the induced current is given in terms of the polarization tensor Πµν(x, y),
+ln Z[Aµ] = i
+2
+�
+dDx
+�
+dDy Aµ(x) Πµν(x, y) Aν(y) + O(A3)
+(299)
+Therefore, the induced current is related to the external ﬁeld by
+⟨Jµ(x)⟩ ≡ Jµ(x) =
+�
+dDy Πµν(x, y) Aν(y)
+(300)
+Expressions of this type are know as a Kubo formula. The polarization tensor can be regarded as
+a generalized susceptibility.
+Although we are using a continuum relativistic notation, these expressions are generally
+valid, even in lattice systems. Gauge invariance requires that the polarization tensor be con-
+served,
+∂µΠµν(x, y) = 0
+(301)
+In general, the (retarded) polarization tensor Πµν(x, y) is related to the the (retarded) current-
+current correlation function DR
+µν(x, y),
+Dµν(x, y) = − i
+ℏΘ(x0 − y0)⟨[Jµ(x), Jν(y)]⟩
+(302)
+by the identity
+ΠR
+µν(x, y) = DR
+µν(x, y) − iℏ
+�δJµ(x)
+δAν(y)
+�
+(303)
+The last term in Eq.(303), usually called a “contact term”. This term vanishes only for a theory
+of relativistic fermions (in the continuum). In all other cases, lattice or continuum, relativistic
+or not, the contact term does not vanish and its form depends on the speciﬁc theory. In non-
+relativistic systems, e.g. in a Fermi liquid, this term is the origin of the f-sum rule [193, 194].
+When the external ﬁeld Aµ represents is a (locally) uniform electric ﬁeld E, the induced
+current is Ji = σi jE j, where σi j is the conductivity tensor, which can be obtained from the
+polarization tensor as the limit
+σi j = lim
+ω→0
+1
+iω lim
+q→0 Πi j(ω, q)
+(304)
+In a metal, which has a Fermi surface, the order of limits in which ω and q vanish matters and
+only the order of limits speciﬁed above is the correct one to take. Ina Dirac systems, that we
+will discuss below, the order does not matter due to relativistic invariance. The other case in
+which the order does not matter is the Chern insulator that we are interested in. In general, in an
+isotropic system, the conductivity tensor has a symmetric part and an antisymmetric part. The
+symmetric part of the conductivity tensor yields the longitudinal conductivity which has all the
+eﬀects of dissipation. The antisymmetric part does not vanish in the presence of a magnetic ﬁeld
+or, more generally, if time reversal symmetry is broken, and yields the Hall conductivity.
+A Chern insulator is an insulator and as such is a state with an energy gap. In such a state the
+longitudinal conductivity vanishes since there is no dissipation in a gapped state. But in a Chern
+insulator time reversal invariance is broken. In the system that we are discussing is broken by
+60
+
+the magnetic ﬁeld. The Hall conductivity can be calculated from the antisymmetric part of the
+polarization tensor as the limit
+σxy = lim
+ω→0
+i
+ωΠxy(ω, 0)
+(305)
+In addition, the contact term does not contribute to the Hall conductivity. As a result the hall
+conductivity can be computed in terms of the antisymmetric component of the current-correlation
+function.
+let us now return to the theory of free fermions on a lattice in a (commensurate) magnetic
+ﬁeld takes. As we saw the electronic states are split into q bands with single particle states
+|ψn(k)⟩ where k takes values on the ﬁrst magnetic Brillouin zone. We will compute the Hall
+conductivity for this system assuming that the Fermi energy EF lies in the gap between the n-th
+and the n − 1-th bands. Let us label by α the occupied bands and by β the unoccupied bands.
+Hence Eα(k) < EF < Eβ(k). In this case, the Kubo formula for the Hall conductivity becomes
+σxy = ie2
+ℏ
+�
+Eα<EF<Eβ
+�
+BZ
+d2k
+(2π)2
+⟨uα(k)|Jy|uβ(k)⟩⟨uβ(k)|Jx|uα(k)⟩ − ⟨uα(k)|Jx|uβ(k)⟩⟨uβ(k)|Jy|uα(k)⟩
+(Eα(k) − Eβ(k))2
+(306)
+In the lattice model the current operator is simply
+J = e
+ℏ∂k �H(k)
+(307)
+Using this form of the current and the Schr¨odinger equation of Eq.(296) we can rewrite the Kubo
+formula for the Hall conductivity, Eq.(306), as a sum over only the occupied states labeled by α.
+The expression for the Hall conductivity now takes the more compact form
+σxy = e2
+ℏ
+�
+α
+�
+BZ
+d2k
+(2π)2ǫjl∂ j⟨uα(k)|∂k|uα(k)⟩ = e2
+h
+�
+α
+1
+2π
+�
+BZ
+F (α)(k) = e2
+h
+�
+α
+C(α)
+1
+(308)
+Therefore, we conclude that each occupied band contributes to the Hall conductivity an amount
+equal to its ﬁrst Chern number (multiplied by e2/h). This is the main result of TKNN [179].
+In the speciﬁc case of the model that we are discussing here the Chern number is given
+explicitly in terms of the solution of the Diophantine equation, Eq.(297)
+C(r)
+1 = 1
+2π
+�
+Γ
+dk jA(r)
+j (k) = ℓr − ℓr−1
+(309)
+where the integers ℓr and ℓr−1 are the solutions of the Diophantine equation for the bands r and
+r − 1 (A detailed derivation can be found in Ref.[9], chapter 12). The TKNN integers ℓr are
+topological invariants of the bands.
+We conclude, with TKNN, that for a system with r ﬁlled bands the Hall conductivity σ(r)
+xy is
+given by the expression
+σ(r)
+xy = e2
+h
+r�
+n=1
+(ℓr − ℓr−1) = e2
+h ℓr
+(310)
+thus showing that this system exhibits and integer quantum Hall eﬀect and that the Hall conduc-
+tivity is (up to the ratio of universal constants e2/h) given by a topological invariant, the ﬁrst
+Chern number of the band.
+We conclude with two comments. The ﬁrst is that when all the bands are occupied the
+Hall conductivity vanishes. This obvious physical constraint is automatically satisﬁed by the
+Diophantine equation. The other comment concerns the case of what happens when q is an even
+integer. In this case it is possible to show that there are band crossings. This is what happens for
+q = 2 which is the π ﬂux lattice. in this case one has a theory of two species of Dirac bi-spinors.
+As we will see in section 10.1.5 what happens depends on how the gapless state is reached.
+10.1.5. The Anomalous Quantum Hall Eﬀect
+We will now discuss a simple and insightful model of a Chern insulator that displays the
+quantum anomalous Hall eﬀect, that is a Hall eﬀect in a system with broken time reversal sym-
+metry in the absence of a uniform magnetic ﬁeld. In order to do that we will reconsider the
+graphene model with additional terms in the Hamiltonian of Eq.(280) that will open up energy
+gaps in its spectrum. There are several ways to do this. One way requires breaking the symme-
+try between the two sublattices A and B, by assigning them a site energy of ±M, which breaks
+61
+
+the mirror reﬂection symmetry of the hexagon. This is what happens with graphene on a boron
+nitrade substrate [178] (there are other ways that involve breaking the point group symmetry
+[195]). Another way to open a gap is to break time reversal symmetry which is accomplished by
+complex hopping amplitudes t2 exp(iφ) between nearest neighbor A sites (and between nearest
+neighbor B sites as well). The phase φ is oriented from rA to rA+ai, where a1 = d2−d3, etc. (and
+similarly for the B sites). Physically, the oriented sum of the phases φ on the triangles generated
+by the hopping terms t1 and t2 represent the magnetic ﬂuxes through each triangle such that the
+total ﬂux on the elementary hexagon is zero. This model was considered by Haldane [196]. With
+these changes the one-particle Hamiltonian becomes 2 × 2 Hermitian matrix
+H(k) = h0(k) 1 + h(k) · σ
+(311)
+where I is the 2 × 2 identity matrix and σ = (σ1, σ2, σ3) is a vector made of the three Pauli
+matrices. The functions h0(k) and h(k) = (h1(k), h2(k), h3(k) are, respectively, given by
+h0(k) =2t2 cos φ
+�
+i=1,2,3
+cos(k · ai),
+h1(k) = t1
+�
+i=1,2,3
+cos(k · di),
+h2(k) =t1
+�
+i=1,2,3
+sin(k · di),
+h3(k) = M − 2t2 sin φ
+�
+i=1,2,3
+sin(k · ai)
+(312)
+Provided |t2/t1| < 1/3 the spectrum of the Hamiltonian of Eq.(280) has two bands that do not
+overlap and have a ﬁnite energy gap unless they cross at the corners K and K′ the Brillouin zone
+where the gap is smallest. The two bands cross at K and/or K′ if M = ±3
+√
+3t2 sin φ (for K and K′,
+respectively). When either one of these conditions are not met the spectrum is gapped at either K
+or K′ or, more generally, at both. For values of the parameters for which the band gaps are small,
+the low energy states of the Hamiltonian of Eq.(280) is that of a theory of two Dirac ﬁelds, that
+we will label as ψ1(x) and ψ2(x), whose one-particle Hamiltonians have the form of Eq.(275) but
+with generally diﬀerent masses m1 ∝ M − 3
+√
+3t2 sin φ and m2 ∝ M + 3
+√
+3t2 sin φ. So, in general,
+both ﬁelds will have non-vanishing masses which can have the same sign or opposite sign.
+We conclude that the eﬀective low-energy theory consist of two species of massive Dirac
+bi-spinors, albeit generally with diﬀerent masses. To examine the electromagnetic response we
+will couple the lattice model and the eﬀective low energy theory to a slowly varying background
+electromagnetic ﬁeld Aµ. The Lagrangian of the low energy theory is
+L = ¯ψ1(i/∂ − m1)ψ1 + ¯ψ2(i/∂ − m2)ψ2 − eAµ( ¯ψ1γµψ1 + ¯ψ1γµψ1)
+(313)
+In general the two masses will be diﬀerent in magnitude and sign.
+Equivalently the Dirac
+fermions may have the same or opposite chirality. We will examine the electromagnetic re-
+sponse for both cases in section 10.1.6 where we will see that in the phase in which the masses of
+both Dirac fermions have the same sign this model describes a topological insulator that exhibits
+the anomalous quantum Hall eﬀect, whereas in the phase where the signs are opposite it is a
+conventional insulator.
+However, by varying the parameters we can tune to special values of these parameters where
+one of the two masses vanishes. In section 10.1.6 we will see that these special values of the
+parameters correspond to quantum phase transitions at which the topological properties of the
+ground state change. We will see that this is a quantum critical point with special properties.
+In particular, time reversal invariance is explicitly broken at this quantum critical point. In this
+regime the heavy Dirac fermion plays the role of the heavy regulating ﬁeld as in the Pauli-Villars
+regularization of the theory of a single Dirac fermion. In the lattice model this is a manifestation
+of the Nielsen-Ninomiya theorem which requires that there should be an even number of Dirac
+fermions [174, 175, 177]. In this context, the heavy fermion is the “doubler”.
+10.1.6. The Parity Anomaly
+We saw in section 7.3 that Dirac fermions in 1+1 dimensions have a chiral anomaly. That
+statement mean that although the theory of free massless Dirac fermions has a global chiral sym-
+metry, the associated current, which we called j5
+µ, is not conserved when the system is coupled
+to an external gauge ﬁeld. This eﬀect happened because the right and left moving (Weyl) com-
+ponents of the massless Dirac fermion cannot be separately conserved in the presence of a gauge
+ﬁeld.
+Analogs of the 1+1-dimensional chiral anomaly exist on 3+1 dimensions (and, more gener-
+ally, in all even spacetime dimensions). The cancelation of these anomalies is a key condition
+62
+
+for the consistency of gauge theories [96]. We will see in section 10 that these anomalies play a
+role on the theory of 3D topological insulators. More generally, anomalies play a central role in
+the theory of symmetry-protected topological (SPT) phases of matter [197, 198, 199].
+Dirac fermions in 2+1 dimensions do not have an analog of the chiral anomaly for the simple
+reason that the chiral symmetry does ot exist. There is, however, an anomaly involving parity
+or, equivalently, time reversal invariance, known as the parity anomaly. The parity anomaly was
+discovered in the computation of the eﬀective action of the electromagnetic ﬁeld in a theory of
+massive Dirac fermions in 2+1 dimensions by Redlich [200, 201].
+Consider theory on a single massive Dirac fermion (a bi-spinor) ψ(x) in 2+1 dimensions
+coupled to a background U(1) gauge ﬁeld Aµ. The Lagrangian is
+L[ψ, ¯ψ, Aµ] = ¯ψi /Dψ − m ¯ψψ
+(314)
+where the covariant derivative is Dµ = ∂µ + iAµ. The eﬀective action S eﬀ[Aµ] of the gauge ﬁeld
+Aµ is formally obtained by computing the path integral
+Z[Aµ] = exp(iS eﬀ[Aµ]) =
+�
+DψD ¯ψ exp(i
+�
+d3x L[ψ, ¯ψ, Aµ]) = Det(i /D[Aµ] − m)
+(315)
+where we used that the theory is free to write the partition function as a functional determinant. In
+1+1 dimensions, in the case of a massless theory, this determinant can be computed exactly using
+the heat kernel (or zeta-function) technique [202, 203]. Aside from important technicalities, the
+reason for why this works is that as a consequence of the chiral anomaly, the eﬀective action
+computed at one loop order in the fermions is exact in 1+1 dimensions. In higher dimensions
+this is no longer true.
+We will sketch the computation of the eﬀective action to one loop order, which means
+quadratic in the gauge ﬁeld Aµ, which has the form
+S eﬀ[Aµ] = 1
+2
+�
+d3x
+�
+d3y Aµ(x) Πµν(x − y) Aν(y) + . . .
+(316)
+where the ellipsis contains the higher order contributions. The polarization operator Πµν(x − y)
+in momentum space is (here pµ is the momentum transfer) by the one-loop result
+Πµν(p) =
+�
+d3q
+(2π)3 tr �S (q + p/2)γµS (q − p/2)γν�
+(317)
+The symbol tr signiﬁes the trace over the spinor labels and S (p) is the (massive) Feynman prop-
+agator
+S (p) =
+1
+/p − m + iǫ
+(318)
+The polarization tensor Πµν(p) has the explicit decomposition into a parity even and a parity odd
+contributions
+Πµν(p) = (p2gµν − pµpν) Π0(p2) − iǫµνλpλ ΠA(p2)
+(319)
+where p2 = p2
+0 − p2. The polarization tensor is manifestly transverse, pµΠµν(p) = 0 and, hence,
+the one-loop eﬀective action of Eq.(316) is gauge-invariant (as it should be). The parity even
+Π0(p2) and parity odd ΠA(p2) kernels are given by
+Π0(p2) = − |m|
+4πp2 +
+1
+8π
+�
+p2
+�4m2
+p2 + 1
+�
+sinh−1
+
+�
+p2
+�
+4m2 − p2
+
+(320)
+ΠA(p2) = −
+m
+2π
+�
+p2 sinh−1
+
+�
+p2
+�
+4m2 − p2
+
+(321)
+These two results are obtained in the limit in which the regulator scales are sent to inﬁnity leaving
+behind these ﬁnite results, which hold provided the Dirac mass m does not vanish.
+In the low energy regime, |p2| ≪ m2, the kernels take the asymptotic values
+Π0(p2) ≃
+1
+16π|m|
+ΠA(p2) ≃ − 1
+4πsgn(m)
+(322)
+This result is the leading contribution to the parity even term (the ﬁrst term) up to corrections
+of the order of p2/m2. However, the second term, which is parity odd, is the exact contribution
+63
+
+in the limit p → 0 for all non-vanishing values of the Dirac mass. In this regime, the eﬀective
+Lagrangian of the gauge ﬁeld Aµ takes the local form of the sum of a Maxwell term and a Chern-
+Simons term
+Leﬀ[Aµ] = −
+1
+4π|m|FµνFµν ± 1
+2
+1
+4π ǫµνλ Aµ∂νAλ
+(323)
+where Fµν = ∂νAν − ∂νAµ is the ﬁeld strength of the gauge ﬁeld. Clearly, since the Maxwell term
+has two derivatives while the Chern-Simons term has one derivative, in the long distance regime
+the Maxwell term is irrelevant (subdominant). Also, while the prefactor of the Chern-Simons
+term is a pure number, the prefactor of the Maxwell term is proportional to
+1
+|m| (as required by
+dimensional analysis).
+The second term in the Lagrangian of Eq.(323) is a Chern-Simons term for the background
+gauge ﬁeld Aµ [163, 164]. This term is gauge-invariant (up to boundary terms) but breaks parity
+and time reversal invariance. This phenomenon is called the parity anomaly [200], although
+Witten [199] has argued that it is better to call this phenomenon a time-reversal anomaly. The
+sign of the prefactor of the Chern-Simons term depends of the sign of the Dirac mass m: positive
+for m > 0 and negative for m < 0 or, alternatively for positive and negative chirality of the Dirac
+(bi-spinor) ﬁeld which ﬁxes the sign of the breaking of time reversal invariance.
+In addition to the sign, the prefactor of the Chern-Simons term is equal to 1
+2, and it is not
+an integer. In section 9.6 we showed that for a dynamical Chern-Simons gauge theory to be
+deﬁned consistently on a closed surface this coeﬃcient must be quantized and should take integer
+values. The fact that the coeﬃcient is half-quantized means that the classical global symmetry of
+a theory of a single Dirac bi-spinor cannot be gauged, which means that the gauge theory cannot
+be quantized. This is an example of an obstruction to the quantization of a gauge theory due to
+an anomaly [96].
+We will now use these results to compute the eﬀective low-energy action for the theories
+of lattice Dirac fermions of section 10.1.2. In those systems the low energy theory is that of
+two Dirac (bi-spinors) with diﬀerent masses m1 and m2. Since both Dirac ﬁelds are coupled
+(minimally) to the same background gauge ﬁeld Aµ, the eﬀective Lagrangian is just the sum of
+the contribution for each Dirac fermion
+Leﬀ[Aµ] = −
+1
+4π|m|eﬀ
+FµνFµν + 1
+2
+�sgn(m1) + sgn(m2)� 1
+4πǫµνλAµ∂νAλ
+(324)
+where |m|eﬀ is
+1
+|m|eﬀ
+=
+1
+|m1| +
+1
+|m2|
+(325)
+This result implies that, as anticipated in section 10.1.2, this theory has two phases: a parity even
+phase and a parity-odd phase. In the regime in which the signs of the two mass terms are equal
+and opposite, sgn(m1) = −sgn(m2), the coeﬃcient of the Chern-Simons term cancels and the
+low-energy eﬀective Lagrangian is a Maxwell term (generally with an eﬀective speed of light
+much smaller than that in vacuum): this phase is a conventional insulator.
+Conversely, in the phase in which the two masses have the same sign, sgn(m1) = sgn(m2),
+the coeﬃcient of the Chern-Simons terms does not cancel and it is given by ± 1
+4π, where the sign
+is the sign of both mass terms. This phase is also an insulator but one in which time-reversal
+invariance is broken. Moreover, in this phase the induced current jµ by the background ﬁeld Aµ
+in the long-distance limit is controlled by the Chern-Simons term and it is given by
+jµ = δLeﬀ
+δAµ(x) = ± e2
+2πℏǫµνλ∂νAλ
+(326)
+From this result we see that in the parity-broken anomalous quantum Hall phase the system has
+a correctly quantized and non-vanishing Hall conductivity
+σxy = ±e2
+h
+(327)
+Therefore, the phase with sgn(m1) = sgn(m2) displays the quantum anomalous Hall eﬀect with a
+Hall conductivity whose sign equals the signs of both masses. Notice that this is true regardless
+the magnitudes of the masses m1 and m2, and that only their signs matter. In section 10.1 we will
+identify this phase with a topological phase of matter known as the Chern Insulator.
+We showed that the Hall conductivity of the anomalous quantum Hall phase is, as expected,
+equal to e2/h using the eﬀective low energy Dirac theory. One may wonder if this approximation
+64
+
+may be missing some contributions to the Hall conductivity. Fortunately there is an alternative
+quite elegant way of computing the Hall conductivity as a property of the entire occupied band.
+This approach, originally introduced by Gregory Volovik [204] in the context of superﬂuid 3He-
+A, and adapted to the theory of the quantum anomalous Hall eﬀect by Viktor Yakovenko [205],
+by Maarten Golterman, Karl Jansen and David Kaplan for a theory of Wilson fermions in odd
+dimensional hypercubic lattices [206], and by Xiao-Liang Qi, Yong-Shi Wu and Shou-Cheng
+Zhang [207], involves the derivation of a topological invariant for a two-band model.
+As we saw above the Hall conductivity is obtained from the xy component of the polarization
+operator as the limit
+σxy = lim
+ω→0
+i
+ω lim
+q→0 Πxy(ω, q)
+(328)
+For a free fermion system Πxy(ω, 0) is given by the current correlator
+Πxy(ω, 0) =
+�
+BZ
+d2k
+(2π)2
+� ∞
+−∞
+dΩ
+2π tr
+�
+Jx(k)G(k, ω + Ω)Jy(k)G(k, Ω)
+�
+(329)
+where G(k, Ω) is the propagator for a two-band free fermion system. A generic two-band free
+fermion system has a one-particle Hamiltonian of the form given in Eq.(311). The (one-body)
+current operator for such a system is obtained as
+Jl(k) = ∂h0(k)
+∂kl
+1 + ∂ha(k)
+∂kl
+σa
+(330)
+The propagator G(k, ω) is the 2 × 2 matrix (in band indices)
+G(k, ω) =
+1
+ω1 − h(k) · σ + iǫ =
+P+(k)
+ω − E+(k) + iǫ +
+P−(k)
+ω − E−(k) + iǫ
+(331)
+where P±(k) are the operators that project onto the (empty) conduction band and the (ﬁlled)
+valence band whose energies are, respectively E±(k),
+P±(k) = 1
+2
+�
+1 ± ˆh(k) · σ
+�
+(332)
+where ˆh(k) is the unit vector deﬁned for every momentum k of the ﬁrst Brillouin zone
+ˆh(k) =
+h(k)
+||h(k)||
+(333)
+Upon performing the frequency integration and the band traces in the expression for Πxy(ω, 0) of
+Eq.(329), we ﬁnd that the Hall conductivity takes the form
+σxy = e2
+2ℏ
+�
+BZ
+d2k
+(2π)2 ǫabc
+∂ˆha(k)
+∂kx
+��ˆhb(k)
+∂ky
+ˆhc(k) (n+(k)) − n−(k))
+(334)
+where n±(k) are the Fermi functions (at zero temperature) for the two bands. Since
+E+(k) − E−(k) = 2||h(k)|| = 2
+�
+h2(k) > 0
+(335)
+there is a non-vanishing gap on the entire Brillouin zone between the occupied valence band,
+with n−(k) = 1, and the unoccupied conduction band, with n+(k) = 0. For the insulating state
+the Fermi energy lies inside this gap and the expression for the Hall conductivity reduces to the
+following
+σxy = − e2
+2ℏ
+�
+BZ
+d2k
+(2π)2 ǫabc
+∂ˆha(k)
+∂kx
+∂ˆhb(k)
+∂ky
+ˆhc(k)
+(336)
+In our discussion of quantum antiferromagnets in one space dimensions in section 5.2 we found
+that their eﬀective low-energy action contained a crucial topological term proportional to the
+integer-valued topological invariant Q (the winding number) of Eq.(103) which classiﬁes the
+smooths maps (homotopies) of the 2D surface (say a sphere S 2) onto the target space of a three-
+component unit vector ﬁeld which also a sphere S 2. These equivalence classes are represented
+by the notation Π2(S 2) ≃ Z. In the case at hand the unit vector ˆh(k) are points on a 2-sphere.
+Hence, ˆh(k) is a map of the ﬁrst Brillouin zone (which is a 2-torus) to the sphere S 2. Such maps
+65
+
+are also classiﬁed by the same integer-valued topological invariant deﬁned in Eq.(103). These
+results imply that the Hall conductivity of the two-band system is
+σxy = e2
+2πℏQ[ˆh]
+(337)
+In other words we have shown that the Hall conductivity is given in terms of a topological in-
+variant of the occupied band (in units of e2/h), the winding number Q. In the two-band model
+the topological invariant Q plays the same role as the Chern number does in the work of TKNN
+[179, 187] When Q � 0, the two-band system exhibits the quantized anomalous quantum Hall
+eﬀect. This si a property of the entire band of occupied states, and not just a consequence of the
+low energy approximation. This result implies that the low energy approximation captures all of
+the topology of the band. It also implies that in these lattice models the Berry curvature is highly
+concentrated near the points in momentum space where the two bands are close in energy.
+.
+We will now consider the problem of the quantum phase transition between the trivial and
+the Chern insulator. To address this problem we will tune the parameters of the lattice model
+discussed in section 10.1.2, the phase φ and the ratio of hopping amplitudes t2/t1, to the point at
+which the mass of one of the two species of Dirac fermions, say the fermion ψ1, is zero, m1 = 0,
+while keeping the fermion ψ2 massive, m2 � 0. In the low energy regime we have a massless
+fermion and a massive fermion. This point in parameter space is a quantum phase transition
+between a trivial insulator and a Chern insulator. In the low energy regime the fermion ψ1 is
+massless. A massless fermion is a scale-invariant system in the sense that the correlators of all
+its observables exhibit power law behavior (free ﬁeld in this case).
+We will now discuss brieﬂy the electromagnetic response of this system at the quantum phase
+transition where one fermion becomes massless. In particular, it is natural to ask if the coeﬃ-
+cient of the parity-odd (Chern-Simons) term non-vanishing at the quantum phase transition. To
+answer this question we will look at the behavior of the parity-even kernel Π0(p2) and the parity-
+odd kernel ΠA(p2) in the massless limit for the light fermion, m1 → 0. We ﬁnd that the total
+contribution of both the light fermion and of the heavy fermion to the polarization kernels is
+(assuming m2/m1 → ∞)
+lim
+m1→0 Π0(p2) =
+i
+16
+�
+p2 ,
+lim
+|m2|→∞ ΠA(p2) = − 1
+4πsgn(m2)
+(338)
+The important conclusion is that at this quantum critical point the parity-even kernel Π0(p2)
+is non-local and that the heavy regulator fermion (the “doubler”) yields the leading ﬁnite non-
+vanishing and local contribution to the parity-odd kernel ΠA(p2).
+We can now use these results to compute the conductivity tensor σi j at the quantum critical
+point where m1 → 0. Since the system is spatially isotropic the conductivity tensor has the form
+σi j =
+� σxx
+σxy
+−σxy
+σxx
+�
+(339)
+where we used that σyy = σxx since the system is isotropic. The longitudinal conductivity σxx
+and the Hall conductivity σxy are
+σxx = π
+8
+e2
+h ,
+σxy = ±1
+2
+e2
+h
+(340)
+in other words, at the quantum critical point the system has a ﬁnite (and universal) longitudinal
+conductivity. This result may seem surprising as there is no disorder in this model. A ﬁnite
+universal longitudinal conductivity is a standard occurrence in 2D systems at a quantum critical
+point. For example, at the superconductor-insulatortransition the conductivity is (conjectured) to
+be) σxx = e2/2h (with σxy = 0). In addition, it also has ﬁnite and also universal Hall conductivity.
+The Hall conductivity at the quantum critical point is due to the heavy fermionic “doubler” and is
+equal to 1/2 (in units of e2/h). So, the quantum critical point is not time-reversal invariant since
+this symmetry is broken at the UV (lattice).
+We can examine the massless theory using a more formal approach [199, 208]. In a gauge-
+invariant regularization of the massless theory the partition function of a Dirac fermion coupled to
+a background (unquantized) U(1) gauge ﬁeld, the partition function is not time-reversal invariant.
+66
+
+It is given by
+Z[Aµ] =
+�
+D ¯ψDψ exp
+�
+i
+�
+d3x ¯ψi /D[Aµ]ψ
+�
+= det(i /D[Aµ]) =
+����det(i /D[Aµ])
+���� exp
+�
+±iπ
+2η[Aµ]
+�
+(341)
+where the sign depends on how time-reversal invariance is broken by the choice of regularization.
+The quantity η[Aµ] that appears in the phase factor is the Atiyah-Patodi-Singer η-invariant [147]
+which we already encountered in the discussion of the fractionally charged solitons in section 8.3.
+With some caveats [199, 208], the phase factor of the partition function of Eq.(341) is commonly
+written in the form of a 1/2-quantized Chern-Simons term
+π
+2η[Aµ] ≡ ±1
+2
+�
+d3x 1
+4πǫµνλAµ∂νAλ
+(342)
+often denoted as a U(1)1/2 Chern-Simons term. This term plays the same role as the contribution
+of the heavy fermion doubler in the lattice theory.
+However, we can also wonder if there is a way to have a time-reversal invariant theory of a
+single massless Dirac fermion, m → 0. This case cannot be realized in a y 2D lattice model but,
+as will see, it can be realized on the surface states of a 3D time-reversal invariant Z2 topological
+insulator, which we will discuss in section 10.1.
+10.2. Three-dimensional Z2 topological insulators
+The classiﬁcation of topological insulators in terms of a Chern number is only possible in
+even space dimensions in systems with broken time reversal symmetry: in d = 2 the Berry
+connection is abelian and the topological invariant is the ﬁrst Chern number while in in d = 4 the
+Berry connection in a non-abelian SU(2) gauge ﬁeld and the topological invariant is the second
+Chern number [135], etc. We will now discuss the time-reversal invariant topological insulators
+[209, 190, 135] and, in particular, those that are invariant under inversion symmetry. Such states
+exist in both two and space dimensional insulator with strong spin-orbit coupling.
+10.2.1. Z2 Topological Invariants
+Let {|un(k)⟩} be the Bloch states. We will represent time reversal by the anti-unitary operator
+Θ that acts on the single particle (Bloch) states by complex conjugating the state and reverses the
+spin. For spin-1/2 fermions Θ = exp(iπσ2)K, where K is the complex conjugation operator. In
+this case Θ2 = −1. Let us assume that we have two occupied Bloch bands for each point k f the
+Brillouin zone. In this case the states form a rank-2 vector bundle over the torus of the Brillouin
+zone. In time reversal invariant systems the anti-unitary time reversal transformation T induces
+an involution in the Brillouin zone that identiﬁes the points k and −k. Time reversal the acts
+on the one-particle (Bloch) Hamiltonian as ΘH(k)Θ−1 = H(−k). The states |un(±k)⟩ are related
+by time reversal as |un(−k)⟩ = Θ|un(k)⟩ which implies that the bundle is real. The condition
+Θ2 = −1 implies that the bundle is real. In algebraic topology these bundles are classiﬁed by an
+integer (here the number of occupied bands) and a Z2 index that will allow us to classify these
+states.
+In a periodic lattice there exists a set of points Qi of the Brillouin zone with the property
+that they diﬀer by their images under the action of time reversal by a reciprocal lattice vector,
+−Qi = Qi + G. In d = 2 there four such points and d = 3 there are eight points, and are given
+by Qi = 1
+2
+�
+j n jb j, where n j = 0, 1, j = 1, 2 in d = 2 and j = 1, 2, 3 in d = 3. Here b j are
+the primitive lattice vectors. Kane and Mele [210] deﬁned the 2N × 2N antisymmetric matrix
+�m,n(k) = ⟨um(−k)|Θ|un(k)⟩. They showed that at each time reversal invariant point Qi one can
+deﬁne an index δi
+δi =
+�
+det[�(Qi)]
+Pf[�(Qi)]
+= ±1
+(343)
+where det[�] and Pf[�] are, respectively, the determinant and the Pfaﬃan of the matrix �, and
+det[�] = Pf[�]2. The sign of the quantities δi can be made unambiguous by requiring that the
+Bloch states be continuous. In addition, the quantities δi are gauge-dependent. However the
+products
+(−1)ν =
+4
+�
+i=1
+δi
+(344)
+67
+
+in d = 2, and
+(−1)ν0 =
+8
+�
+i=1
+δi,
+(−1)νk =
+�
+nk=1,nj�k=0,1
+δi(n1, n2, n3)
+(345)
+are gauge and are also topological invariant. The Z2-valued indices ν and ν0 are robust to disorder
+and are called strong topological indices. Furthermore, Fu and Kane showed that if ξn(Qi) = ±1
+are the parity eigenvalues of the occupied parity eigenstates, the quantities δi are given by δi =
+�N
+m=1 ξ2m(Qi). In d = 3 the index ν0 does not rely on the existence of inversion symmetry.
+In the case of a two-band model in d = 2 and in d = 3, the states are four-component spinors
+reﬂecting the two bands and the two spin components. In the context of these systems with
+strong spin-orbit coupling spin is actually the z-component of the atomic total angular momentum
+J of the electrons with energies close to the Fermi energy. In these systems the one-particle
+Hamiltonian H(k) is a 4 × 4 Hermitian matrix which can be expanded as a linear combination
+of Dirac matrices. A simple and very useful model of systems of this type is the Wilson fermion
+model (with continuous time) [55, 211, 135] of a square (cubic) lattice with 4 states per site
+(parity and spin) whose Hamiltonian three dimensions is
+H(k) = sin k · α + M(k) β
+(346)
+where α = γ0γ and β = γ0 are the conventional 4 × 4 Dirac matrices. The γ matrices satisfy
+the Cliﬀord algebra {γµ, γν} = 2gµν1, where gµν = diag(1, −1, −1, −1) is the metric tensor of
+four-dimensional Minkowski space time. An additional γ matrix of interest is γ5 = iγ0γ1γ2γ3.
+The (Wilson) mass term M(k) in two dimensions is M(k) = M + cos k1 + cos p2 − 2, and in
+three dimensions is M(k) = M + cos k1 + cos k2 + cos k3 − 3. In Eq.(346) we see that, consistent
+with the requirements of the Nielsen-Ninomiya theorem [174, 175], in addition to a possible low
+energy Dirac fermion (if M is small) there are three more massive Dirac fermions (in d = 2) and
+seven other Dirac fermions in d = 3, so that the total number of Dirac fermions is always even, 8
+in this case. With Wilson’s mass term the additional Dirac fermions (the “doublers”) are always
+heavy even if the Dirac fermion near the Γ point Q = 0 is light.
+In the Dirac basis time reversal is the operation Θ = (iσ2 ⊗ I)K, where K is complex con-
+jugation, and parity is P = β. The matrices α and β commute with PΘ. At the time-reversal
+and parity invariant points of the Brillouin zone {Qi} the Hamiltonian only depends of the the
+matrix β, H(Qi) = M(Qi)β. Since the parities of the spinors are the eigenvalues of the matrix
+β, we conclude that in the two-band models the quantities δi are simply equal to the sign of
+the mass of the fermions deﬁned at the time-reversal-invariant points Qi of the BZ [209, 135],
+δi = δ(Qi) = −sgn M(Qi). Using this result it follows that in two dimensions the system is a Z2
+topological insulator with index ν = 1 (mod 2) if 0 < M < 4 while it is trivial for other values
+of M, i.e. ν = 0 mod 2. Similarly, in three dimensions a Z2 (strong) topological insulator exists
+only if 0 < M < 2 (in the other regimes this system is in a weak topological insulator state or in
+a trivial one).
+We conclude that both in two and three dimensions the low energy theory of the Z2 topolog-
+ical insulators consists of a single Dirac fermion whose mass is small compared to that of the
+fermion doublers and has the opposite sign. In both cases there is a quantum phase transition
+between a trivial insulator and the time-reversal invariant topological insulator at the point where
+he mass of the light fermion vanishes.
+10.2.2. The Axial Anomaly and the Eﬀective Action
+We will now look at the electromagnetic response of a Z2 topological insulator in 3+1 dimen-
+sions. This problem can be addressed more easily using the continuum ﬁeld theory description
+which is valid in the regime where the mass M is weak. In this regime one species of Dirac
+fermions is light (i.e. its mass is small) while the Dirac doublers remain heavy. Much as we did
+in our discussion of the Chern insulators and the parity anomaly in section 10.1.6 we will keep
+in mind that the fermion doublers play the role of heavy regulators, such as in the Pauli-Villars
+scheme, in Quantum Field Theory. Here too, we will see that a ﬁeld-theoretic anomaly, known
+as the axial anomaly plays a central role in the physics. The analysis is very similar to what we
+did with the chiral anomaly in 1+1 dimensions in section 7.3.
+Let us begin with a theory of a single massive Dirac fermion in 3+1 dimensions. The La-
+grangian of the free massive Dirac theory is
+L = ¯ψ � i/∂ − m� ψ
+(347)
+68
+
+The equation of motion of the spinor ﬁeld operator ψ(x) (I omit the spinor indices here) is the
+Dirac equation
+�i/∂ − m� ψ = 0
+(348)
+The Dirac Lagrangian has a global gauge symmetry ψ → eiθψ which requires that the Dirac
+current jµ = ¯ψγµψ is locally conserved, ∂µ jµ = 0. The massless Dirac theory can be decomposed
+into a theory of two Weyl bi-spinor ﬁelds which obey separate Dirac Lagrangians. Furthermore,
+the massless theory has the additional global symmetry under the transformation ψ → eiθγ5ψ and
+the additional formally locally conserved axial current j5
+µ = i ¯ψγµγ5ψ. However the conservation
+law of the axial current is violated in the massive Dirac theory
+∂µ j5
+µ = −2mi ¯ψγ5ψ
+(349)
+since the two Weyl bi-spinors transmute into each other in the presence of a mass term. This is
+the origin of the phenomenon of neutrino oscillations. In the condensed matter physics context
+there is a similar phenomenon in systems of Weyl semimetals which have crossings between the
+valence and conducting bands at two locations ±Q of the BZ, with each crossing associated with
+each Weyl bi-spinor. A charge density wave with ordering wavevector 2Q mixes the two Weyl
+fermions which become gapped, becoming eﬀectively a single massive Dirac fermion [212].
+This state is often called an “axionic”-charge-density-wave.
+We will now show that the axial symmetry has an anomaly and cannot be gauged. Thus, we
+will consider the problem of a Dirac theory coupled to a background U(1) gauge ﬁeld Aµ and
+reexamine the putative conservation of the axial current j5
+µ. This question can be addressed in
+diﬀerent ways. Quite early on Steven Adler [133], and John Bell and Roman Jackiw [134] exam-
+ined this problem by computing a Dirac fermion triangle diagram for the process of a neutral pion
+decaying into two photons, π0 → 2γ. In particle physics the pion is the Goldstone boson of the
+spontaneously broken chiral symmetry. The analog of this problem in condensed matter physics
+is the phase mode of an incommensurate charge density wave. The relativistic Lagrangian for
+this problem is a theory of Dirac fermions coupled to a complex scalar ﬁeld φ = φ1 +iφ2 through
+two Yukawa couplings
+L = ¯ψi /Dψ + gφ1 ¯ψψ + igφ2 ¯ψγ5ψ − V(φ1, φ2) = ¯ψi /Dψ + g|φ| ¯ψeiγ5θψ − V(|φ|2)
+(350)
+where |φ|2 = φ2
+1 + φ2
+2, tan θ = φ2/φ1, and Dµ = ∂µ + ieAµ is the covariant derivative. Here we are
+regarding the gauge ﬁeld Aµ as a background probe ﬁeld.
+The triangle Feynman diagram computes the polarization tensor for the electromagnetic ﬁeld
+Aµ with an insertion of the coupling to the complex scalar ﬁeld in an otherwise massless theory.
+Assuming a gauge-invariant regularization of the diagram, this computation ﬁnds that the axial
+current j5
+µ is anomalous and is not conserved even in the massless theory [133, 134]
+∂µ j5
+µ = − e2
+16π2 FµνF∗
+µν
+(351)
+where F∗
+µν = 1
+2ǫµνλρFλρ is the dual of the electromagnetic ﬁeld tensor. This is the axial anomaly.
+To see how the axial anomaly arises we will follow the physically transparent approach of
+Nielsen and Ninomiya [176], that we also employed in section 7.3 in 1+1 dimensions. We will
+consider a theory of free massless dirac fermions coupled to a background electromagnetic ﬁeld.
+Since the theory is massless, the Dirac equation decouples into an equation to the right handed
+Weyl fermion ψR (with positive chirality γ5ψR = +ψR) and a left handed Weyl fermion ψL (with
+negative chirality, γ5ψL = −ψL). In the gauge A0 = 0, the Dirac equations become
+[i∂0 − (−i∂ − eA) · σ]ψR = 0,
+[i∂0 − (i∂ − eA) · σ]ψL = 0
+(352)
+Let us now consider look at the solutions of the Weyl equation for right-handed fermions ψR,
+Eq.(352). The left-handed fermions ψL are analyzed similarly. Wd will consider a gauge ﬁeld
+A1 = 0 and A2 = Bx1 representing a uniform static magnetic ﬁeld of strength B pointing along
+the x3 direction. In this this gauge the eigenstates are plane waves along the directions x2 and
+x3 and harmonic oscillator states along the direction x1. The eigenvalue spectrum consists of
+Landau levels with energies
+E(n, p3, σ3) = ±
+�
+2eB
+�
+n + 1
+2
+�
++ p2
+3 + eBσ3
+�1/2
+(353)
+69
+
+for n = 0, 1, 2, . . ., except for the zero mode with n = 0 and σ3 = −1, for which
+E(n = 0, p3, σ3 = −1) = ±p3
+(354)
+where the + sign holds for ψR and the − sign for ψL. Just as in the case of non-relativistic fermions
+the relativistic Landau levels are degenerate. We will consider a system of Dirac fermions at
+charge neutrality and, hence, EF = 0. The positive and negative energy states are charge conju-
+gate of each other and the negative energy states are ﬁlled. For right-handed fermions, the zero
+mode states with p3 < 0 are ﬁlled while for right handed states the zero modes with p3 > 0 are
+ﬁlled. The density of states of the zero modes is LeB/(4π2) (where L is the linear size of the
+system).
+Let us consider now turning on an external electric ﬁeld E parallel to the magnetic ﬁeld B.
+Just as we saw in 1+1 dimensions in section 7.3, the electric ﬁeld leads to pair creation by shifting
+the Fermi momentum to pF for the zero modes. There is no particle creation for the states with
+n � 0 and they do not contribute to the anomaly. The rate of creation of right-handed fermions
+NR,
+dNR
+dx0
+= 1
+L
+LeB
+4π2
+pF
+dx0
+= e2
+4π2 EB
+(355)
+The annihilation rate of left handed particles is
+dNL
+dx0
+= − 1
+L
+LeB
+4π2
+pF
+dx0
+= − e2
+4π2 EB
+(356)
+and the creation rate of left handed anti-particles is
+d ¯NL
+dx0
+= 1
+L
+LeB
+4π2
+pF
+dx0
+= e2
+4π2 EB
+(357)
+the axial anomaly is the total rate of creation of right handed particles and of left-handed antipar-
+ticles:
+dQ5
+dx0
+= dNR
+dx0
++ d ¯NL
+dx0
+= e2
+2π2 EB
+(358)
+which agrees with the expression of Eq.(351).
+We now turn to the Z2 topological insulator. As we saw this is a system with two species of
+(4 component) Dirac spinors. In the topological phase the sign of the Dirac mass term one of the
+Dirac fermions (the one near the Γ point in the lattice model) is opposite (negative) to the sign
+of the mass term of the of the other Dirac fermion (the fermion doubler). Explicit calculations
+[135, 213, 214] on the lattice model obtain the result that the eﬀective low-energy action for the
+electromagnetic gauge ﬁeld Aµ in the topological phase is
+S eﬀ[Aµ] =
+�
+d4x
+�
+− 1
+4e2 FµνFµν +
+θ
+32π2e2ǫµνλρFµνFλρ
+�
++ . . .
+(359)
+In a time-reversal invariant system the allowed values of the θ angle of Eq.(359) are restricted to
+be θ = nπ, with n ∈ Z. The case θ = 0 (mod 2π) represents a trivial insulator whereas θ = π (mod
+2π) holds for a Z2 time-reversal invariant topological insulator.
+The second term in the eﬀective action of the electromagnetic gauge ﬁeld of Eq.(359), known
+as the θ term, has been extensively discussed in the high-energyphysics literature [202, 215, 216].
+The derivation of this term is subtle. As it stands, unless θ is varying in space-time (in which case
+this is known a the axion ﬁeld) this term is a total derivative. In non-abelian Yang-Mills gauge
+theories this term is proportional to a topological invariant known as the Pontryagin index which
+counts the instanton number of the gauge ﬁeld conﬁgurations [99, 93]. In the context of the
+Lagrangian of Eq.(350) this term is induced by the coupling of the Dirac fermion to the Yukawa
+coupling of the complex scalar ﬁeld φ to the Dirac and γ5 mass terms. In this case, the lowest
+order contribution is given by the triangle diagram. The fact that this term is exact at lowest order
+reﬂects the fact that the axial anomaly is in fact a non-perturbative eﬀect which is the same in
+both the weak coupling and the strong coupling regimes [96].
+In the phase where the potential V(|φ|2) in the Lagrangian of Eq.(171) has a minimum at
+|φ0| exp(iθ) the chiral symmetry is spontaneously broken and the phase ﬁeld θ(x) is the Goldstone
+boson of the spontaneously broken chiral symmetry. In this phase the Dirac fermions become
+massive through the Yukawa couplings to the complex scalar ﬁeld, and the phase of the ﬁeld φ
+enters in the eﬀective action as an axion ﬁeld which couples to the gauge ﬁeld Aµ through the θ
+term of the eﬀective action. We should note that in a recently studied axionic CDW state of a
+Weyl semimetal [212] the phase of the CDW plays the role of the axion ﬁeld.
+70
+
+10.2.3. Theta terms, and Domain walls: Anomaly and the Callan-Harvey Eﬀect
+We will now discuss some remarkable behaviors of three-dimensional Z2 topological in-
+sulators. We will begin with the electromagnetic response encoded in the eﬀective action of
+Eq.(359). We will assume that the θ angle is eﬀectively a slowly varying Goldstone mode of the
+spontaneously broken U(1) chiral symmetry (i.e. the axion ﬁeld) present in the Lagrangian of
+Eq.(350). If the ﬁeld θ is constant, the θ term is a total derivative and it does not contribute to
+the local equations of motion. We will see below that this term [lays a key role in the physics of
+a domain wall, which we will regard as the interface of a Z2 topological insulator and a trivial
+insulator. In the general case in which θ varies slowly (as Goldstone modes do) its presence leads
+to interesting modiﬁcation of Maxwell’s equations, known as axion electrodynamics [215]:
+▽ · E =˜ρ − e2 ▽ θ · B,
+▽ × E = − ∂tB
+▽ × B =∂tE + ˜j + e2 (∂tθ + ▽θ × E) ,
+▽ · B =0
+(360)
+where ˜ρ and ˜j are external probe electric charge and current densities. The equations of axion
+electrodynamics have many remarkable properties. Here we will focus on eﬀects: the topological
+magnetoelectric eﬀect [135] and the Witten eﬀect [217].
+Let us consider a Z2 topological insulator with a ﬂat open boundary (the x1 − x2 plane)
+perpendicular to the direction x3. We will assume that the topological insulator lies at x3 < 0.
+This means that for x3 < 0, and far from the surface, θ(x3) → π, while in the trivial vacuum, and
+also far from the surface, θ(x3) → 0. We will assume that the change of θ from 0 to π occurs
+on a short distance ξ. We will call this conﬁguration an axion domain wall. In the region where
+θ(x3) is changing, |x3| ≲ ξ, an applied uniform magnnetic ﬁeld B induces a uniform electric
+ﬁeld E parallel to B whose magnitude is proportional to the change in θ. This is the topological
+magnetoelectric eﬀect [135].
+A similar striking eﬀect is obtained by considering the case of a magnetic monopole of mag-
+netic charge 2π/e (as required by Dirac quantization) inside a sphere of the trivial region of radius
+R, with θ = 0, surrounded by a region with θ � 0. The two regions are separated by a thin (axion)
+wall in which θ changes for 0 to π in a narrow shell of thickness ξ ≪ R. The equations of axion
+electrodynamics imply that the magnetic monopole induces an electric charge Qe on the surface
+of the sphere
+Qe = e∆θ
+2π
+(361)
+Thus, a magnetic monopole acquires an electric charge and becomes a dyon. This is the Wit-
+ten eﬀect [217]. In the particular case of a Z2 topological insulator, time reversal invariance
+requires that ∆θ = π, which implies that a monopole with unit magnetic charge has an elec-
+tric charge e/2. A similar argument implies that an external magnetic ﬁeld perpendicular to the
+open surface of the Z2 topological insulator (which is essentially an axion domain wall) induces
+an electric charge polarization on the surface proportional to the total magnetic ﬂux, which is
+another manifestation of the topological magnetoelectric eﬀect.
+The reader should readily recognize that the result of Eq.(361) for the electric charge induced
+my the magnetic monopole is the same as the Goldstone-Wilczek equation for the fractional
+charge for a one dimensional soliton of Eq.(215). We will now see that this is not just an analogy.
+We will follow here the general approach of Curtis Callan and Jeﬀrey Harvey [218] who extended
+the earlier work of Goldstone and Wilczek [141]. Let us consider the bulk of a Z2 topological
+insulator and assume that θ(x) is slowly varying. We can use the triangle diagram calculation to
+ﬁnd that an electromagnetic gauge ﬁeld Aµ induces a current ⟨Jµ(x) given by
+⟨Jµ(x)⟩ = −i
+e
+16π2 ǫµνλρ
+φ∗(x)∂νφ(x) − φ(x)∂νφ∗(x)
+|φ(x)|2
+Fλρ(x) =
+e
+8π2 ǫµνλρ∂νθ(x)Fλρ(x)
+(362)
+This result implies that, in the case of a domain wall in the x1 − x2 plane, a magnetic ﬁeld
+perpendicular to the wall induces a current towards the wall and, hence, a charge accumulation
+on the wall. Where does this charge come from? To understand this problem we will consider
+a Z2 topological insulator occupying a slab of macroscopic size L between a wall at x3 = 0 and
+a far way “anti-wall” at x3 = L. In this conﬁguration a magnetic ﬁeld normal to the wall(s)
+induces a transfer of charge from one wall to the other. Similarly, an electric ﬁeld parallel to the
+wall induces a current also parallel to the wall and perpendicular to the electric ﬁeld, i.e. a Hall
+current.
+In 1+1 dimensions we saw that soliton conﬁgurations acquire a fractional charge associated
+to states bound to the soliton (which are zero modes when ∆θ = π). We will see that the surfaces
+71
+
+of the 3D Z2 topological insulators also have zero modes which ar Weyl fermions propagating
+on the wall. To see how this works we will consider a 3+1 dimensional Dirac fermion with a
+Dirac mass that changes sign at x3 = 0. The Lagrangian now ill be
+L = ¯ψi/∂ψ + gφ(x) ¯ψψ
+(363)
+where φ(x) is now a real scalar ﬁeld that has the asymptotic behaviors φ(x3) = φ0 f(x3) such
+that limx3→±∞ f(x3) = ±1. We will assume that f(x3) is a monotonous function of x3, but its
+actual dependence on x3 is immaterial aside from the requirement that it should change sign at
+some point which we will take to be x3 = 0. This is a special case of Eq.(350) with φ1 = φ and
+φ2 = 0. We will now recognize that this just a Dirac fermion with a position-dependent Dirac
+mass m(x3) = gφ(x3). The one-particle Dirac Hamiltonian for this system is
+H = −iα · ▽ + m(x3)β
+(364)
+where α and β are the four 4 × 4 Dirac matrices. By symmetry, this Hamiltonian can be split into
+two Hamiltonians, H = Hwall + H⊥,
+Hwall = −iα1∂1 − iα2∂2,
+H⊥ = −iα3∂3 + m(x3)β
+(365)
+Let the spinor ψ± be and eigenstate of the anti-hermitian Dirac matrix γ3 = βα3 with eigenvalues
+±i, respectively
+γ3ψ± = ±iψ±
+(366)
+We seek a spinor solution ψ± of the Dirac equation Eq.(364) such that H⊥ψ± = 0
+± ∂3ψ± + m(x3)ψ± = 0
+(367)
+which is also a solution of
+(iγ0 − iγ1∂1 − iγ2∂2)ψ± = 0
+(368)
+In other words, it is a solution of the massless Dirac equation in 2+1 dimensions. The require-
+ment that ψ± be an eigenstate of γ3 reduces the number of spinor components from four to two.
+The full solution has the form
+ψ± = η±(x0, x1, x2)F±(x3),
+±∂3F±(x3) = −m(x3)F±(x3)
+(369)
+For a domain wall with limx3→∞ m(x3) = +m, the normalizable solution is F+(x3) is
+F+(x3) = F(0) exp
+�
+−
+� ∞
+0
+dx′
+3 m(x′
+3)
+�
+(370)
+where F(0) is a constant. For the anti-domain wall, for which limx3→∞ m(x3) = −m, the normal-
+izable solution is F−(x3).
+We conclude that there is a 2+1-dimensional massless Dirac theory that describes the quan-
+tum states bound to the wall which propagate along the wall. These states are the generalization
+of the fractionally charged mid-gap states of solitons in one dimension discussed in section 8.3.
+The energy of these states is E(p) = ±|p|, where p = (p1, p2) (where I set the Fermi velocity to
+unity). Experimental evidence for 2+1 dimensional massless Dirac fermions on the surface of the
+3D Z2 topological insulator Bi2Te3 has been found in spin-polarized angle-resolved photoemis-
+sion studies of the surface states which showed that they have a linear energy-momentumrelation
+(expected of massless Dirac fermions) as well as the spin-momentum locking characteristic of
+these spinor states [219].
+Having succeeded in showing that the surface of the Z2 time-reversal invariant 3D topological
+insulator has a two-component massless Dirac spinor we now want to determine its electromag-
+netic response. In fact we have already discussed this problem in our discussion of the parity
+anomaly in section 10.1.6 where we showed that the eﬀective action for the electromagnetic ﬁeld
+Aµ of Eq.(342) of a single massless Dirac bi-spinor is a Chern-Simons term with a prefactor
+which is 1/2 of the allowed value. In that purely 2+1-dimensional context we saw that time re-
+versal invariance is actually broken. However, the 3D problem is time-reversal invariant so there
+must be a contribution that cancels this time-reversal anomaly. The answer is that the requisite
+cancellation is supplied by the bulk.
+To see how this can happen we return to the bulk eﬀective action of the 3D topological
+insulator of Eq.(359) where we observed that the θ-term is a total derivative. Suppose that the
+72
+
+system has a boundary at x3 = 0 and that the topological insulator exists for x3 > 0 (where the
+fermion mass is negative, m < 0) and trivial for x3 < 0 (where m > 0). Only the region with
+m < 0 contributes to the θ-term. The region of four-dimensional space time occupied by the
+topological insulator is M and in this region the θ-term becomes
+S θ[A] =
+θ
+32π2
+�
+M
+d4x ǫµνλρFµνFρλ =
+θ
+8π2
+�
+M
+d4x ǫµνλρ∂µAν∂λAλ =
+θ
+8π2
+�
+∂M
+d3x ǫµνλAµ∂νAλ
+(371)
+where ∂M ≡ Σ × R is the boundary of the region M, Σ is the surface of the 3D topological
+insulator and R is time.
+Thus we see that the θ-term of the eﬀective action S θ[A] integrates to the boundary where it
+has the form of a 2+1-dimensional Chern-Simons term. Since for a time-reversal-invariant 3D
+topological insulator θ = π, we see that in this case the boundary Chern-Simons term becomes
+S θ=π[A] = 1
+8π
+�
+∂M
+d3x ǫµνλAµ∂νAλ
+(372)
+with a coeﬃcient which is 1/2 of the allowed value. This bulk contribution either cancels the
+parity anomaly of the boundary state, rendering the full system time-reversal invariant. In other
+words, in this system time-reversal symmetry is realized by cancellation of the anomaly between
+the bulk of the topological insulator and the boundary or, equivalently, by an inﬂow of the parity
+anomaly between the boundary and the bulk of the system. Another way to phrase this result is
+the statement that a single Dirac fermion cannot exist on its own in 2+1 dimensions but it can
+as the boundary state of a 3+1-dimensional system whose anomaly cancels the anomaly of the
+boundary.
+10.3. Chern-Simons Gauge Theory and The Fractional Quantum Hall Eﬀect
+We will now turn to the problem of the quantum Hall eﬀects. This is a problem that revealed
+the existence of profound and far reaching connections between condensed matter physics, quan-
+tum ﬁeld theory, conformal ﬁeld theory and topology. In particular, the fractional quantum hall
+eﬀect is the best studied and best understood topological phase of matter. As such, it has become
+the conceptual springboard for its manifold generalizations.
+The integer (IQHE) and fractional (FQHE) quantum Hall eﬀects are fascinating phenomena
+observed in ﬂuids of electrons in two dimensions in strong perpendicular magnetic ﬁelds. The
+integer quantum Hall eﬀect was discovered by Klaus von Klitzing in 1980 [220] in transport
+measurements of the longitudinal and Hall resistivity of the surface states of metal oxide ﬁeld-
+eﬀect transistors (MOSFET) in magnetic ﬁeld of up to 15 Tesla. The eﬀect that von Klitzing
+discovered (for which he was awarded the 1985 Nobel Prize in Physics) was that in the high
+ﬁeld regime the measured Hall conductivity showed a series of sharply deﬁned plateaus at which
+took the values σxy = ne2/h, where n is an integer, and the longitudinal conductivity appeared
+to vanish σxx → 0 as the the temperature was lowered down to T ≃ 1.5 K. Remarkably the
+measured value of the Hall conductivity was obtained with a precision of ∼ 10−9. To this date
+the measurement of the Hall conductivity in the IQHE yields the most precise deﬁnition of the
+ﬁne structure constant.
+Subsequent transport experiments in ultra-high-purity GaAs-AlAs heterostructures by Dan
+Tsui, Horst St¨ormer and Art Gossard found that, in addition to the IQHE, two-dimensional elec-
+tron ﬂuids in high magnetic ﬁelds exhibit the fractional quantum Hall eﬀect meaning that there
+are equally sharply-deﬁned plateaus of the Hall conductivity at the values σxy = p
+q
+e2
+h , where p
+and q are co-prime integers [221]. Much as in the IQHE case, in the FQHE the longitudinal
+conductivity vanishes at low temperatures. It is important to note that both in the IQHE and in
+the FQHE the observed temperature dependence in the highest purity samples of the longitudinal
+conductivity is activated, σxx ∼ exp(−W/T). The observed value of the energy scale W is the
+experimental estimate of an energy gap in the electron ﬂuid in the quantum Hall states.
+In addition to MOSFETS and GaAs-AlAs heterostructures both the IQHE and the FQHE
+have been seen in several other experimental platforms, particularly in graphene and other 2D
+materials [222, 223]. The explanation of these eﬀects and of a panoply of startling consequences
+that were uncovered in the course of understanding this phenomenon is the focus of this section.
+10.3.1. Landau levels and the Integer Hall eﬀect
+At some level the integer quantum Hall eﬀect can be explained by the Landau quantization of
+the energy levels of a free charged moving in two dimensions in a perpendicular magnetic ﬁeld
+73
+
+[224]. The Hamiltonian for a non-relativistic particle of charge −e and mass M in a perpendicular
+magnetic ﬁeld B is
+H =
+1
+2M
+�
+−iℏ ▽ +ie
+c A(x)
+�2
+(373)
+for a uniform perpendicular magnetic ﬁeld B = B ˆez = ▽ × A(x).
+In the circular gauge the vector potential is Ai = − 1
+2 Bǫi jx j. We will assume that the 2D plane
+has linear size L. The total magnetic ﬂux is Φ = BL2 and we will assume that there is an integer
+number Nφ of magnetic ﬂux quanta piercing the plane, φ = Nφφ0, where φ0 = hc/e is the ﬂux
+quantum. In units such that ℏ = e = c = 1 the ﬂux quantum is φ0 = 2π and Φ = 2πNφ.
+In the presence of a magnetic ﬁeld the components of the canonical momentum operator
+p = −iℏ ▽ − e
+c A do not commute with each other,
+[pi, p j] − ieℏ
+c Bǫi j
+(374)
+This means that translations in two directions do not commute with each other. However, the
+components of the operator k = p(−B) commute with p (and hence with the one-particle Hamil-
+tonian) but do not commute with each other: the commutator is the same as in Eq. (374). Since
+[k, H] = 0 they act as symmetry generators of the group of magnetic translations. For arbitrary
+displacements a and b the translation operators t(a) = exp(ia· k/ℏ) (and similarly with b) satsify
+t(a)t(b) = exp(ia × b · ez/ℓ2
+0)t(b)t(a)
+(375)
+Magnetic translations only commute with each other is the area subtended by a and b contains
+an integer number of magnetic ﬂux quanta.
+Given the rotational symmetry of the circular gauge it is natural to work in complex coordi-
+nates z = x1 + ix2. We will also use the notation ∂z = (∂1 − i∂2)/2 and ∂¯z = (∂1 + i∂2)/2. In this
+gauge, up to a normalization, the eigenstate wave functions have the form
+ψ(z, ¯z) = f(z, ¯z) exp
+− |z|2
+4ℓ2
+0
+
+(376)
+where ℓ0 =
+�
+ℏc
+e|B| is the magnetic length. In this gauge (and in complex coordinates) the angular
+momentum operator Lz = −iℏ(x1∂2 − x2∂1) = ℏ(z∂z − ¯z∂¯z).
+Any analytic function f(z) is an eigenstate with energy E0 =
+1
+2ℏωc. A complete basis of
+analytic functions are the monomials fn(z) = zn and have energy E0 and angular momentum
+Lz = nℏ. This is the lowest Landau level whose wave functions are ψn(z) = zn exp(−|z|2/4ℓ2
+0). On
+the other hand, an anti-analytic function fN = ¯zN is an eigenstate of energy EN = ℏωc
+�
+N + 1
+2
+�
+,
+where ωc =
+e|B|
+Mc is the cyclotron frequency, and angular momentum Lz = −Nℏ. States with
+angular momentum nℏ have the same energy and the degeneracy is equal to the number of ﬂux
+quanta Nφ. For the most part we will be interested in the states in the lowest Landau level.
+In the absence of disorder the Landau levels have an extensive degeneracy equal to the num-
+ber of ﬂux quantum Nφ. If we consider a system of N electrons in a Landau level the natural
+measure of density is not the areal density ρ = Ne
+L2 but the fraction ν = Ne
+Nφ the states in the Landau
+level which are occupied by electrons. The many-body state in which all Nφ states of the lowest
+Landau level has ﬁlling fraction ν = 1. The wave function for this state is the Slater determinant
+of the Landau states in the m = 0 level. After some simple algebra the wave function of this state
+is found to be
+Ψν=1(z1, . . . , zNe) =
+�
+i< j
+(zi − zj) exp
+−
+Ne
+�
+i=1
+|zi|2
+4ℓ2
+0
+
+(377)
+This is the ground state of the non-interacting system and it is non-degenerate. It is easy to
+see that this state has a ﬁnite energy gap. Indeed, in the Hilbert space with a ﬁxed number
+of electrons, the lowest energy excitation is a particle-hole pair in which the particle is in an
+unoccupied state of the ﬁrst excited Landau level, with m = 1 and energy E1 = 3
+2ℏωc and a hole
+a single-particle state in the lowest Landau level, with energy E0 = ℏωc. The excitation energy of
+the electron-hole pair is just ℏωc which is ﬁnite. However, in the free particle system this excited
+states has a huge degeneracy as the particle and the hole can be in any single particle state of the
+Landau level.
+74
+
+At a very naive level one can estimate the Hall conductivity for a translationally invariant
+system ﬁlling up n Landau level by the following simple argument. If n Landau levels are ﬁlled,
+the number of electrons N = nNφ and the total charge is then Q = eN = e n Nφ = e n BL2/φ0 =
+ne2BL2/hc. If a weak and uniform in-plane electric ﬁeld E is applied in the presence of a per-
+pendicular magnetic ﬁeld B = B ez, then the entire system (its center of mass) moves at the drift
+velocity � such that � × B = −Ec. Thus, there is a Hall current J = Q�. The current density
+is j = J/L2 = Q�/L2. Hence, ji =
+Qc
+BL2 ǫi j E j, which implies that the Hall conductivity is
+σxy = Qc/BL2 = ne2/h.
+While the above argument is formally correct and yields the correct value of the Hall conduc-
+tivity in this highly idealized setting, at a very basic level it is faulty. To begin with, it assumes
+exact translational invariance and, hence, Galilean symmetry, which are not obeyed in any realis-
+tic system. The second objection is that, even in this idealized system, as the number of electrons
+is varied, the chemical potential (and hence the Fermi energy) jumps discontinuously from one
+Landau level to the next resulting in a linearly increasing Hall conductivity (as predicted classi-
+cally) without any of the observed plateaus. Furthermore, this argument does not explain which
+the Hall conductivity is so precisely quantized whereas, a priori, one would expect that being a
+transport coeﬃcient the Hall conductivity would depend on lots of complicated materials details.
+But, it it turns out that it does not!
+Let us ﬁrst discuss the observed universality of the Hall conductivity, namely its robustness
+and independence of microscopic details. In a remarkable paper Qian Niu, David Thouless and
+Yong-Shi Wu showed that, provided the ground state is separated by a ﬁnite energy gap from
+the excited states, the Hall conductivity is actually a topological invariant [225]. The argument
+has a similarity with what we discussed in the case of the Hall eﬀect on a 2D lattice (in section
+10.1.4) but in this more general system one considers the full many-body wave function, rather
+than the one-particle states. To show that this true they considered a system of N electrons
+with periodic boundary conditions, i.e. on a two-dimensional torus, T 2 = S 1 × S 1, with Nφ
+ﬂux quanta going through the torus. The torus is a rectangle of dimensions L1 and L2 with
+opposite ends identiﬁed. This is necessary to allow for a Hall current to be globally allowed. The
+electromagnetic gauge ﬁeld for a system ona torus with uniform non-vanishing ﬂux cannot obey
+periodic boundary conditions but rather accross the torus the gauge ﬁelds must diﬀer by (large)
+gauge transformations
+A1(x1, x2 +L2) = A1(x1, x2)+∂1β2(x1, x2),
+A2(x1 +L1, x2) = A2(x1, x2)+∂2β1(x1, x2) (378)
+while the wave functions themselves obey twisted boundary conditions
+Ψ([x(j) + L1e1]) = exp
+−i e
+ℏc
+Ne
+�
+j=1
+β1([x(j)]) + iθ1
+ × Ψ([x(j)])
+Ψ([x(j) + L2e2]) = exp
+−i e
+ℏc
+Ne
+�
+j=1
+β2([x(j)]) + iθ2
+ × Ψ([x(j)])
+(379)
+where e j (with j = 1, 2,) are two unit vectors on the torus, and θ1 and θ2 are two angles that twist
+the boundary conditions of the wave functions.
+In order to have a current on the torus they assumed that, in addition to the uniform ﬂux going
+through the torus, there a weak uniform electric ﬁeld E on the torus represented by the constant
+in space vector potential δA = E t = ▽[U(x)t] whose circulation on the two non-contractible
+circles Γ1 and Γ2, which wrap around the directions x1 and x2 of the torus T 2, are
+I j =
+�
+Γj
+δA · dx = t
+�
+Γj
+E · dx = itE jL j
+(380)
+Line integrals of a gauge ﬁeld on non-contractible loops in space (or space-time) are called
+holonomies. By inspection we see that the angles θ1 and θ2 are given by
+θj = e
+ℏcI j
+(381)
+Alternatively, the angles θ = (θ1, θ2) can be interpreted as magnetic ﬂuxes (in units of the ﬂux
+quantum φ0) through the two non-contractible circles of the torus T 2.
+The angles θ are deﬁned mod 2π since they are phase factors that twist the phase of the
+wave functions, and deﬁne a 2-torus of boundary conditions. By comparing with what we did in
+75
+
+section 10.1.4 in the case of the single-particle wave functions in the lattice model, we see that
+the many-body wave function Ψθ and the twist angles θ is the same as the relation between the
+phase of the Hofstadter wave functions with the momentum k of the magnetic BZ (which is also
+a 2-torus). Indeed, as it is always the case, while the phase of (in this case) the many-body wave
+function Ψθ is arbitrary, the changes of this phase as the the twist angle θ is changed are not. To
+quantify this dependence, for a given many-body state Ψ(α)
+θ , where α labels the state, we deﬁne a
+Berry connection A(α)(θ) on the torus of boundary conditions θ
+A(α)(θ) = i
+�
+Ψ(α)
+θ
+����∂θ
+����Ψ(α)
+θ
+�
+(382)
+Under a redeﬁnition of the phase of the state
+Ψ(α)
+θ
+→ exp(if(θ)) Ψ(α)
+θ
+(383)
+the Berry connection A(α)(θ) changes by a gauge transformation
+A(α)(θ) → A(α)(θ) − ∂θ f(θ)
+(384)
+Niu, Thouless and Wu [225] showed that the expression of the Kubo formula for the Hall
+conductivity of the state Ψ(α)
+θ , averaged over the boundary conditions θ, is given in terms of the
+ﬂux of the Berry connection A(α)(θ) through the 2-torus of boundary conditions, by the gauge-
+invariant quantity
+⟨(σxy)α⟩θ = e2
+ℏ
+� 2π
+0
+dθ1
+2π
+� 2π
+0
+dθ2
+2π
+�
+∂1A(α)
+2
+− ∂2A(α)
+1
+�
+= e2
+h
+1
+2π
+�
+γ
+A(α)(θ) · dθ
+(385)
+where γ is the square contour with corners at (0, 0), (2π, 0), (0, 2π) and (2π, 2π), that deﬁnes the
+2-torus of boundary conditions θ. This result is known as the Niu-Thouless-Wu formula.
+Eq.(385) implies that for the Hall conductivity to be non-vanishing the Berry connection
+A(α)(θ) must have a non-vanishing ﬂux through the torus of boundary conditions. For this to be
+true, just as we did in section 10.1.3, we must consider large gauge transformations of the form
+A1(θ + 2πe2) =A1(θ) + ∂1 f2(θ)
+A2(θ + 2πe1) =A2(θ) + ∂2 f1(θ)
+Ψ(α)({x(j)}; θ + 2π e1) = exp(if1(θ)) Ψ(α)({x(j)}; θ)
+Ψ(α)({x(j)}; θ + 2π e2) = exp(if2(θ)) Ψ(α)({x(j)}; θ)
+(386)
+where e1 and e2 are two orthogonal unit vectors on the torus of boundary conditions. We can
+now repeat the analysis we did in section 10.1.3 which, in this context, implies that the wave
+function Ψ(α)
+θ
+cannot be globally well deﬁned on the torus of boundary conditions, where it must
+have zeros, and where it should be deﬁned in patches. The end result is that the wave functions
+labeled by α are classiﬁed by a topological invariant, the ﬁrst-valued Chern number C(α)
+1 , which
+here it is given by
+C(α)
+1
+= 1
+2π
+�
+γ
+A(α)(θ) · dθ
+(387)
+and, hence, that the Hall conductivity in he state α (averaged over the twisted boundary condi-
+tions) exhibits the integer quantum Hall eﬀect,
+⟨(σxy)(α)⟩θ = C1
+e2
+h
+(388)
+Expressing the Hal conductivity in terms of the ﬁrst Chern number, which is a topological invari-
+ant, proves that the value of the conductivity cannot be changed by local physics eﬀects, such
+as disorder, etc. The topological nature of the Hall conductivity is the reason for the robustness
+and high precision of the measured value of the Hall conductivity. In subsequent work Niu and
+Thouless showed that, provided the state has a ﬁnite energy gap to all excitations, in the thermo-
+dynamic limit the Hall conductivity averaged over boundary conditions and for a given boundary
+condition are equal.
+The result of Eq.(388) seemingly implies that there can only be an integer quantum Hall
+eﬀect which, as we see shortly, it is not the case: there is a fractional quantum Hall eﬀect in
+76
+
+which the Hall conductivity is a fraction, σxy = p
+q
+e2
+h , where p and q are co-prime integers. This
+value of the Hall conductivity is also found experimentally to be obeyed with the same precision
+as in the integer quantum Hall eﬀect. In other words, the Hall conductivity in the fractional
+case must also have a topological character. To understand why this can be the case we need an
+implicit assumption that we made in our derivation. In fact, in deriving the result of Eq.(388) we
+assumed (implicitly) that for each value of θ there is only one many body state Ψ(α)
+θ
+(up to gauge
+transformations). As we will see below, in the fractional quantum Hall eﬀect on a torus (and, in
+fact on any closed surface except the sphere) the ground state is degenerate in the thermodynamic
+limit. We will also see that traversing the torus of boundary conditions once maps one degenerate
+state to another one. In general, we will ﬁnd that on the 2-torus in the thermodynamic limit there
+are m ∈ Z exactly degenerate states. In this case, one returns to the original state after sweeping
+m times the torus of boundary conditions. We will also see below that the degenerate states are
+labeled by quantum numbers related to the anyons that they support.
+While the topological argument proves the robustness (and universality) of the value of the
+Hall conductivity, it does not provide an insight for why there are plateaus in its magnetic ﬁeld
+dependence. If for some value of the ﬁlling fraction ν = N/Nφ the electron ﬂuid is exactly at a
+ground state Ψ(α) upon changing either the number of particles N or the magnetic ﬁeld B (but not
+both) the ﬂuid will now will be at the state Ψ(α) plus or minus some number of electrons. The
+existence of the plateau in σxy can be understood if the extra electrons (or holes) do not contribute
+to the Hall conductivity. This happens in the presence of disorder as these extra particles will
+be localized by the disorder and localized states, whose wave functions decay exponentially are
+insensitive to boundary conditions and, hence, these states do not contribute to the conductivity.
+This argument was ﬁrst put forth by Laughlin [226] who build on the result that in a disordered
+system in two dimensions all states are localized [63] but made the key assumption that the
+state at the center of the Landau level (broadened by disorder) is extended and contributes to
+the conductivity if this state is ﬁlled. This picture implies that this is actually a quantum phase
+transition from an insulator (dubbed a Hall insulator) to a state with a quantized Hall conductivity.
+This intuitive and appealing proposal has been the focus of research for many years and
+it is largely an unsolved problem. There is in fact a proposed ﬁeld theory for this quantum
+phase transition based on a a non-linear sigma model (in replica space) with a topological θ-term
+[227, 228], analogous to the theory we discussed in section 5.2 for quantum antiferromagnets in
+1+1 dimensions. in this proposal the (Boltzmann) longitudinal conductivity σxx plays the role of
+the inverse of the coupling constant of the non-linear sigma model while σxy plays the role of the
+coeﬃcient of the θ-term. Although the RG ﬂows suggested by this approach qualitatively agree
+with currently existing experimental data, and actual derivation is still wanting.
+However, aside from the fact that is is still an unsolved problem, there is the problem that this
+explanation ignores the fact that in a Landau level interactions are dominant and are the key to
+the understanding of the fractional case even in the same samples! A symptom of this problem
+is that, at least in the cleaner samples, as we noted above the longitudinal conductivity vanishes
+exponentially with temperature. This dependence means that there is a clean energy gap (and not
+just a mobility gap) which suggests that the state with additional excitations may have some sort
+of at least local order which would provide for a local energy gap. An actual theory based on this
+picture has yet to be developed.
+10.3.2. The Laughlin Wave Function
+We will turn now to the fractional quantum Hall eﬀect (FQHE). The FQHE is observed in
+fractionally ﬁlled Landau levels. These states have fractional ﬁlling fractions, namely that ν = Ne
+Nφ
+is a fraction. Most of the observed states are in the lowest Landau level, with N = 0, but a few
+states with very intriguing properties are seen in the ﬁrst excited Landau Level with N = 1.
+Let us consider for now the states in the lowest Landau level, N = 0. We will assume that
+there is no disorder and, for now, we will ignore the eﬀects of boundaries of the sample. Physi-
+cally, the important interaction here is the Coulomb interaction although, in some samples with
+screening layers, short-range interactions may be important as well. Since there is a large mag-
+netic ﬁeld, the Zeeman interaction is typically a large energy scale and the electros are expected
+to be fully polarized. However, there are situations under which the gyromagnetic factor can be
+made small (and even zero) and spin unpolarized states have to be considered. In some platforms,
+such as graphene, additional quantum numbers are present, such as “ﬂavor” associated with the
+diﬀerent Dirac states. In our discussion we will not cover these richer possibilities.
+Under these circumstances we have a problem in which all Nφ available one-particle states are
+77
+
+exactly degenerate (the Landau level is “ﬂat”) but only a fraction of them are ﬁlled. In this limit,
+this system as as strongly interaction as it gets. Not surprisingly, in a system of this type time-
+honored standard approximations fail, such as Hartree-Fock. Given the macroscopic degeneracy
+and the nature of the states in a magnetic ﬁeld, a system of this type may harbor many diﬀerent
+types of actual ground states depending of the types of interactions that are considered.
+Tsui, St¨ormer and Gossard [221] did transport experiments in the two-dimensional electron
+gas (2DEG) in high-purity GaAs-AlAs heterostructures and reported the discovery of a state with
+a highly precise value of the Hall conductivity of σxy = 1
+3
+e2
+h . Such a state cannot be understood
+in any obvious way in terms of the integer quantum Hall eﬀect. The observation of a clean gap
+in the low temperature longitudinal resistivity σxx → 0 as T → 0 implied that the 2DEG is
+incompressible and quite likely uniform.
+The ﬁrst breakthrough was Laughlin’s unique insight which led to his explanation of the
+experiments in terms of a novel quantum liquid of fully spin polarized electrons at ﬁlling fractions
+ν = 1
+m (with m and odd integer) whose wave function he proposed to be
+Ψm(z1, . . ., zN) =
+�
+i< j
+(zi − zj)m × exp
+−
+Ne
+�
+i=1
+|zi|2
+4ℓ2
+0
+
+(389)
+where N = Ne is the number of electrons, {zi} are their (complex) coordinates and m is an odd
+integer (which makes the wave function antisymmetric, as required by Fermi statistics).
+Laughlin extracted the physics encoded in this wave function by considering the probability
+����Ψm(z1, . . ., zN)
+����
+2
+of ﬁnding the N electrons in the locations {zi} (with i = 1, . . ., N). The norm of
+the Laughlin wave function is
+����
+����Ψm
+����
+����
+2
+=
+�
+d2z1 . . . d2zN
+����Ψm(z1, . . ., zN)
+����
+2
+(390)
+The Laughlin wave function Ψm(z1, . . ., zN) is an eigenstate of angular momentum with eigen-
+value Lm = 1
+2mN(N − 1). This remarkable wave function has a large overlap (∼ 98%) with the
+exact wave function with V(R) = e2
+εR (Coulomb) interactions in systems of up to eight particles
+(ε is the dielectric constant).
+The norm of the state can ten be interpreted as the classical partition function of a system
+of a system of N particles at the locations {zi} at inverse temperature β = m with the eﬀective
+interaction
+U(z1, . . ., zN) = −2
+�
+1≤i< j≤N
+ln |z1 − zj| +
+1
+2mℓ2
+0
+N
+�
+j=1
+|zj|2
+(391)
+This classical partition function is a one-component plasma, of a gas of particles with unit (neg-
+ative) charge interacting with each other with the repulsive 2D classical (logarithmic!) Coulomb
+interaction, VCG = − ln |zi − zj| (not 1/R!). The last term represents the contribution of a neu-
+tralizing background positive charge (which here it originates in the gaussian factor of the wave
+function). In other words, this is a one-component plasma. We can check that this is correct
+since ▽2
+�
+|z|2
+2mℓ2
+0
+�
+=
+2
+mℓ2
+0 which corresponds to a uniform (areal) charge density ρ0 =
+1
+2πmℓ2
+0 . Clas-
+sical Monte Carlo simulations of this probability distribution showed that this wave function
+describes an incompressible ﬂuid state if m ≲ 7, while for larger values it represents a crystalline
+state. This approach is known as the plasma analogy.
+10.3.3. Quasiholes have fractional charge
+Laughlin considered a vortex-like excitation in the ﬂuid created by the adiabatic insertion of
+an inﬁnitesimally thin solenoid at some coordinate z0 carrying a magnetic ﬂux Φ, a ﬂuxoid. In
+the presence of this solenoid the single particle state zn exp(|z|2/4ℓ2
+0) acquires a branch cut and
+becomes the state zn+α exp(|z|2/4ℓ2
+0) where α = Φ/φ0, with φ0 = hc/e being the ﬂux quantum.
+This analytic structure is the Aharonov-Bohm eﬀect [155].
+However, if the solenoid carries just one ﬂux quantum α = 1 and the net eﬀect is that the state
+becomes zn+1 exp(−|z|2/4ℓ2
+0) which has one more unit of angular momentum. For these reasons,
+in the presence of a solenoid with one ﬂux quantum inserted at z0, the Laughlin wave function
+for the 2DEG in a large magnetic ﬁeld now becomes
+Ψqh
+m (z0; z1, . . ., zN) =
+N
+�
+i=1
+(zi − z0) × Ψm(z1, . . . , zN)
+(392)
+78
+
+A calculation using the plasma analogy reveals that the net eﬀect of the solenoid is to expel a
+charge equal to −e/m from the vicinity of the location of the solenoid or, equivalently, to create a
+fractionally charged quasihole at z0 with charge Qqh = +e/m. The charge distribution is uniform
+at long distances but is depleted (hence a quasihole) on a length scale ξ. One can check that this
+state costs a ﬁnite energy ε0 which depends on the particular interaction.
+On the other hand, if we consider a system in which the solenoid is inserted adiabatically
+so that after a long time t there is a quasihole at z0, an amount of charge equal to −e/m will
+ﬂow radially outwards to the outer edge of the 2DEG. The adiabatic insertion of the solenoid
+generates a azimuthal electric ﬁeld (an e.m.f.) and a radial Hall current, and a Hall conductivity
+σxy = 1
+m
+e2
+h . Hence, the Laughlin wave function exhibits the fractional quantum Hall eﬀect and
+its excitations are vortices with fractional charge e/m.
+It is important to note the non-local nature of the Laughlin quasihole. The non-locality is
+inherited from the ﬂuxoid (the solenoid) inserted at z0: even though the magnetic ﬁeld of the
+ﬂuxoid vanishes away from it, B(x) = Φ δ2(x − x0), its vector potential A(x) does not since its
+circulation on any non-contractible loop γ that contains the location of the ﬂuxoid inside must
+be equal to the ﬂux Φ carried by the ﬂuxoid. Although in principle one can choose a singular
+gauge in which A(x) vanishes locally, this cannot be true everywhere as it must leave behind a
+Dirac-like string on a curve Γ ranging form the location z0 of the ﬂuxoid to the boundary of the
+system with A taking a singular value on Γ. This feature has the same form as the 2D vortex that
+we discussed in section 5.1.1. Kivelson and Ro˘cek [229] showed that the existence of this Dirac
+string causes the phase of a quantum state that carries charge e∗ to have a branch cut on the curve
+Γ and to change by exp(i2π(e∗/e)Φ/φ0), as required by the Aharonov-Bohm eﬀect [155]. This
+eﬀect is unobservable for integer-charged states, i.e. electrons.
+10.3.4. The Jain States
+The construction of the Laughlin quasihole motivated Jainendra Jain [230, 231] to rewrite
+the Laughlin wave function in the form
+Ψm(z1, . . . , zN) =
+�
+i< j
+(zi − zj)m−1 × Ψν=1(z1, . . ., zN)
+(393)
+Here the prefactor represents particles each attached with an even number, m − 1, of ﬂux quanta.
+The second factor,Ψν=1(z1, . . ., zN), is the wave function of fermions ﬁlling up a lowest Landau
+level, i.e. the Vandermonde determinant of Eq.(377). In this form, the inverse of the ﬁlling factor,
+1/ν, which is the number of ﬂux quanta per particle, is written as 1
+ν = (m − 1) + 1 = m. The
+interpretation now is that the (m − 1)φ0 magnetic ﬂux carried by each particle, which partially
+screens the uniform ﬁeld down to an eﬀective ﬁeld Beﬀ = B − (m − 1)φ0 corresponding to one
+eﬀective ﬂux quantum per particle, i.e. a ﬁlled lowest Landau level of the eﬀective ﬁeld Beﬀ. In
+other words, the wave function for the ν = 1/m FQH state is reinterpreted as the νeﬀ = 1 integer
+QH state of N composite fermions, each being an electron carrying attached an even integer m−1
+ﬂux quanta.
+This procedure is known as ﬂux attachment. Physically it means that the strong interactions
+between the electrons for other electrons to be further away which, in virtue of being in a strong
+magnetic ﬁeld, is equivalent to an increase of their relative angular momentum, as if there was a
+local change in the magnetic ﬁeld. There is no reason to believe that the composite fermions are
+weakly coupled, even though this is frequently stated in the literature without any real supporting
+evidence.
+Jain then generalized this reinterpretation of the Laughlin state to a whole sequence of FQH
+states obtained by ﬁlling p levels of these partially screened magnetic ﬂuxes
+Ψm,p(z1, . . . , zN) = PLLL
+�
+i< j
+(zi − zj)m−1Ψν=p(z1, . . ., zN)
+(394)
+where Ψν=p(z1, . . ., zN) is the wave function of p ﬁlled Landau levels (of composite fermions),
+and PLL is an operator that projects onto the lowest Landau level. By counting ﬂuxes we see that
+the ﬁlling fractions for the Jain states satisfy
+1
+ν(m, p) = m − 1 ± 1
+p
+(395)
+79
+
+where we allowed for the possibility that the external ﬂuxes may be over-screened by the com-
+posite fermions. Alternatively we can write
+ν(m, p) =
+p
+p(m − 1) + ±1
+(396)
+The Jain states with p = 1 are the laughlin states and each is called the primary state of a
+Jain sequence. Almost all the observed FQH states belong to one of these sequences (and its
+generalizations). In particular, the most prominent states (i.e. those with larger plateaus) belong
+to the sequence 1
+3, 2
+5, 3
+7, 4
+9, 5
+11, . . . and to the reversed sequence 1, 2
+3, 3
+5, 4
+7, . . .. For p → ∞, the Jain
+sequences converge to the values of the ﬁlling fraction limp→∞ ν(m, p) =
+1
+m−1. In this limit, the
+m − 1 ﬂuxes attached to each particle exactly cancels th external ﬂux leading us to a (possibly)
+Fermi liquid of composite fermions [232].
+10.3.5. Quasiholes have fractional statistics
+The non-locality and topological nature of the Laughlin quasihole implies that this state must
+be regarded as a soliton (or vortex) of the charged quantum ﬂuid. The vortex-like state with wave
+function Ψqh
+m (z0; z1, . . . , zN) is called the Laughlin quasihole. This state represents a composite
+object of a ﬂux quantum and a fractional charge. This structure of this quantum state is strongly
+reminiscent of the ﬂux-charge composite objects considered by Frank Wilczek who showed that
+such objects should be anyons, states that exhibit fractional statistics [156]. Although at a con-
+ceptual level anyons were proposed (almost) prior to the discovery of the FQHE, it is in this
+setting that fractional statistics entered actual physics. The actual experimental observation took
+work by many people, and was only conﬁrmed in 2020 [233].
+Fractional statistics dictates the analytic form of the wave functions of two or more quasiholes
+[234]. Let us consider a laughlin-type wave function for two quasiholes located at (complex)
+coordinates u and �. Naively we expect the wave function for two quasiholes in the ν = 1/m
+Laughlin state to have the form
+Ψ(u, �; z1, . . . , zN) = N(u, �)
+N
+�
+j=1
+(zj − u)(zj − �) Ψm(z1, . . ., zN)
+(397)
+The prefactor N(u, �) must be chosen to account for the fact that theres is an additional change of
+angular momentum due to the additional ﬂuxoid at �. We also expect that as the quasihole with
+charge e/m is adiabatically carried around the the other quasihole (also with charge e/m) there
+will be ac accrued phase in the wave function due to the Aharonov-Bohm eﬀect of the charge
+of one quasihole circling around the ﬂux of the other (or, equivalently, crossing the branch cut)
+[229]. The requirement of translation invariance and analyticity are met by the choice [234]
+(ignoring a multiplicative normalization constant)
+N(u, �) = (u − �)1/m exp
+−
+1
+4ℓ2
+0m(|u|2 + |�|2)
+
+(398)
+which has a branch cut stretching from u to �. A calculation using the plasma analogy represents
+this state as a set of N charge −1 particles interacting with two additional particles of charge −1/m
+at u and � (and a neutralizing background). The branch cut implies that dragging a quasihole
+around the other during a π rotation followed by a translation (i.e. an exchange), induces a
+monodromy in the wave with a jump in its phase of exp(±iπ/m) (with the sign determined by the
+orientation of the monodromy) in such a way the that wave function changes by a phase factor
+Ψ(u, �; z1, . . ., zN) �→ e±i π
+m Ψ(u, �; z1, . . . , zN)
+(399)
+In other words, the quasihole is an anyon with fractional statistics π/m. Daniel Arovas, J. Robert
+Schrieﬀer and Frank Wilczek [235] further reﬁned this argument by showing that under such
+a slow adiabatic change the wave function of two quasiholes acquires a Berry phase which ac-
+counts for the fractional statistics. An explicit path-integral derivation of this eﬀect, which uses
+the fact that the quasihole states are coherent states can be found in Ref. [9], chapter 13.
+10.3.6. Hydrodynamic Eﬀective Field Theory
+We will now turn to the approaches to the FQHE using the methods of quantum ﬁeld theory.
+As we will see the concept of ﬂux attachment plays a key role. In section 9.5 we showed that
+80
+
+Chern-Simons gauge theory is in fact a theory of ﬂux attachment. Thus we expect that Chern-
+Simons gauge theory should play a key role as well.
+It is useful to ask ﬁrst why should Chern-Simons gauge theory play a central role at least in
+the description of the low energy physics. There is a simple, yet powerful, phenomenological
+argument due to J¨org Fr¨ohlich and Anthony Zee that shows why this should be the case [236].
+This in essence a hydrodynamic argument. The fractional quantum Hall eﬀect occurs in a 2DEG
+in the presence of a large magnetic ﬁeld which breaks explicitly time reversal invariance. In the
+regime of the FQHE the 2DEG is a uniform charged incompressible ﬂuid in which charge is
+conserved. This means that the charge 3-current jµ = (j0, j) (where j0 is the charge density and
+j is the charge current density) must be locally conserved,
+∂µ jµ = 0
+(400)
+which is the continuity equation. The solution of this equation is that the conserved current can
+be written in terms of a dual (in the Hodge sense) vector ﬁeld Bµsuch that
+jµ = 1
+2πǫµνλ∂νBλ
+(401)
+The factor of
+1
+2π is introduced for later convenience. The current ﬁeld jµ is not changed by a
+smooth redeﬁnition of the vector ﬁeld Bµ → Bµ + ∂µΦ, where Φ(x) is a non-singular ﬁeld. This
+means that the vector ﬁeld Bµ is a gauge ﬁeld.
+The eﬀective action of this theory is a functional of the current distribution µ and, hence, of
+the gauge ﬁeld Bµ. It should be gauge-invariant, and odd under parity and time-reversal. Since
+the ﬂuid is incompressible and uniform, at long distances and low energies the action must be a
+local functional of the gauge ﬁeld which should be at least Galilean invariant. In addition, the
+coupling of the ﬂuid to an external electromagnetic probe ﬁeld Aµ must be of the usual form
+−ejµAµ. To lowest orders in derivatives, there the unique local and gauge-invariant Lagrangian
+density which is odd under time reversal (and parity) is the Chern-Simons theory
+Leﬀ[Bµ] = m
+4πǫµνλBµ∂νBλ − 1
+4g2 F 2
+µν − e
+2πAµǫµνλ∂νBλ + . . .
+(402)
+The ﬁrst term is the Chern-Simons term of the Lagrangian. The coeﬃcient m is a dimensionless
+integer which, as we saw in section 9.3 is required for the theory to be invariant on a closed
+manifold. The second is a Maxwell term. Here Fµν = ∂µBν −∂νAµ is the ﬁeld strength tensor of
+the gauge ﬁeld Bµ. By power counting we see that the coupling constant g2 has units of length−1
+and, hence this term is irrelevant at long distances and will be neglected. The third term is the
+coupling of the current jµ to the electromagnetic ﬁeld Aµ. Upon an integration by parts this term
+can be written as BµJ µ where
+Jµ = − e
+2πǫµνλ∂νAλ
+(403)
+is a current minimally coupled to Bµ.
+The equation of motion of the (dynamical) gauge ﬁeld Bµ is
+δL
+δB = 0
+(404)
+which yields the relation
+m
+2πǫµνλBµ∂νBλ = Jµ
+(405)
+From the deﬁnition of the current Jµ, Eq.(403), we see that the solution of the equation of
+motion of Eq.(405), up to a gauge transformation, is
+mBµ = −eAµ
+(406)
+By plugging this relation back into the Lagrangian for the gauge ﬁeld Bµ, Eq (402) (which
+si equivalent to integrate out the gauge ﬁeld Bµ) we ﬁnd that the eﬀective Lagrangian of the
+electromagnetic ﬁeld Aµ is just a Chern-Simons term
+Leﬀ[Aµ] =
+e2
+4πmǫµνλAµ∂νAλ + . . .
+(407)
+81
+
+This result allows us to read-oﬀ the Hall conductivity of the ﬂuid
+σxy = 1
+m
+e2
+2πℏ
+(408)
+where we restored units such that ℏ is not unity. In other words, this ﬂuid exhibits the fractional
+quantum Hall eﬀect for a ﬂuid at ﬁlling fraction ν = 1/m.
+We should note that this heuristic argument applies equally to systems of fermions, for which
+m is odd, as well as to systems of bosons, for which m is even.
+10.3.7. Composite Boson Field Theory
+We will now discuss tow ﬁeld-theoretic approaches to the FQHE. These approaches use the
+concept of ﬂux attachment as a mapping of fermions to bosons and as a mapping of fermions
+to fermions. These approaches are a form of duality transformation which engineers a form
+of statistical transmutation. We will ﬁrst show that Chern-Simons theory can be used to eﬀect
+such a mapping. That this is possible should not be surprising since, as we saw in section 9.7,
+Chen-Simons theory is a theory of fractional statistics and, hence, of anyons.
+We should make some important comments on both approaches before we get into a detailed
+description. While the mapping of fermions to composite bosons and fermions to composite
+bosons is correct, their approximate mean ﬁeld descriptions violate symmetries of the 2DEG in
+a magnetic ﬁeld. At the root level is the fact that the ﬂux attachment is local in space time and
+approximate descriptions of these theories bring about a large amount of Landau level mixing,
+even in the large ﬁeld limit. For instance, at the mean ﬁeld theory level the composite boson
+approach is equivalent to a Bose-Einstein condensate which is invariant under translations and
+under time reversal. In this picture the breaking of time reversal is present in the coupling to a
+Chern-Simons gauge ﬁeld. Likewise, the mean ﬁeld theory of the composite fermion theory is a
+problem of composite fermions in a partially screened. While a partially screened magnetic ﬁeld
+still breaks time reversal, it does not behave properly under magnetic translations. As we will
+see, these problems will be solved, at least in the low energy regime, by quantum ﬂuctuations
+which restore the symmetries. In both cases, in the fractional states one recovers an eﬀective
+topological ﬁeld theory which captures the universal physics of these states. Needless to say it,
+both approaches do poorly in the computation of non-universal dimensionful quantities such as
+energy gaps, etc. These diﬃculties become very severe in the compressible states whose low
+energy behavior is not topological.
+Unlike the wave function approaches which project these states into a speciﬁc Landau level
+(usually the lowest), the ﬁeld theoretic approach does not eﬀect such projection. Several papers
+have been written attempting to do a ﬁeld theory projected into a Landau level. These approaches
+transformed the problem into quantum ﬁeld theory on a non-commutative plane, which is inher-
+ent the nature of the states in a magnetic ﬁeld. Although some signiﬁcant progress has been made
+in this direction, these theories remain poorly understood [237, 238, 239, 240, 241, 242, 243]
+In this section we will focus on the composite boson theory. Let us consider a theory in 2+1
+dimensions with two dynamical abelian gauge ﬁelds Aµ and Bµ, whose Lagrangian is a sum of
+a Chern-Simons term at level k and a BF term:
+L = k
+4πǫµνλA µ∂νA λ + 1
+2πǫµνλA µ∂νBλ
+(409)
+The equation of motion for the ﬁeld Aµ is
+k
+2πǫµνλ∂νA λ + 1
+2πǫµνλ∂νBλ = 0
+(410)
+whose solution (up to a gauge transformation) is kAµ = −Bµ. Plugging this relation into the
+Lagrangian of Eq.(409) we ﬁnd that the eﬀective Lagrangian for the ﬁeld Bµ is
+L[Bµ] = − 1
+4πkǫµνλBµ∂νBλ
+(411)
+which states that exchanging Aµ ↔ Bµ is equivalent to the “duality” k ↔ −1/k.
+In section 9.7 we showed that particles coupled to a Chern simons gauge ﬁeld at level k
+have fractional statistics exp(±iπ/k). Hence, we see that particles coupled to the ﬁeld Bµ have
+fractional statistics exp(±iπk). In other words, for k odd this coupling amounts to exchanging
+a theory of fermions to a theory of bosons coupled to the gauge ﬁeld Bµ. This is a form of
+82
+
+bosonization. Alternatively, for k even it maps a theory of fermions to another theory of fermions
+coupled to a gauge ﬁeld Bµ. These observations have led to distinct, but ultimately equivalent
+description of the quantum Hall eﬀect.
+We will begin the approach of mapping fermions to bosons and use it in the problem of the
+FQHE. This is the Landau-Ginzburg theory (or composite boson theory) of Shoucheng Zhang,
+Hans Hansson and Steven Kivelson [244] (ZHK). In this approach the problem of fermions
+coupled to a magnetic ﬁeld interacting with each other through a two-body potential (that we
+will assume is ultra with coupling constant λ, for simplicity) becomes the same problem but for
+a theory of bosons which are also coupled to a Chern-Simons gauge ﬁeld, which we will denote
+by Aµ. The Lagrangian density for this equivalent system of composite bosons is
+LCB = φ∗(x)[iD0 + µ]φ(x) + 1
+2M |Dφ(x)|2 − λ(|φ(z)|4 +
+1
+4πmǫµνλA µ∂νA λ
+(412)
+where x = (x0, x) are the space-time coordinates, µ is the chemical potential, M is the mass of
+the fermions, and m is an arbitrary odd integer. The covariant derivative
+Dµ = ∂µ + i e
+ℏcAµ + iAµ
+(413)
+which eﬀects the minimal coupling of the complex scalar ﬁeld φ(x) to the background electro-
+magnetic ﬁeld Aµ and to the Chern-Simons gauge ﬁeld Aµ, known in this context os known as
+the statistical gauge ﬁeld.
+ZHK showed that the FQH state can be thought as being closely related to to a phase in which
+the complex scalar ﬁeld condenses, much as in the case of a superﬂuid even though the FQH iis
+not a superﬂuid. To see how this works we will write
+φ(x) =
+�
+ρ(x) exp(iω(x))
+(414)
+The classical equations of motion of the theory of composite bosons, Eq.(412), are
+δLCB
+δφ∗(x) =0
+⇒
+(iD0 + µ)φ(x) − 1
+2M D2φ(x) − 2λ|φ(x)|2φ(x) = 0
+(415)
+LCB
+δA0(x) =0
+⇒
+1
+2πmǫi j∂iA j + |φ(x)|2 = 0
+(416)
+LCB
+δAi(x) =0
+⇒
+1
+2πmǫiαβ∂αA β +
+i
+2M
+�φ∗(x)Diφ(x) − (Diφ(x))∗φ(x)� = 0 (417)
+LCB
+δµ =ρ0L2T
+⇒
+�
+d3x |φ(x)|2 = ρ0L2T
+(418)
+where ρ0 is the areal density of electrons.
+A uniform solution of Eqs. (415)-(418) with constant amplitude of the composite bosons,
+i.e. a uniform and static condensate ¯φ, with uniform statistical gauge ﬁeld strength ¯
+B, requires
+that the there should be no current in the ground state and that each term of Eq.(417) should be
+zero. This condition implies that the external ﬁeld is canceled by the average statistical ﬁeld,
+eB
+ℏc + ¯
+B = 0. This condition together with Eqs.(416) and (418) imply that
+ρ0 = 1
+m
+eB
+ℏc = 1/m
+2πℓ2
+0
+(419)
+and we conclude that the ﬁlling fraction is ν = 1
+m, which are the Laughlin states. In addition we
+get that the chemical potential is µ = 2λρ0 and that |¯φ|2 = ρ0, with ρ0 given in Eq.(419).
+However, these mean ﬁeld results seeming imply that this system is a Bose-Einstein conden-
+sate. To understand why this is incorrect, and to determine what it actually is, we need to go
+beyond the mean ﬁeld theor that we just described and evaluate the eﬀects of quantum ﬂuctua-
+tions and write
+φ(x) = (ρ0 + δρ(x))1/2 exp(iω(x)),
+eAµ
+ℏc + Aµ = δAµ
+(420)
+The partition function of this theory is a path integral over the ﬂuctuating density ﬁelds δρ, the
+Goldstone ﬁeld ω and the ﬂuctuation of the Chern-Simons gauge ﬁeld δAµ. Upon integrating out
+83
+
+the massive density ﬂuctuations to lowest (quadratic) order we ﬁnd that the eﬀective Lagrangian
+for ω and δAµ is
+Leﬀ[ω, δAµ] = κ
+2 (∂0ω − δA0)2 − ρs
+2 (∂iω − δAi)2 +
+1
+4πmǫµνλδA µ∂νδA λ + . . .
+(421)
+The ﬁrst two terms of Eq.(421) is the eﬀective Lagrangian of the Goldstone mode ω in the
+Bogoliubov theory of superﬂuidity a compressibility κ and a superﬂuid stiﬀness ρs given by
+κ = 1
+2λ,
+ρs = ρ0
+M = ν
+2πℏωc
+(422)
+However, as we see, this is not a superﬂuid since the phase ﬁeld is “eaten” by the Chern-Simons
+gauge ﬁeld δAµ which now acquires a mass term. In this sort of Higgs mechanism the ﬂuctuating
+gauge ﬁeld has a massive longitudinal mode and a massive transverse mode. Thus, this state doe
+not have any massless modes.
+To see that this theory describes the fractional quantum Hall eﬀect we need to compute the
+linear response to a weak electromagnetic perturbation δAµ(x). This can be accomplished by
+considering the new Lagrangian
+Leﬀ[ω, δAµ, δAµ] = κ
+2
+�
+∂0ω − δA0 − e
+ℏcδA0
+�2
+−ρs
+2
+�
+∂iω − δAi − e
+ℏcδAi
+�2
++ 1
+4πmǫµνλδA µ∂νδA λ+. . .
+(423)
+and integrate out the phase ﬁeld ω (which can be set to zero in the London/unitary gauge) and
+the ﬂuctuation of the statistical ﬁeld δAµ to ﬁnd that eﬀective electromagnetic action is
+S eﬀ[δAµ] =
+1
+4πm
+e2
+ℏ
+�
+d3x ǫµνλδAµ∂νδAλ + . . .
+(424)
+from which we conclude that the Hall conductivity predicted by the composite boson theory is
+σxy = 1
+m
+e2
+h
+(425)
+as it should be for a FQHE at ﬁlling fraction ν = 1/m.
+We conclude by looking at vortex states in the composite boson theory. A vortex state is
+a time-independent solution of the equations of motion Eqs.(415)-(418) with the asymptotic
+behavior (in the temporal gauge δA0 = 0)
+lim
+|x|→∞φ(x) = √ρ0 eiϕ(x)
+(426)
+lim
+|x|→∞δAi(x) = ± ∂iϕ(x) = ±ǫi j
+x j
+|x|2
+(427)
+where ϕ(x) is the azimuthal angle on the plane
+ϕ(x) = tan−1
+� x2
+x1
+�
+(428)
+The energy of a neutral vortex (i.e. not coupled to a gauge ﬁeld) is logarithmically divergent,
+Evortex ≃ ρs
+2 ln(R/a0) where R is the linear size of the system and a0 is a short-distance cutoﬀ
+(see section 5.1.1). However, the situation here is diﬀerent since the complex scalar ﬁeld φ(x) is
+coupled to a dynamical Chern-Simons gauge ﬁeld Aµ. Except for the Chern-Simons nature of
+this gauge ﬁeld, the problem we are dealing with is similar to that of a superconductor coupled
+to a dynamical gauge ﬁeld or to the Abelian-Higgs model in quantum ﬁeld theory. In the case of
+interest here we have ﬁnite energy vortex solutions which satisfy the asymptotic condition
+lim
+|x|→∞
+����
+�
+i∂ j − δA j
+�
+φ(x)
+����
+2
+= 0
+(429)
+which is obeyed by conﬁgurations that obey Eq.(427). Thus, at long distances, the circulation of
+the Chern-Simons gauge ﬁeld on a large closed contour Γ that contains the vortex satisﬁes
+�
+Γ
+δA jdx j = ±2π
+(430)
+84
+
+This vortex carries charge. To see this we compute the local charge density j0(x)
+j0(x) = − δS eﬀ
+δA0(x) = −e δS eﬀ
+δδA0(x) = +e δS CS
+δδA0(x) =
+e
+2πmǫi j∂iδA j(x)
+(431)
+and compute the total charge of the vortex on a larger region Σ whose boundary is Γ and obtain
+Q = e
+�
+d2xj0(x) =
+e
+2πm
+�
+Σ
+d2x ǫi j∂iδA j(x) = e
+m
+1
+2π
+�
+Γ
+dx jδA j(x) = ± e
+m
+(432)
+Therefore, the vortex of the composite boson theory has the same charge as the Laughlin quasi-
+hole.
+To determine the statistics of the vortex we go back to the eﬀective Lagrangian written in the
+form of Eq.(409), (from now on we set δA µ ≡ A µ)
+Leﬀ = κ
+2(∂0ω−A0)2−ρs
+2 (∂ jω−A j)2+ 1
+2πǫµνλA µ∂νBλ− e
+2πǫµνλAµ∂νBλ− m
+4πǫµνλBµ∂νBλ (433)
+where Aµ is the external electromagnetic probe ﬁeld, and where we used units with ℏ = c = 1.
+The ﬁrst two terms make the statistical gauge ﬁeld Aµ massive. In the low energy limit the
+statistical gauge ﬁeld is frozen to the vortex conﬁgurations of the complex scalar ﬁeld φ or,
+equivalently, to the vortex singularities of its phase ﬁeld ω. The vorticity current Ωµ is
+Ωµ = ǫµνλ∂νA λ
+(434)
+The eﬀective Lagrangian for the ﬁeld Bµ is
+Leﬀ[Bµ] = − m
+4πǫµνλBµ∂νBλ − e
+2π Aµǫµνλ∂νBλ + ΩµBµ
+(435)
+which, except for the coupling to the electromagnetic ﬁeld, has the same form as in Eq.(224),
+and of the eﬀective action introduced in section 10.3.6 on phenomenological grounds.
+The eﬀective topological ﬁeld theory of Eq.(435) encodes al the universal data of fractional
+quantum Hall ﬂuids. Being topological this eﬀective ﬁeld theory has no energy scales and de-
+scribes the physics at energies low compared to any excitation energy gap. In addition to yield-
+ing the correct Hall conductivity σxy = νe2/h (with ν = 1/m) and the fractional vortex charge
+Q = e/m, this theory will allow us to draw important additional results about the low energy
+physics.
+By using the results of section 9.7, we conclude that the vortices of the composite boson are
+anyons with fractional statistics exp(±iπ/m), consistent with the conclusions of section 10.3.5.
+In addition, on a 2-torus the ground state of this system is m-fold degenerate. This feature,
+characteristic of systems with topological order, is that the ground state degeneracy depends on
+the topology of the surface on which the 2DEG resides. The ground state on a torus degeneracy
+was shown by Duncan Haldane and Edward Rezayi [245] by an explicit construction of a model
+wave function for the Laughlin states on a torus. On a surface with g handles (i.e. genus g) the
+degeneracy is mg. Xiao-Gang Wen and Qian Niu [246] showed that these results hold for any
+FQH state which are, quite generally, topological quantum ﬂuids.
+Finally, we may ask how many distinct types of vortices does this theory have. The funda-
+mental (Laughlin) vortex has charge e/m and statistics π/m. In principle we could have a vortex
+with any integer topological charge (winding number) n ∈ Z. Such a vortex has charge ne/m and
+statistics πn2/m. However, a vortex with topological charge n = m has charge e and statistics mπ.
+This vortex is indistinguishable from a hole (a missing electron) since it has the same charge and
+statistics. We conclude that a Laughlin state has m distinct vortices with n = 1, 2, . . ., m − 1. We
+notice that this number is the as the number of degenerate states on a torus. In section 10.3.10
+we will see that this fact is closely related to the number of primary ﬁelds of a conformal ﬁeld
+theory associated with the FQH states.
+10.3.8. Composite Fermion Field Theory
+At the beginning of section 10.3.7 we noted that ﬂux attachment can be used to deﬁne equiv-
+alent theories: a) by attaching an odd number, m, ﬂuxes to each electron thereby becoming a
+composite boson, and b) by attaching an even number, m − 1, ﬂuxes by which the electrons
+turn into composite fermions, as in Jain’s construction [230]. In this section we discuss the most
+salient features of the ﬁeld theory approach to composite fermions, introduced by Ana L´opez and
+85
+
+me in 1991[247, 248] as a theory of all the Jain states. This approach was extended in 1993 by
+Bertrand Halperin, Patrick Lee and Nicholas Read [232] to the case of the compressible states.
+By following the same approach that we used in section 10.3.7, but adapted to the case in
+which we map to a theory of a composite Fermi ﬁeld ψ(x), we ﬁnd their dynamics is described
+by the eﬀective action [247]
+SCF =
+�
+d3x
+�
+ψ∗(x)[iD0 + µ]ψ(x) +
+1
+2M |Dψ(x)|2
+�
++
+1
+4πn
+�
+d3xǫµνλA µ∂νA λ
+− 1
+2
+�
+d3x
+�
+d3x′ (|ψ(x)|2 − ρ0)V(x − x′)(|ψ(x′)|2 − ρ0)
+(436)
+where n = m − 1 is an even integer. Here V(x − x′) = δ(x0 − x′
+0)V(|x − x′|) is the instantaneous
+electron-electron repulsive interaction, and ρ0 is the average areal neutralizing charge density.
+As before, the covariant derivative is Dµ = ∂µ + ieAµ + Aµ, where Aµ is the electromagnetic ﬁeld
+(including the uniform magnetic ﬁeld B), and Aµ is the statistical gauge ﬁeld. We are using units
+such that ℏ = c = 1.
+We can simplify somewhat the form of the actions for the composite fermions by using the
+fact that the Gauss law of Chern-Simons gauge theory states that the particle density and the
+gauge ﬂux are rigidly tied together, |ψ(x)|2 =
+1
+2πnB ≡ ǫi j∂iA j (this is the ﬂux attachment) as a
+operator identity. Thus, we can write the action as
+SCF =
+�
+d3x
+�
+ψ∗(x)[iD0 + µ]ψ(x) +
+1
+2M |Dψ(x)|2
+�
++
+1
+4πn
+�
+d3x ǫµνλA µ∂νA λ
+− 1
+2
+�
+d3x
+�
+d3x′
+�B(x)
+2πn − ρ0)
+�
+V(x − x′)
+�B(x′)
+2πn − ρ0
+�
+(437)
+Since this action is a quadratic form in the Fermi (Grassmann) ﬁelds ψ, we can integrate them
+out to obtain the following eﬀective action for the statistical gauge ﬁeld Aµ
+S eﬀ[Aµ] = −itr
+�
+iD0 + µ + D2
+2M
+�
++
+1
+4πn
+�
+d3x ǫµνλ (A µ − eAµ)∂ν (A λ − eAλ) + S int[Aµ − eAµ]
+(438)
+where Aµ is a probe external electromagnetic ﬁeld (with vanishing average) and does not include
+the uniform magnetic ﬁeld. The interaction term is
+S int[Aµ − eAµ] = −1
+2
+�
+d3x
+�
+d3x′
+�(B(x) − eB(x))
+2πn
+− ρ0)
+�
+V(x − x′)
+�(B(x′) − eB(x′))
+2πn
+− ρ0
+�
+(439)
+Here too B(x) is an external probe with vanishing average.
+We will investigate the properties of the path integral for the statistical gauge ﬁeld
+Z[Aµ] =
+�
+DAµ exp(iS eﬀ[Aµ, Aµ])
+(440)
+using a saddle point expansion. The saddle-point condition of the eﬀective action of Eq.(438)
+δS eﬀ
+δAµ(x) = 0
+(441)
+leads to the equation of motion for the ﬁeld Aµ
+⟨jF
+µ (x)⟩ +
+1
+4πnǫµνλ
+�
+F νλ(x) − eFνλ�
+= 0
+(442)
+where ⟨jF
+µ ⟩ is the expectation value of the fermionic current. In addition we need to impose the
+condition for the particle density to be uniform and equal to the neutralizing background charge
+⟨jF
+0 (x)⟩ = ρ0.
+We will assume that the electromagnetic ﬁeld Aµ describes just a static uniform magnetic
+ﬁeld of strength B. Under these conditions, the ground state should also be static and uniform.
+This means that the ﬁeld strength of the statistical gauge ﬁeld should be constant and uniform
+value ⟨B⟩, and that the current ⟨jF⟩ should vanish in the ground state. On the other hand, the
+Gauss Law of the Chern-Simons gauge ﬁeld implies that ⟨B⟩ = −2πn ρ0, while the zero current
+condition requires the the (statistical) electric ﬁeld vanishes, ⟨E ⟩ = 0. Since the composite
+86
+
+fermion couples in the same way to the electromagnetic ﬁeld Aµ and to the statistical ﬁeld Aµ we
+conclude that they experience an eﬀective magnetic ﬁeld
+Beﬀ = B + 1
+e⟨B⟩ = B − 2πn ρ0
+e
+(443)
+If the total number of electrons is N, then ρ0/e = N/L2 where L is the linear size of the system.
+Let us denote by Neﬀ
+φ the total number of ﬂux quanta of the eﬀective magnetic ﬁeld (in units of
+ℏ = c = 1 in which the ﬂux quantum is φ0 = 2π), and 2πNφ = BL2 is the total ﬂux. Then, the
+total eﬀective ﬂux is
+2πNeﬀ
+φ = 2πNφ − 2πn N
+(444)
+Let ν = N/Nφ be the ﬁlling fraction and νeﬀ = N/Neﬀ
+φ
+the eﬀective ﬁlling fraction. Eq.(444)
+implies that these ﬁlling fractions are related by
+1
+νeﬀ = 1
+ν − n
+(445)
+Recall that n is an even integer. However, the system will be incompressible only if νeﬀ = p ∈ Z.
+In other words, the composite fermions ﬁll an integer number p of the partially screened Landau
+levels, which is Jain’s condition. For this condition to be satisﬁed the ﬁlling fraction ν(p, n) is
+ν±(n, p) =
+p
+np ± 1
+(446)
+which are the Jain fractions. In the special case p = 1 the Jain fractions are the Laughlin fractions,
+with n = m − 1. Similarly we ﬁnd that Beﬀ is
+Beﬀ = ±
+B
+np ± 1
+(447)
+So we see that the external ﬁeld B is partially screened to the smaller value Beﬀ which can be
+parallel (screened) to B or anti-parallel (overscreened) to B. For the same reason, at this mean
+ﬁeld level the cyclotron frequency ωc is reduced by the same amount to ωeﬀ
+c = ωc/(np ± 1).
+We will discuss now the eﬀects of quantum ﬂuctuations for the action of Eq.(438) about the
+saddle-point solutions. We will work to the lowest order (quadratic) in the ﬂuctuations, which
+can be regarded as a semi-classical (or “RPA”) treatment of this theory. The quadratic eﬀective
+action for the ﬂuctuations of statistical gauge ﬁeld, which we will still denote by Aµ is
+S (2)
+eﬀ[Aµ] =1
+2
+�
+d3x
+�
+d3y A µ(x)ΠCF
+µν (x, y) A ν(y)
++
+1
+4πn
+�
+d3x ǫµνλ(Aµ(x) − eAµ(x))∂ν(Aν(x) − eAν(x)) + S int(Aµ − eAµ)
+(448)
+Here, Πµν
+F (x, y) is the polarization tensor of the composite fermions for p ﬁlled eﬀective Landau
+levels. The interaction term of the action is given in Eq.(439).
+As required by gauge invariance, the composite fermion polarization tensor is transverse,
+∂µΠCF
+µν = 0, which ﬁxes the tensorial structure. In momentum and frequency space the com-
+posite fermion polarization tensor ΠCF
+µν (q, ω) depends on three kernels ΠCF
+0 (q, ω), ΠCF
+1 (q, ω) and
+ΠCF
+2 (q, ω), where ΠCF
+0
+and ΠCF
+2
+contain the parity-even response of the composite fermions and
+ΠCF
+1
+is their parity-odd response. Each kernel is given as a series terms representing particle-hole
+processes with simple poles at the excitations energies ωrs = (r − s)ωeﬀ
+c , with r > p (particles)
+and s ≤ p (holes). Each term has a residue which is an integer power of q2 times a Laguerre
+polynomial in q2. Details of these kernels can be found in Ref.[247] and in chapter 13 of Ref.[9].
+However, what will be important here is that, in the gapped states these kernels have a low energy
+and low momentum limit which is local. This will enable us to ﬁnd a local topological eﬀective
+action only for the gapped states.
+After integrating out the statistical gauge ﬁeld Aµ we ﬁnd the eﬀective action for the external
+electromagnetic probe ﬁeld Aµ, which has the standard form
+S eﬀ[Aµ] = 1
+2
+�
+d3x
+�
+d3y Aµ(x) Πµν(x − y) Aν(y)
+(449)
+which encodes the full electromagnetic response of the system, not just of the composite fermions.
+Once again, gauge invariance requires that the full polarization tensor be transverse, ∂µΠµν = 0.
+87
+
+In section 10.1.4 we showed that there is a relation between the polarization tensor Πµν and
+the current-current correlation function Dµν. There we mentioned the these correlators are re-
+quired (by gauge invariance) to satisfy a Ward identity known as the f-sum rule:
+� ∞
+−∞
+dω
+2π iω DR
+00(ω, q) = ρ0
+M q2
+(450)
+where the (retarded) density-density correlation function is related to the (retarded) polarization
+tensor, DR
+00(ω, q) = ΠR
+00(ω, q). In the limit of low momentum q, at ﬁxed frequency ω, The
+expression for ΠR
+00(ω, q) is
+ΠR
+00(ω, q) ≃ q2ΠR
+0(ω, q) = −ρ0
+M
+q2
+(ω + iǫ)2 − ω2c
+(451)
+where ωc = eB/(Mc) is the cyclotron frequency of the electrons. This expression saturates the
+sum rule. This means that whichever corrections Eq.(451) may have must vanish in the low q
+limit. In other words, in this limit the approximation that we made is actually exact. This is well
+known result in the theory of the Fermi liquid [194].
+In addition, the result of Eq.(451) implies that there is a particle-hole (density) collective
+mode which at low momentum has energy ℏωc, without corrections. This result is known as
+Kohn’s theorem [249] which states that in a Galilean invariant system the 2DEG must have an
+exact eigenstate at the cyclotron frequency. More intuitively, the center of mass of the 2DEG exe-
+cutes a cyclotron motion regardless of whether the particles are free, interacting or not, fermions,
+bosons, or anyons. This is a general result. Notice that the composite fermions do not satisfy
+Kohn’s theorem as they cyclotron mode would be at the eﬀective cyclotron frequency ωeﬀ
+c , and
+that the leading quantum ﬂuctuations have restored the magnetic symmetry.
+In the limit of low energy ω → 0 and low momentum q → 0 of the kernels ΠCF
+0 , ΠCF
+1
+and
+ΠCF
+2
+become
+ΠCF
+0 (0, 0) = p
+2π
+M
+Beﬀ
+≡ εCF,
+ΠCF
+1 (0, 0) = ± p
+2π ≡ σCF
+xy ,
+ΠCF
+2 (0, 0) = − 1
+2π
+p2
+M ≡ −χCF
+(452)
+where εCF is the “dielectric constant” of the composite fermions, χCF is their eﬀective “per-
+meability”, and σCF
+xy is the (integer) composite fermion Hall conductivity. We then ﬁnd that
+low-energy eﬀective action of the statistical gauge ﬁeld Aµ is
+S eﬀ[Aµ] =1
+2
+�
+σCF
+xy +
+1
+2πn
+� �
+d3x ǫµνλA µ(x) ∂ν A λ(x)
+− e
+2πn
+�
+d3x ǫµνλA µ(x) ∂ν Aλ(x) + e2
+4πn
+�
+d3x ǫµνλAµ(x) ∂ν Aλ(x)
++
+�
+d3x1
+2
+�
+εCF Ei
+2(x) − χCF B2(x)
+�
+−
+1
+8π2n2
+�
+d3x
+�
+d3y (B(x) − eB(x)) V(x − y) (B(y) − eB(y))
+(453)
+Notice that, except possibly for the last term, the eﬀective action is a local. This is possible
+because the mean ﬁeld theory state is gapped.
+In what follows we will focus only on the leading terms of the eﬀective action and neglect the
+subleading terms, the least two terms of the eﬀective action which have the form of a Maxwell
+term and an additional (possibly non-local) term. The remaining leading terms have a Chern-
+Simons form and a BF form whose eﬀective Lagrangian is
+Leﬀ[Aµ] = 1
+4π
+�
+±p + 1
+n
+�
+ǫµνλA µ(x) ∂ν A λ(x)− e
+2πn ǫµνλA µ(x) ∂ν Aλ(x)+ e2
+4πn ǫµνλAµ(x) ∂ν Aλ(x)
+(454)
+If now now integrate ot the statistical gauge ﬁeld Aµ we ﬁnd the the electromagnetic ﬁeld Aµ has
+the eﬀective Lagrangian
+Leﬀ[Aµ] = σxy
+2
+ǫµνλAµ(x) ∂ν Aν(x)
+(455)
+from where we ﬁnd that the Hall conductivity σxy of the 2DEG is
+σxy = e2
+2πℏ
+p
+pn ± 1
+(456)
+88
+
+which is the Hall conductivity of the Jain states (in standard units). Here, as before, n is an even
+integer.
+We will now return to the eﬀective Lagrangian of Eq.(454). As it stands, since the Chern-
+Simons term has fractional level, it is not invariant under large gauge transformations and, as
+such, cannot be deﬁned on a closed manifold. To remedy this problem we can now write this
+Lagrangian in terms of a theory with two dynamical gauge ﬁelds Aµ and Bµ as follows:
+Leﬀ[Aµ, Bµ] = p
+4πǫµνλA µ(x) ∂ν A λ(x) − n
+4πǫµνλBµ(x) ∂ν Bλ(x)
++ 1
+2πǫµνλA µ(x) ∂ν Bλ(x) − e
+2πǫµνλBµ(x) ∂νAλ
+(457)
+This is a topological quantum ﬁeld theory for all the Jain states [250]. This theory can be deﬁned
+on any manifold no matter its topology. It is easy to see that upon integrating out the ﬁeld Bµ
+we recover the eﬀective ﬁeld theory for Aµ of Eq.(454). Also, in the special case of the Laughlin
+states, for which p = 1, we can integrate out the ﬁeld Aµ, and arrive to the eﬀective topological
+ﬁeld theory for the ﬁeld Bµ of Eq.(435) that we derived in the composite boson theory in section
+10.3.7.
+Let us rewrite the Lagrangian of Eq.(457) in a more compact yet general form. To this end we
+deﬁne a multicomponent gauge ﬁeld A I
+µ with I = 1, . . ., L, an L-component charge vector t, and
+an L × L second rank symmetric matrix KIJ. Wen and Zee have given a general classiﬁcation of
+a large class of fractional quantum Hall states, said to be abelian, whose topological ﬁeld theory
+is deﬁned by the Lagrangian [251, 252]
+L = 1
+4πKIJǫµνλA I
+µ (x) ∂ν A J
+λ (x) − e
+2πtIǫµνλAµ(x) ∂ν A λ
+I (x)
+(458)
+In the case of the theory of Eq.(457) which has two components, L = 2, with A 1
+µ = Aµ and
+A 2
+µ = Bµ. The two-component vector is t = (0, 1) and the matrix 2 × 2 matrix is
+K =
+�−p
+1
+1
+n
+�
+(459)
+Wen and Zee showed that, quite generally, a theory of the K-matrix form has a vacuum degener-
+acy on a torus of |detK| and, on a surface of genus g, the degeneracy is |detK|g. They also showed
+that the Hall conductivity is σxy = ν e2/h where the ﬁlling fraction is
+ν =
+L
+�
+I,J=1
+tI K−1
+IJ tJ
+(460)
+The properties of the quasiparticles can be determined by computing the appropriate correla-
+tor. In the composite fermion theory, whose Lagrangian is given in Eq.(437), the gauge-invariant
+composite fermion propagator is
+GCF(x − y; γ(x, y)) =
+�
+ψ†(x) exp(i
+�
+γ(x,y)
+dzµ(eAµ(z) + A µ))ψ(y)
+�
+(461)
+which, as in any gauge theory, it is gauge-invariant but path-dependent. Here γ is an oriented
+open path with endpoints that the space-time locations x and y. The composite fermion propa-
+gator has information about many of the properties of the quasiparticles, including their electric
+charge. To ﬁnd what is the statistics of the quasiparticles we need to compute a two-particle cor-
+relator (a four-point function) which is deﬁned in terms of two open oriented paths, say γ1(x, y)
+and γ2(u, w), with endpoints at the locations of the two particles, respectively.
+The computation of these correlators is done using a feynam sum over trajectories with dif-
+ferent weights which depend on the gauge ﬁeld conﬁgurations. To do these calculations is, in
+general, a complicated problem. However, in a gapped state, the quasiparticles are massive and
+in the low energy regime these expressions are dominated by a classical trajectory on an oriented
+path ˜γ(x, y) with the same endpoints x and y. The end result reduces to the computation of the
+expectation value of a Wilson loop operator
+W[Γ] =
+�
+exp(i
+�
+Γ
+dzµ(eAµ(z) + A µ(z))
+�
+A µ
+(462)
+89
+
+on the closed oriented path Γ = γ � ˜γ− (where ˜γ−(y, x) has the opposite orientation as ˜γ(x, y) and
+runs from y to x. The explicit dependence on the external electromagnetic ﬁeld yields the value
+of the charge of the quasiparticle through the Aharanov-Bohm eﬀect. Likewise, the two-particle
+correlator is reduced, in the asymptotic low energy regime, to the computation of an expectation
+value of two Wilson loops in the eﬀective Chern-Simons theory, as was done in section 9.7,
+which yields the fractional statistics of the quasiparticles. The interested reader can ﬁnd details
+in chapter 13 of Ref.[9].
+A system described by a theory of the form of Eq.(458) has diﬀerent types of quasiparticles.
+In general we can assign an integer-valued quantum number ℓI ∈ Z as the charge of the quasi-
+particle with respect to the gauge ﬁeld A µ
+I . The general coupling of a quasiparticle current has
+the form Lqp = jµℓIA µ
+I . Then, the charge Q[ℓ] and the statistics δ[ℓ] of a general quasiparticle
+deﬁned by the integer-valued L-component vector ℓ are
+Q[ℓ] = −etIK−1
+IJ ℓJ,
+δ[ℓ]
+π
+= ℓIK−1
+IJ ℓJ + 1
+(463)
+where the +1 is required to refer the statistics to fermions (not bosons) [250]. These results are
+general and hold for any abelian FQH state [251].
+In the case of the Jain states, the charge vector is t = (0, 1) and the quasiparticle is labeled by
+the vector ℓ = (1, 0) which yield the results for the charge Q and the statistics δ
+Q =
+−e
+np + 1,
+δ = π
+�
+−n
+np + 1 + 1
+�
+(464)
+which are the quantum numbers of the quasiparticles of the Jain states. For the Laughlin state at
+ν = 1/3 we obtains Q = −e/3 and δ = π/3 (as we should), which for the ﬁrst Jain state at ν = 2/5
+the charge Q and the statistics δ are Q = −e/5 and δ = 3π/5. With some caveats, the predictions
+for the quasiparticle charge in the FQH states with ν = 1/3 and ν = 2/5 have been conﬁrmed in
+noise measurements at a quantum point contact by R. de Picciotto and coworkers [253] and by
+L. Samindayar and coworkers [254]. Fractional statistics has been measured experimentally in
+both FQH states at ν = 1/3 and ν = 2/5 in (Fabry-Perot) interferometers by J. Nakamaura, S.
+Liang, G. Gardner, and M. Manfra [233].
+10.3.9. The Compressible States
+At large values p → ∞ the Jain sequences converge to the even-denominator fractions
+ν∞(n) = 1/n. In this limit, Beﬀ → 0 and ωeﬀ
+c
+→ 0. In other words, at the ﬁlling fractions
+ν∞(n) the average eﬀective ﬁeld experienced by the composite fermions is zero. This mean
+ﬁeld theory then would predict that this state is essentially a Fermi liquid of composite fermions
+which is a compressible state. In addition, one can compute the eﬀective charge of the composite
+fermion and ﬁnd that it is e/(np ±1), which also vanishes in the compressible limit. It is straight-
+forward to see that the Fermi sea of the composite fermions is a disk with Fermi momentum
+pF = (2ν∞(n))1/2ℏ/ℓ0 and that the Fermi energy is EF = ν∞(n)ℏ/(Mℓ2
+0). Halperin, Lee, and Read
+[232] did an in-depth analysis of the consequences of this compressible state which are largely
+in agreement with experiment.
+In our discussion of the incompressible FQ states, both in the composite boson and in the
+composite fermion approaches in sections 10.3.7 and 10.3.8 we saw that the mean ﬁeld approxi-
+mation violated symmetries that were restored by quantum ﬂuctuations. This fact allowed us the
+derive eﬀective ﬁled theories which are asymptotically exact in the low energy IR regime. What
+allowed us to obtain these exact results is the existence of a gap. However, the compressible
+states are gapless and in some sense should be regarded theories of a quantum critical system.
+In fact, if we reexamine the ﬁeld theory of composite fermions of section 10.3.8 we see
+that, in the compressible limit p → ∞, the action of Eq.(437) describes a system of fermions at
+ﬁnite density coupled to the ﬂuctuations Aµ of the statistical gauge ﬁeld with a Chern-Simons
+term, but without an external uniform ﬁeld (which has been canceled by the ﬂux of the average
+statistical gauge ﬁeld). This theory has two salient features. one is that the only trace left of
+the explicitly broken time-reversal symmetry is in the presence of the Chern-Simons term. This
+means that while the mean ﬁeld theory looks like a Fermi liquid, the consequences of the broken
+time reversal invariance only enters in the quantum ﬂuctuations. In addition, in this regime this
+theory also breaks the particle-hole symmetry of a half-ﬁlled Landau level (in the high magnetic
+ﬁeld regime).
+However, these contributions, already at one-loop order, typically contain IR divergencies.
+These IR divergencies are due to singular forward scattering processes of the composite fermions
+90
+
+by the gauge ﬁelds. They are most severe in the computation of the fermion self energy and de-
+pend on the form of the electron-electron interactions. For instance if the interaction potential
+V(R) is short range, the one-loop contribution to the fermion self-energy has an imaginary part
+withe the behavior Σ′′(ω) ∼ ω2/3 which is much larger than the real part Σ′(ω) as ω → 0 and the
+the Fermi energy EF is approached [232]. This behavior, which is found in many metallic sys-
+tems at quantum criticality [255, 256, 69] implies that the width of the quasiparticle is asymptot-
+ically much larger than the energy as the Femi surface is approached. This one-loop result means
+that perturbation theory fails in the IR and that actual behavior may be non-perturbative. This
+problem is present even in the case of unscreened e2
+R (Coulomb) interactions where the singular
+behavior is milder with Σ′′(ω) ∼ ω ln ω (known as Marginal Fermi Liquid scaling [257]).
+At any rate, these IR singular behaviors imply that the quasiparticle picture is inadequate and
+the that in these regimes there may be no quasiparticles. These systems are generally known as
+“Strange Metals”. Many strongly interacting physical systems of interest, ranging form high Tc
+superconductors to (largely conjectured) spin liquids with spinon Fermi surfaces to even quark-
+gluon plasmas share these types of singular IR behaviors [257, 258]. In spite of a large body
+of theoretical work, it has remained largely unsolved problem. Perturbative renormalization
+group methods used to justify the Landau theory of the Fermi liquid [259, 260] generally fail
+in these regimes, although in some cases long range interactions have allowed some degree of
+control [261, 262]. Remarkably, the AdS/CFT Gauge/Gravity Duality [263] has brought new
+insights into strange metal behaviors [264]. We will see below that relativistic dualities have
+given additions insights into the physics of compressible states.
+10.3.10. Fractional Quantum Hall Wave Functions and Conformal Field Theory
+The Laughlin wave function and its generalizations share the remarkable feature of being
+essentially universal. Aside form the dependence on the magnetic length ℓ0 in the gaussian
+factors which ensure that the function is integrable at long distances (a feature inherited from
+the single-particle states in the Landau level), the FQH “ideal” wave functions do not have any
+length scales. In addition, in almost all cases, the ideal wave functions are the exact ground state
+wave functions of some suitable ultra-local Hamiltonian projected onto the lowest Landau level.
+Another feature, closely related to their universality and analyticity, is that these wave functions
+look similar to correlation functions in two-dimensional critical (chiral) critical systems. We will
+now see that these features are not accidental and point to a deep relation between quantum Hall
+states and Conformal Field Theory (CFT).
+Although the relation between the simpler case of the Laughlin state was known to several
+people, such as Sergio Fubini [265], this deep connection with rational CFT was articulated in
+considerable generality in a somewhat earlier paper by Gregory Moore and Nicholas Read [266].
+This approach showed its full power in the formulation of the non-abelian quantum Hall states.
+Let us begin with the simpler case of the Laughlin states with ν = 1/m, where m is odd for
+fermions and even for bosons. We will consider a U(1) Euclidean CFT of a compact boson (a
+scalar ﬁeld) φ(x) closely related to our discussion of bosonization in 1+1 dimensions in section
+7.4. This theory has been extensively studied in the CFT literature [103, 102, 104] The Euclidean
+action for this theory is
+S = 1
+8π
+�
+d2x
+�
+∂µφ
+�2
+(465)
+This theory is compactiﬁed by the condition that the only allowed observables are invariant under
+the discrete symmetry
+φ(x) �→ φ(x) + 2πRn
+(466)
+where R is the compactiﬁcation radius and n is an arbitrary integer. The observables that satisfy
+this condition are vertex operators of the form
+Vp(x) = exp
+�
+i p
+Rφ(x)
+�
+(467)
+where p is an integer.
+The correlators of this theory are products of holomorphic (right-moving in Minkowski
+spacetime) and antiholomorphic (left-moving in Minkowski spacetime) factors. In the context
+of the FQH states we will be interested in a it chiral theory, whose correlators are holomorphic
+(analytic) functions in complex coordinates z = x1+ix2. Formally, the ﬁeld φ(z, ¯z) is decomposed
+into a sum of a holomorphic ﬁeld φ(z) and and antiholomorphic ﬁeld φ(¯z), so that the propagators
+is
+⟨φ(z, ¯z) φ(�, ¯�)⟩ = − ln(z − �) − ln(¯z − ¯�)
+(468)
+91
+
+In what follows we will focus on the chiral(holomorphic) ﬁeld φ(z) whose propagator is
+⟨φ(z)φ(�) = − ln(z − �)
+(469)
+The correlator of the chiral current operator
+j(z) = i
+R∂zφ(z)
+(470)
+is given by
+⟨i∂zφ(z) i∂�φ(�)⟩ ∼
+1
+(z − �)2
+(471)
+which tells us that this operator has (chiral) scaling dimension ∆ = 1. Similarly the correlator of
+two chiral vertex operators
+Vp(z) = exp
+�
+i p
+Rφ(z)
+�
+(472)
+is
+⟨Vp(z) V−p(�)⟩ =
+1
+(z − �)p2/R2
+(473)
+whose (chiral) scaling dimension is ∆p = p2/(2R2). These expressions are very similar to the op-
+erators that we used in section 5.1.1 to describe vortices in 2D superﬂuids (and planar magnets).
+The main diﬀerence is that here the ﬁelds and their correlators are holomorphic where is section
+5.1.1 the correlators are the product of a holomorphic and antiholormophic factors.
+For general p and R the correlator of Eq.(473) has a branch cut from z to �. This means that
+the monodromies of the operators, i.e. dragging an operator around the other, induces a change
+in the phase of the correlator equal to 2πp2/R2. This behavior is reminiscent of the analytic
+structure the the quasiparticle wave function in the FQH states and of the braiding operation
+between anyons.
+Moore and Read [266] showed that the following correlator in a U(1) compactiﬁed boson
+CFT (with compactiﬁcation radius R = √m) is equal to the Laughlin wave function at ﬁlling
+fraction ν = 1/m,
+Ψm(z1, . . . .zN) =
+� 
+N
+�
+i=1
+exp
+�
+i √mφ(zi)
+� exp
+�
+−i
+�
+d2z′ √m ρ0 φ(z′)
+� �
+(474)
+=
+�
+i< j
+(zi − zj)m exp
+−1
+4
+N
+�
+i=1
+|zi|2
+
+(475)
+where we have used units in which the magnetic length is ℓ0 = 1.
+In section 5.1.1 we saw that correlators of vertex operators are non-vanishing only if their
+total charge (vorticity) is zero. The reason for this condition is that the (Euclidean) action is
+invariant under arbitrary uniform shifts of the ﬁeld φ. Now, in the correlator on the right hand side
+of Eq.(475) we have a product of N vertex operators, each with the same charge √m. Then, under
+an arbitrary shift by α of the ﬁeld φ(z), the product of vertex operators change by a phase equal
+to N √mα. On the other hand the other operator in the correlator, which describes a continuum
+background charge, changes by a phase − √mρ0L2α, where L2 is the area. Since ρ0 is the areal
+particle density, N/L2, the two changes of the phase cancel each other exactly. Thus, the operator
+in the expectation value of Eq.(475) is charge neutral and has a non-vanishing expectation value.
+The other important feature of the correlator of Eq.(475) is the choice of compactiﬁcation
+radius, R = √m, and of the vertex operator Vp(z) with p = m:
+Vm(z) = exp(i √mφ(z))
+(476)
+whose correlation function is
+⟨Vm(z) V−m(�)⟩ =
+1
+(z − �)m
+(477)
+Which is invariant under a 2π rotation (since m ∈ Z). However under a rotation by π (i.e. an
+exchange) it changes by (−1)m. Hence, for m odd it changes sign, and the vertex operator Vm
+behaves as a fermion, while for m even the vertex operator behaves as a boson. We will call the
+vertex operator Vm ≡ Ve the electron operator.
+92
+
+In other words, the correlator of Eq.(475) is the expectation value of N vertex operators each
+describing an electron (for m odd) and a neutralizing background charge. Then, an elementary
+calculation yields the expression of the Laughlin wave function.
+In this formulation the ground state wave function for the integer QH state at ν = 1, which
+has the for of a Vandermonde determinant shown in Eq.(377), is interpreted as the correlator of
+N electron operators in the U(1) CFT with compactiﬁcation radius R = 1. In this simple case,
+the electron operator is the vertex operator V1(z) = exp(iφ(z)), and there are no other operators
+(aside from the current j(z) = i∂zφ). The propagator of this vertex operator is
+⟨V1(z)V−1(�)⟩ =
+1
+z − �
+(478)
+which indeed represents a fermion with electric charge e.
+We will now see that the vertex operator of Eq.(472) with p = 1 (and R = √m)
+V1(z) = exp
+� i√mφ(z)
+�
+(479)
+represents the Laughlin quasihole. The correlator of this vertex operator is
+⟨V1(z) V−1(�)⟩ =
+1
+(z − �)1/m
+(480)
+which has a branch cut stretching from z to � and, as a result, under a rotation by ±π its phase
+changes by ±π/m, just as the Laughlin anyons do. To see that this operator indeed is related to
+the Laughlin quasihole we will compute the eﬀect of inserting the vertex operator V1(z) in the
+correlator of Eq.(475) and ﬁnd
+�
+exp
+� i√mφ(�)
+�
+N
+�
+i=1
+exp
+�
+i √mφ(zi)
+� exp
+�
+−i
+�
+d2z′ √m ρ0 φ(z′)
+� �
+=
+=
+N
+�
+i=1
+(zi − �)
+�
+i< j
+(zi − zj)m exp
+−1
+4
+N
+�
+i=1
+|zi|2 − 1
+4m|�|2
+
+= Ψm(�; z1, . . ., zN)
+(481)
+which is, indeed, the wave function for the Laughlin quasihole of Eq.(392). The insertion of
+an additional vertex operator V1(u) at u inside the expectation value of Eq.(481) yields the two-
+quasihole wave function of Eq.(397), proposed by Halperin [234] and discussed in section 10.3.5,
+with the same branch cut shown in Eq.(398) and, therefore carry fractional statistics π/m.
+The operator product expansion discussed in section 3.3.2 of the vertex operators of this CFT
+yields new insight on the quasiholes. Indeed the OPE of two vertex operators of charges p and q
+is
+lim
+�→u Vp(�)Vq(u) = lim
+�→u
+1
+(� − u)∆p+∆q−∆[p+q]m V[p+q]m(u)
+(482)
+where [p]m is the integer p modulo a multiple of m, i.e. if p = mr + s (with r a non-negative
+integer and 0 ≤ s < m, then [p]m = s. Eq.(482) is understood in the sense that the contribution
+of all additional operators to the right hand side vanish as � → u. In other words, the vertex
+operators are primary ﬁelds of this CFT and there are only m primary ﬁelds. The physical process
+described by the OPE is called fusion.
+A CFT with a ﬁnite number of primaries is called a rational CFT [103, 102]. This result is,
+of course, the same statement that we made is section 10.3.7 that the FQH state has m distinct
+anyons (vortices). This result also means that a vertex operator of charge m, i.e. an electron, is
+indistinguishable from the identity operator, V0. In this sense, the allowed primaries are ﬁelds
+which are local respect to the electron operator Vm. From the point of view of the FQH state, an
+operator that creates or destroys an electron (at ﬁxed ﬁlling fraction) has no eﬀect, as the FQH
+state is a ﬂuid made of electrons This also means that all physical operators must braid trivially
+with the electron operator. In a CFT an operator of this type is said to generate and extended
+symmetry algebra [267, 268].
+Furthermore, the OPE of the chiral current j(z) with the vertex operator V1(z) is
+lim
+�→u j(�)V1(u) = 1/m
+z − wV1(u)
+(483)
+93
+
+which implies that the vertex operator represents a state with charge 1/m (in units of the electric
+charge e).
+The CFT approach to construct FQH states has become a very powerful tool. We will see in
+section 10.3.11 that FQH states on an open geometry (e.g. a disk) have edge states which can
+be understood a chiral CFTs. In addition to providing a deeper understanding of more general
+abelian (muliti-component) FQH states with a K-matrix structure, new classes of of FQH states
+with non-abelian (multi-dimensional) representations of the Braid Group has been proposed.
+The anyons of these states are of particular interest since they have been proposed originally
+by Kitaev in 1997 [169] that anyons can be used as physical qubits for topologically-protected
+quantum computation [169, 269, 270, 271, 170].
+So far we discussed the case of abelian anyons. Abelian anyons have the property that the
+fusion of two anyons is another anyon. Similarly, a braiding operation between two anyons is
+equivalent (up to a phase factor) to another anyon. Mathematically this means that abelian anyons
+are one-dimensional representations of the Braid group. The fractional statistics of an anyon is
+then a label of the representation of the Braid group. As we saw, the same holds under fusion.
+However, the Braid group generally admits multi-dimensional representations. In this more
+general case the fusion (or braiding) of two anyons is represented by a linear combination (su-
+perposition) of anyon states. Linear combinations of anyon states are represented by unitary
+matrices of rank greater than 1. Since matrices generally do not commute braiding and/or fusion
+processes correspond to a multiplication of these matrices. Such unitary processes are then re-
+garded as quantum gates. This concept is the physical basis of topological quantum computation.
+It is currently the cutting edge of research in the ﬁeld.
+We will now discuss the simplest non-abelian FQH states, the Moore-Read (MR) states [266].
+The Moore-Read wavefunctions are
+ΨMR(z1, . . . , zN) = Pf
+�
+1
+zi − zj
+� �
+i< j
+(zi − zj)n exp(− 1
+4ℓ2
+0
+N
+�
+i=1
+|zi|2)
+(484)
+which describes a FQH of electrons at ﬁlling fraction ν = 1/n, with n an even integer. Here
+Pf
+�
+1/(zi − zj
+�
+is the Pfaﬃan of the matrix 1/(zi − zj).
+This state was motivated by the discovery of an unexpected FQH state in the N = 1 Landau
+level with an even denominator ﬁlling fraction ν = 5/2 by Willett and coworkers in 1987 [272,
+273, 274]. Until then (and since then) this is the only FQH state with an even denominator
+ﬁlling fraction. The Pfaﬃan factor has a simple pole when two coordinates approach each other.
+However, provided n ≥ 1 the “Laughlin factor” in the MR wavefunction cancels the singularity.
+We will shortly discuss the CFT construction of the Moore-Read state. The poles in the MR
+wavefunction suggest that in this state the electrons can get closer to each other than in a Laughlin
+state. This observation suggests that there may be some form of attraction (or suppression of
+repulsion) between the electrons in the MR state and motivated the notion that the observed
+even-denominator plateau may be physically related to some sort of pairing interaction. Shortly
+after the Moore-Read proposal was made, a paired wave function at ν = 1/2 with a Pfaﬃan factor
+was also proposed by Greiter, Wen and Wilczek [275, 276, 277] who suggested that it reﬂects
+electrons pairing in the ℓ = 1 angular momentum channel in time-reversal breaking condensate
+with symmetry px + ipy.
+Although, following the logic behind the Kohn-Luttinger mechanism for paring in angular
+momentum state ℓgeq1 with repulsive interactions [278, 279, 280] it may possible to get a paired
+state even in the lowest Landau level (although there is no evidence for it, so far), the N = 1
+Landau level may be more hospitable to such a state. Indeed, in the N = 1 Landau level the
+single-particle states i have angular momentum greater or equal than 1, and their probability
+distributions are suppressed both at short and long distances; these states look like smoke-rings.
+The matrix elements of the Coulomb potential in the N = 1 Landau level are parametrically
+suppressed at short distances (compared with the states in the lowest, N = 0, Landau level). In
+this scenario, the ν = 5/2 plateau is interpreted as a ν = 1/2 fully polarize state of the N = 1
+landau level, with an N = 0 Landau level ﬁlled with electrons with both spin polarizations in
+a state with ν = 2. There is numerical evidence that an instability into a p-wave paired state
+such as the MR state is favored if the short-distance repulsion between the composite fermions is
+softened [281].
+In fact, in 1983 Halperin [282] considered generalizations of the Laughlin state in which
+due to very strong attractive interactions electrons would form clusters which then condense into
+a Laughlin-type state. The simplest example was Laughlin state of bosons (paired electrons)
+94
+
+a paired state at ν = 1/2 whose Laughlin quasihole carries charge e/4. Extensive numerical
+calculations are more compatible with the MR state than with the Halperin state of pairs [283].
+We will see shortly that the Halperin state is an abelian relative of the MR state.
+Moore and Read derived the wave function of Eq.(484) by considering a CFT with two
+sectors: a chiral boson CFT with compactiﬁcation radius R = √n, and a chiral Majorana fermion
+CFT, i.e. the chiral part of the 2D Ising CFT, discussed in section 4.2. The chiral Ising CFT has
+central charge c = 1/2 and several primary ﬁelds: the identity I, the twist ﬁeld σ, and the
+(chiral) Majorana fermion χ with (chiral) scaling dimensions 0, 1/16, and 1/2, respectively. The
+propagator of the Majorana fermion, which is a free ﬁeld, is
+⟨χ(z) χ(�)⟩ =
+1
+z − �
+(485)
+The primary ﬁeld σ, the “twist ﬁeld”, is non-local with respect to the Majorana fermion χ, and
+changes its boundary conditions from periodic to antiperiodic.
+The fusion rules of the chiral Ising CFT are
+χ ⋆ χ = I,
+σ ⋆ σ = I + χ,
+σ ⋆ χ = χ
+(486)
+What will be important below is that the σ ﬁeld has two fusion channels. As in the analysis of
+the Laughlin states, the compactiﬁed boson n primaries (anyons), the vertex operators Vp(x) =
+exp(ipφ(x)/n), with n = 0, 1, . . .n and there are n types of anyons, and a (charge) current J =
+i√n∂zφ(z).
+The ﬁrst task is to identify the electron operator which has to be a fermion and has to
+carry electric charge. This means that it is a composite of an operators (with Fermi statis-
+tics) the (neutral) chiral ising CFT and a vertex operator whose charge is an integer multi-
+ple of e.
+The desired electron operator is ψe(z) = χ(z) exp(i √nφ(z)) whose correlator is
+⟨χ(z)Vn(z)χ(�)V−n(z)⟩ = 1/(z − �)1+n, which is a fermion with charge e.
+The n-point function of the chiral Majorana fermions is
+⟨χ(z1) . . .χ(zN)⟩ = Pf
+�
+1
+zi − zj
+�
+(487)
+which follows fro applying Wick’s theorem. Then we see that the MR wave function is the
+expectation value
+ΨMR(z1, . . ., zN) = ⟨χ(z1) . . .χ(zN)⟩ ×
+� 
+N
+�
+i=1
+exp
+�
+i √nφ(zi)
+� exp
+�
+−i
+�
+d2z′ √n ρ0 φ(z′)
+� �
+(488)
+Next we need to identify the primary ﬁelds of the tensor product of the chiral Ising CFT, Z2,
+and the chiral U(1) CFT with compactiﬁcation radius R = √n. The allowed primary ﬁeld must
+belocal with respect to the electron operator, ψe. This leaves us with four primary ﬁelds: a) the
+identity I, b) the non-abelian vortex σ(z) × exp
+�
+i
+2 √nφ(z)
+�
+with charge e/(2n) and non-abelian
+statistics, c) the charge-neutral Majorana fermion χ, and d) the abelian vortex (the Laughlin
+quasihole) exp
+�
+i√nφ(z)
+�
+, with charge e/n and abelian statics δ = π/n.
+The he new feature here is the non-abelian fractional statistics of the non-abelian vortex
+(sometimes called a “half-vortex”. Its non-abelian character is a consequence of the fact that the
+fusions of two twist ﬁelds has two channels, see Eq.(486). This implies that the wave function of
+four non-abelian vortices can be expressed as a linear combination of two so-called conformal
+blocks. In other words, this wave function belongs to a degenerate two-dimensional Hilbert
+space, each component labeled by a conformal block [102]. A braiding operation between two
+non-abelian vortices is a monodromy of the wave function that induces a unitary transformation
+U in this Hilbert space
+U =
+1√
+2
+exp
+�
+iπ
+�1
+8 + 1
+4n
+��
+×
+� 1,
+1
+−1
+1
+�
+(489)
+Therefore, the degenerate Hilbert space of four quasiholes provides a two-dimensional represen-
+tation of the Braid Group. This observation is the basis for regarding a state of four non-abelian
+quasiholes as a qubit. Furthermore, Nayak and Wilczek showed that a state of 2p quasiholes
+spans a 2p−1-dimensional degenerate Hilbert space [284]. This means that, for large p, the de-
+generacy per quasihole is
+√
+2. In other words, this degeneracy is not due to a local degree of
+freedom attached to each quasihole but that it is shared in a non-local fashion!
+95
+
+In what sense are the Moore-Read states paired? In an insightful paper Read and Green [285]
+used a BCS theory approach to investigate the pairing properties of a condensate of composite
+fermions in the px + ipy channel. The mean-ﬁeld BCS Hamiltonian is
+HF =
+�
+d2k
+�
+(ε(k) − µ) ψ†(k)ψ(k) + 1
+2
+�
+∆∗(k)ψ(−k)ψ(k) + ∆(k)ψ†(k)ψ†(−k)
+��
+(490)
+where ψ(k) is the composite fermions ﬁeld, ε(k) =
+k2
+2M, and ∆(k) is the gap function. For a
+px + ipy condensate the gap function the gap function transforms under rotations as an ℓ = −1
+angular momentum eigenstate, and for k → 0 it behaves as ∆(k) = (kx − iky)∆, where ∆ is a
+constant pairing amplitude.
+The BCS ground state has the form
+|G⟩ =
+�
+k
+�
+u(k) + �(k)ψ†(k)ψ†(−k)
+�
+|0⟩
+(491)
+where |u(k)|2 = |�(k)|2 = 1. The amplitudes u(k) and �(k)| satisfy the Bogoliubov-de Gennes
+Equation (BdG)
+� ξ(k)
+−∆∗(k)
+−∆(k)
+−ξ∗(k)
+� �u(k)
+�(k)
+�
+≡ E(k) ˆn(k) · σ
+�u(k)
+�(k)
+�
+= E(k)
+�u(k)
+�(k)
+�
+(492)
+where ξ(k) = ε(k)−µ, σ = (σx, σy, σz) are the Pauli matrices, and ˆn(k) = (−Re∆(k), Im∆(k), ξ(k))
+is a unit vector deﬁned for every k. The eigenvalues E(k) and the eigenvectors (u(k), �(k) are
+E(k) =
+�
+ξ2(k) + |∆(k)|2,
+�(k)
+u(k) = −E(k) − ξ(k)
+∆∗(k)
+(493)
+The spinor amplitudes are |u(k)|2 = 1
+2
+�
+1 + ξ(k)
+E(k)
+�
+and �(k) = 1
+2
+�
+1 − ξ(k)
+E(k)
+�
+.
+In the low momentum limit and in real space the BdG Equations become
+i∂tu = − µu + ∆∗i(∂x + i∂y)�
+i∂t� =µ� + ∆i(∂x − i∂y)u
+(494)
+which is just the Dirac Equation in 2+1 dimensions, with µ playing the ole of the mass, and with
+the restriction that the spinor (u, �) obeys the Majorana condition
+(u, �)
+�0
+1
+1
+0
+�
+=
+�u
+�
+�
+(495)
+The Majorana condition is obeyed in all superconductors and reﬂects that fact that in these con-
+densates only the fermion parity (−1)NF (with NF being the number of fermions) is conserved,
+while the fermion number NF is not.
+This is a good BCS state in that it is fully gapped and it is chiral. Assuming that the pair
+ﬁelds in the px and py channels are equal, they showed that the BCS ground state |G⟩ has the
+form
+|G⟩ ∼ exp
+
+1
+2
+�
+k
+g(k)ψ†(k)ψ†(−k)
+ |0⟩
+(496)
+Projected onto a state with N fermions with real space coordinates xi the wave function is a
+Pfaﬃan
+Ψ(x1, . . ., xN) = ⟨x1, . . ., xN|G⟩ = Pf(g(xi − x j))
+(497)
+The long distance behavior of this wave function depends on whether the chemical potential
+µ > 0 (this is the weak-pairing or BCS regime), or µ < 0 (which is the strong pairing BEC-
+like regime). While in the case of an s-wave superconductor these two regimes are smoothly
+connected by a crossover, in the px + ipy case they are separated by a quantum phase transition
+at µ = 0. In the weak pairing regime the function g(k) has the asymptotic long distance form, as
+k → 0,
+g(k) ≃ −
+2µ
+(kx + iky)∆∗
+(498)
+where the pair ﬁeld behaves as ∆(k) ≃ (kx − iky)∆. In real space (in complex coordinates) the
+function g(z) becomes
+g(z) =
+� iµ
+π∆∗
+� 1
+z
+(499)
+96
+
+which is the form used in the Pfaﬃan wave function. On the other hand, in the strong pairing
+regime, µ < 0, g(k) has the asymptotic behavior
+g(k) ≃ −A(kx − iky)
+a−2
+0 + k2
+(500)
+where A and a0 are functions of ∆ and µ,
+A =
+2|µ|M∆
+2|µ| + m|∆|2 ,
+a0 =
+1
+2|µ|
+�
+2|µ|
+M + |∆|2
+(501)
+In real space, at distances |x| ≫ a0 the Fourier transform of g(k) decays exponentially, and as
+a0 → ∞, µ → 0 and the Fourier transform g(z) has the power-law behavior
+g(z) ≃
+� i|∆|
+2π∆∗
+�
+1
+z|z|
+(502)
+This means that µ = 0 is a quantum critical point that separates the weak pairing phase from the
+strong pairing phase, which have distinct properties.
+Read and Green then used a topological argument, originally formulated by Grisha Volovik
+[286] in the context of superﬂuid 3He ﬁlms. To see this we notice that the three component
+unit vector ˆn(k) takes values on the surface of a sphere S 2. In addition, since �(k) → 0 for
+k → ∞(and, hence u(k) → 1), we can wrap the momentum space k onto a sphere S 2. Thus
+ˆn(k) are smooth functions of S 2 �→ S 2. such maps are classiﬁed into the homotopy classes of
+π2(S 2) ≃ Z given in terms of the integer-valued topological invariant
+Q = 1
+4π
+�
+d2k ˆn(k) · ∂kx ˆn(k) × ∂ky ˆn(k)
+(503)
+In the strong pairing phase µ < 0 and ξ(k) > 0 and ˆn(k) takes values only on the northern
+hemisphere of the target space S 2. Such maps can be deformed continuously to the North Pole
+and, hence, are topologically trivial, Q = 0. On the other hand, in the weak pairing phase, µ > 0,
+ξ(k) takes both positive and negative values. In this phase ˆn(k) is topologically non-trivial and
+the topological invariant Q takes non-trivial values ±1.
+The upshot of this analysis is that, in the weak pairing phase, the paired FQH state is, at this
+mean ﬁeld level, a two-dimensional topological superconductor. Read and Green [285] further
+showed that this pairing state has vortices with an interesting fermionic spectrum. A vortex is a
+state win which at long distances the (complex) amplitude of the px + ipy condensate behaves as
+∆ exp(iϕ), where ϕ is the azimuthal angle, which winds by 2π on a circumference of large radius
+R. The BdG equation, Eq.(494), has zero mode solutions. In polar coordinates (r, ϕ) the BdG
+Equation the zero modes satisfy
+∆ieiϕ �
+∂r + i
+r∂ϕ
+�
+� =µu
+∆ie−iϕ �
+∂r − i
+r∂ϕ
+�
+u = − µ�
+(504)
+The normalizable zero mode spinor solution is
+�u(r, ϕ)
+�(r, ϕ)
+�
+= f(r)
+√r
+
+1√
+ieiϕ/2
+1
+√
+−ie−iϕ/2
+
+(505)
+where f(r) is given by
+f(r) ∼ exp
+�
+−
+� r
+0
+dr′ µ(r′)
+|∆|
+�
+∼ exp(−µr/|∆|)
+(506)
+Under a 2π rotation this spinor solution is double-valued
+(u(r, ϕ + 2π), �(r, ϕ + 2π)) = −(u(r, ϕ), �(r, ϕ))
+(507)
+which follows form the global phase invariance. I fact, in all paired states, topological or not, un-
+der a global transformation of the pair ﬁeld by a phase θ, the (composite) fermion must transform
+with a phase of θ/2,
+∆(x) �→ eiθ∆(x),
+ψ(x) �→ eiθ/2ψ(x),
+ψ†(x) �→ e−iθ/2ψ†(x)
+(508)
+97
+
+In other words, in a paired state the fermion is non-local to the vortex. Hence, under a 2π phase
+transformation of the pair ﬁeld, the fermions must change sign, and the spinor zero mode solution
+must be double-valued. This means that this state has a brach cut.
+The structure of the vortex and of the fermion zero mode is closely related to the problem of
+solitons with fractional charge that we discussed in section 8.3. In fact, Roman Jackiw and Paolo
+Rossi [287] investigated a closely related problem of a theory of Dirac ﬁeld in 2+1 dimensions
+coupled to a charged scalar ﬁeld through a Yukawa coupling with the form of a Majorana mass.
+They showed that such a theory admits vortex solutions with fermion zero modes. In a subsequent
+paper Erik Weinberg [288] showed that these zero modes are counted by an index theorem which
+relates the number of zero modes to the vorticity. More recently, Nishida, Santos and Chamon
+[289] that the relativistic theory of Jackiw and Rossi in the non-relativistic approximation reduces
+to the theory of a px + ipy superconductor.
+There is, however, a subtle but profound diﬀerence between the fermion zero modes of the
+one-dimensional fractionally charged solitons and the fermion zero modes of a superconductor.
+In the case of teh 1D solitons, the zero modes can be either occupied or empty rendering a
+soliton charge of −e/2 or +e/2. However, in the case of the superconductor, the BdG fermions
+are charge-neutral since their charge has gone into the condensate. Thus, in a superconductor
+fermion number is not conserved, and only the fermion parity is conserved. This means that
+the ﬁeld operator associated with the vortex zero mode, which we will denote by γ, must be a
+self-adjoint operator, γ† = γ.
+Ivanov [290] showed that this behavior is the origin of the non-abelian fractional statistics in
+this system. Indeed, if one considers a conﬁguration with 2n vortices, each will carry a Majorana
+zero mode γi. Since these are fermion operators they satisfy the usual anticommutator algebra,
+{γi, γ j} = 2δi j. We can (arbitrarily) group the 2n Majorana fermion operators into n pairs. For
+each pair we can deﬁne a complex Dirac operator ψ j = (γ2 j + iγ2 j+1)/2 (and its adjoint), which
+satisﬁes the usual fermionic algebra, {ψ j, ψ†
+k} = δ jk. Each Dirac fermion can be either in an
+empty state or in an occupied state. Thus a system of 2n vortices supports a degenerate Hilbert
+space of dimension 2n−1, which agrees with the results of Nayak and Wilczek [284]. However,
+the assignment of Majorana operators to speciﬁc pairs is actually arbitrary, which amounts to
+a particular deﬁnition of the branch cuts. Changing the assignment of the operators into pairs
+is then equivalent to a rearrangement of the cuts. In 2006 Michael Stone and Suk Bum Chung
+[291] showed that the these vortices obey the braiding and fusion properties of Ising anyons.
+These properties follow from the branch cut conﬁgurations which aﬀect the monodromies of the
+vortices. It is important to stress that the vortices with zero modes are the non-abelions. Majorana
+fermions are fermions and, as such, are abelian.
+In addition to the Moore-read Pfaﬃan state, the CFT approach has led to the formulation of
+new non-trivial FQH states. Read and Rezayi [292] proposed a series of so-called cluster states,
+which generalize the concept of paired states. They investigated a particular type of cluster states
+in which the Pfaﬃan factor of the MR state is replaced by a correlator of parafermion primary
+ﬁelds of a Zk CFT of Zamolodchikov and Fateev [293]. Parafermions were originally introduced
+by Fradkin and Kadanoﬀ [152] (see also Ref.[294]) as a generalization of the fermions of the 2D
+Ising model to the Zk clock model. In this model one can deﬁne several types of parafermions
+each consisting of the fusion of a charge operator and a magnetic (disorder) operator. These
+operators obey an algebra of the type AB = exp(ip2π/k)BA, where the integer p depends on the
+electric and magnetic charges of the parafermions, which is reminiscent of fractional statistics. In
+close analogy with the Majorana fermions of the 2D Ising model, Zk CFT describes the behavior
+of these systems their self-dual points.
+The Zk CFT is actually much richer and has more primary ﬁelds than the Z2 case of the
+Ising model. As a CFT, the Zk theory is the same as the CFT on the coset SU(2)−k/U(1). A
+coset means that a U(1) sector has been projected out of the spectrum. Here we will focus on a
+special case of the Z3 CFT. This CFT has a parafermion primary ﬁeld, which we will call ψ, and
+a non-abelian primary that we will call τ. Read and Rezayi [292] proposed to replace the Pfaﬃan
+factor of the MR state with a correlator of N parafermions of a Z3 CFT and a Laughlin factor
+with an exponent of n + 2/3. Such states require that the number of electrons N be a multiple of
+3. Thus, these states can be viewed as sates in which the electrons cluster in groups of 3. They
+also considered the more general case of the Zk CFT in which case N is a multiple of k. The
+resulting ﬁlling fractions are ν = k/(mk + 2) (with m ≥ 0). The Read-Rezayi k = 3 parafermion
+state is a leading candidate for the observed FQH plateau at ν = 12/5 = 2 + 2/5 [295, 296] (the
+compering state being the 2/5 Jain state). There is strong numerical support for the 12/5 state
+98
+
+being a Z3 parafermion [297].
+The quasiholes obtained by inserting the primary ﬁelds τ in the correlator of parafermions
+have interesting properties which stem from their basic fusion rule
+τ ⋆ τ = I + τ
+(509)
+Read and Rezayi [292] showed that the number of conformal blocks for 3n quasiholes, i.e. the
+number of degenerate states, in the parafermion theory is the Fibonacci number F3n−2, where
+F1 = 1, F2 = 2, F3 = 3, F4 = 5. In general Fm = Fm−1 + Fm−2. For m → ∞, the rate of
+increase approaches the limit limm→∞ Fm/Fm−1 = (1 +
+√
+5)/2 which is known as the Golden
+Mean. In other words, for large m, the dimension of the Hilbert space increases exponentially
+as ((1 +
+√
+5)/2)m. Aside from these interesting mathematical curiosities, the signiﬁcance of this
+fusion rule is that these states, regarded as“qubits”, deﬁne unitary transformations that cover
+uniformly the Bloch sphere which is required for universal quantum computation [270].
+At the level of topological quantum ﬁeld theory the Moore-Read and Read-Rezayi states are
+related to the non-abelian Chern-Simons gauge theory with gauge group SU(2) at Chern-Simons
+level k. In the more general case of the gauge group SU(N) on a manifold M action is given by
+S CS[Aµ] = k
+4π
+�
+M
+d3x ǫµνλ
+�
+Aµ
+a∂νAλ
+a + 2
+3 fabcAµ
+aAν
+bAλ
+c
+�
+(510)
+where k ∈ Z is the level, the gauge ﬁelds Aµ = Aµ
+ata take values in the algebra of SU(N), with {ta}
+being the N2 − 1 generators, and fabc the structure constants of SU(N). The expectation values
+of Wilson loop operators of this theory compute the Jones polynomials which are topological
+invariants of knot theory [165]. The connection with SU2)k Chern-Simons gauge theory has
+been essential in formulating eﬀective low energy quantum ﬁeld theories for the non-abelian
+states [298, 299, 300, 301, 302, 303].
+10.3.11. Edge States and Chiral Conformal Field Theory
+We will now discuss the edge states of the FQH states. As we saw the FQH states are
+incompressible in the bulk and all bulk excitations are gapped. The edge of the region occupied
+by the ﬂuid (in many cases that edge of the physical sample) is where the bulk gap collapses and
+hence where the system has low energy excitations.
+The role of edges and their nature can be seen already in the simple case of the integer Hall
+eﬀect treated as a system of free fermions in the lowest Landau level. In this state all single
+particle states are occupied, and the bulk ground state wave function is a Slater state which takes
+the form of the Vandermonde determinant of Eq.(377). The potential that keeps the electrons
+inside the sample increases monotonously near the edge. As we discussed in section 10.3.1, in
+this region the electrons experience an electric ﬁeld E that pushes them into the sample, and in
+the presence of the perpendicular magnetic ﬁeld B the electrons move along the edge at the drift
+velocity � = c|E|/|B|. At some spatial location, corresponding to the locus of a single particle
+Landau state, the potential crosses the Fermi energy and in that region potential is essentially a
+linear function of the coordinate and hence of momentum (or angular momentum depending on
+the gauge that is being used). In other words, the low energy states are one branch of a one-
+dimensional chiral excitation, such as in the right-moving states of our discussion of the chiral
+anomaly in section 7.2
+In section 9.5 we showed that a Chern-Simons gauge theory on a manifold with a boundary
+projects onto a chiral CFT on that boundary. In that section we showed that an abelian Chern-
+Simons theory U(1)m integrates to the boundary, the 1+1-dimensional Minkowski spacetime
+S 1 × R, as a chiral U(1)m CFT for a compactiﬁed boson with compactiﬁcation radius R = 1
+whose action is given by
+S [ϕ] =
+�
+S 1×R
+d2x 1
+4π
+�
+∂0φ∂1φ − �(∂1φ2)
+�
+(511)
+where we rescaled the compactiﬁed scalar by a factor of √m to that its compactiﬁcation radius
+R = √m as in section 10.3.10. The only diﬀerence between the theory of Eq.(241) and what we
+did in section 10.3.10 is that this theory is in a 1+1-dimensional Minkowski spacetime S 1 × R
+while before we were in Euclidean spacetime.In the case of the integer quantum Hall eﬀect ν = 1
+(and hence m = 1) and the (time-ordered) correlator of the vertex operator, V1(x, t) = exp(iφ(x, t))
+is
+⟨T exp(iφ(x, t)) exp(−iφ(0, 0))⟩ =
+1
+x − �t − iε
+(512)
+99
+
+which is, indeed, the propagator of a chiral free fermion.
+On the other hand, for the Laughlin states at ﬁlling fraction ν = 1/m (and compactiﬁcation
+radius R = √m) the propagator of the electron ψe ∼ exp(i √mφ) is
+⟨T exp(i √mφ(x, t)) exp(−i √mφ(0, 0))⟩ ∼
+1
+(x − �t − iε)m
+(513)
+while the propagator of the quasihole exp
+�
+i√mφ
+�
+is
+�
+T exp
+� i√mφ(x, t)
+�
+exp
+�
+− i√mφ(0, 0)
+��
+∼
+1
+(x − �t − iε)1/m
+(514)
+Therefore, the CFT of the edge is the same as the CFT of the ideal wavefunction with the only
+diﬀerence that the former is in Minkowski spacetime while the latter is in Euclidean spacetime.
+In this sense there is a one-to-one correspondence between the bilk and the edge. We can
+also see how that works by considering a fundamental Wilson arc in the bulk along a path γ(x, y)
+where x and y are at the boundary
+⟨W[γ]⟩ = ⟨exp(i
+�
+γ(x,y)
+dxµA µ) ≡ ⟨exp( i√mφ(x)) exp(− i√mφ(y))⟩
+(515)
+where on the right hand side we have rescaled the ﬁeld by √m, as before. The scaling dimensions
+of the electron, the quasihole and the current are ∆e = m
+2 , ∆qh =
+1
+2m and ∆current = 1. These results
+will be important shortly.
+The same structure applies to the multi-component abelian FQH states. The only diﬀerence is
+that in multi-component FQH states the edge consists, in general, of a charge ﬁeld which couples
+to the electromagnetic ﬁeld and one or more neutral edge states. An example is the theory of the
+non-abelian Pfaﬃan states at ﬁlling fraction ν = 1/n. In this case the edge theory consists of a
+compactiﬁed chiral boson φ of radius R = √n and a chiral Majorana (neutral) fermion χ. At a
+formal level the Lagrangian for the edge state(s) is a sum of two terms
+L = 1
+4π(∂xφ∂tφ − �c(∂xφ)2) + χi(∂t − �n∂x)χ
+(516)
+where �c and �n are the velocities of the (charged) compactiﬁed chiral scalar φ and of the neu-
+tral Majorana ﬁeld χ. Numerically (and experimentally) it is known that the charge mode is
+(substantially) faster than the neutral mode(s), �c > �n, and often by signiﬁcant factors.
+A superﬁcial reading of this Lagrangian suggests that these degrees of freedom are decou-
+pled. However this is not correct. Only operators from both sectors which are local with respect
+to the electron ψe ∼ χ exp(i √nφ) are physically allowed. Here an operator is local with respect
+to the electron means that the operator braids trivially with the electron. This condition restricts
+the allowed observables. In the case of the fermionic MR state, with n = 2, the allowed operators
+are the non-abelion σ exp(iφ/2
+√
+2) with scaling dimension is ∆ = 1/8 and electric charge is
+Q = 1/4, the Majorana fermion with scaling dimension ∆ = 1/2 and no electric charge Q = 0,
+and the Laughlin quasiparticle with scaling dimension ∆ = 1/4 and charge Q = 1/2. Its electron
+operator has scaling dimension ∆ = 3/2 and charge Q = 1.
+Conformal ﬁeld theory has an important deﬁning universal quantity called the central charge
+which is closely related to the energy-momentum tensor of the theory [102]. For pedagogical
+introductions to CFT see Refs.[304, 45] and chapter 21 of Ref.[10]. The energy-momentum
+tensor Tµν is a locally conserved current, ∂µTµν = 0. Its local conservation implies the global
+conservation of energy and momentum. As such, the energy-momentum tensor is a fundamental
+observable of any quantum ﬁeld theory. In a conformal ﬁeld theory, the energy-momentum is
+also the generator of local scale and conformal transformations. In the special case of 1+1-
+dimensional systems, such as the edge states of the FQH ﬂuids, the energy momentum tensor
+has special and crucially important properties. In addition of being locally conserved, conformal
+invariance requires that Tµν must be traceless, T µ
+ν = 0, since its trace is the generator of dilations.
+In this case, if the theory is chiral, the energy momentum tensor has only one (right-moving)
+component T = (T00 − T11)/2. In complex coordinates of the Euclidean metric, the correlator of
+the energy momentum tensor T = Tzz is [305]
+⟨T(z)T(�)⟩ =
+c/2
+(z − �)4
+(517)
+100
+
+where c is a universal quantity known as the it central charge of the CFT. In the case of the
+compactiﬁed boson the central charge is c = 1 whereas for a massless Majorana fermion c = 1/2.
+The central charge of the CFT enters in many observables of fundamental physical impor-
+tance. For example, for a 1+1-dimensional system of length L, such as the edge state of a FQH
+droplet, the ground state energy Egnd for large L has the behavior [306, 307]
+Egnd = ε0L − c π�
+6L + O(1/L2)
+(518)
+where ε0 is the ground state energy density, which is a non-universal quantity, c is the central
+charge of the CFT and � is the velocity of the massless modes. The second term in Eq.(518)
+is known as the Casimir energy. Similarly, the free energy density f(T) of a CFT has the low
+temperature Stephan-Boltzmann behavior (in one dimension)
+f(T) = ε0 + cπT 2
+6� + O(T 3)
+(519)
+where we set the Boltzmann constant to unity. The low-temperature speciﬁc heat c(T) is
+c(T) = cπT
+3� + O(T 2)
+(520)
+In a chiral CFT, such as the edge states of the FQH ﬂuids, the central charge enters in the low
+temperature behavior of the (Hall) thermal conductivity κxy, [285]
+κxy = cπT
+6ℏ
+(521)
+Much of what is known about FQH states comes form experiments involving their edge
+states. In a series of exquisite experiments Granger, Eisenstein and Reno [308] measured κxy in
+the edge states of a ν = 1 quantum Hall ﬂuid and showed that the heat transport is indeed chiral,
+i.e. the edge state behaves as a heat “diode”. This eﬀect was conﬁrmed in the FQH states by
+Bid and coworkers [309] who, in addition, where used this eﬀect to detect the neutral modes in
+several Jain states at ﬁlling fractions 2/3 and 2/5 and in the non-abelian state at ν = 5/2.
+The simplest experimental probes are quantum point contacts and typically are of two types:inter-
+edge tunneling inside a FQH liquid or tunneling of electrons into the edge state of a FQH liquid.
+Since the bulk is gapped, only the edge states participate in these tunneling processes. In the
+general case we will have two edges and a tunnel process at a point contact. Here the two edges
+can be either the two edges of the same FQH state, in which case the process happens inside the
+FQH liquid and involves tunneling of quasiparticles, or the edges of diﬀerent liquids, in which
+case this process is external and involves tunneling of electrons. Problems of these types were
+ﬁrst investigated by Charles Kane and Matthew Fisher [310] which led to a considerable amount
+of work on these problems.
+Let us consider ﬁrst the case of tunneling into the edge state of a ν = 1/m Laughlin state
+from a Fermi liquid, which we will take to be a QH ﬂuid in the ν = 1 state. The point contact is
+at x = 0 The total Lagrangian is
+L = Ledge + LFL − Γ eiω0t ψ†
+e,edge(0, t) ψe,FL(0, t) + h.c.
+(522)
+where ω0 = eV/ℏ, and V is the bias voltage between the two ﬂuids, and Γ is a tunneling matrix
+element. The local electron spectral density (the density of states) N(ω) at energy ω is
+N(ω) = Im lim
+x→0+
+� ∞
+−∞
+dt Ge(x, t) eiωt = const. |ω|m−1
+(523)
+where Ge(x, t) = ⟨Tψ†
+e(x, t)ψe(0, 0)⟩ is the electron correlator in the FQH edge, shown in Eq.(512).
+This result, combined with Fermi’s Golden Rule for a point contact with voltage bias V, predicts
+a tunneling current from a Fermi liquid (FL) into the chiral Luttinger liquid (CLL) of the edge
+state to be [311]
+I(V) = 2π e
+ℏ|Γ|2
+� 0
+−eV
+dE NCLL(E, T)NFL(E + eV, T) ∝ Vm
+(524)
+and a tunneling diﬀerential conductance G(V)
+G(V) = dI
+dV = 2π e
+ℏ|Γ|2NFL(0)NCLL(V, T) ∝ Vm−1
+(525)
+101
+
+which is non-Ohmic and vanishes as V → 0.
+Early experiments by Albert Chang and cowerkers , on a geometry that (most likely) had
+many point contacts, conﬁrmed the predicted CLL behavior [312, 313], although there were
+discrepancies in the measured exponents. The behavior seen in more recent experiments by
+Cohen and coworkers [314], on a point contact in graphene, are consistent with the theoretical
+predictions [311]. Interestingly, these newer class of experiments [315, 314] also show evidence
+of an analog of Andreev reﬂection predicted to exist near the strong coupling ﬁxed point of the
+theory of Sandler, Chamon and Fradkin [316] as a consequence of electron fractionalization.
+Most experiments are done on a geometry call a Hall bar in which the QH ﬂuid occupies a
+rectangular region with its length L larger than its width W. In this geometry, there are chiral
+edge states at opposite sides of the Hall bar propagating in opposite directions. One type of point
+contact consists in creating a constriction in the quantum Hall ﬂuid by applying a gate, a bias
+potential on a narrow strip accross the Hall bar. This gate repels the electrons in the QH ﬂuid,
+forcing the opposite edges to approach each other in the proximity of the gate. In the absence
+of the gate, momentum is conserved on each edge which forbids tunneling accross the Hall bar
+since the edge states have opposite Fermi momentum. However, the gate breaks translation in-
+variance on both edges and tunneling between them is now allowed. In other words, the gate
+creates a point contact between the opposite propagating edge states leading to a tunneling pro-
+cess between the edges of the QH liquid. Thus, this is internal tunneling as oppose the process
+of tunneling between two diﬀerent liquids that we discussed above.
+The tunneling Lagrangian for this system has the same form as in Eq.(522) except that now
+the two edges are identical. The theory Kane and Fisher shows that in the factional case the most
+relevant process is the tunneling of FQH quasiparticles. The Fermi Golden Rule argument used
+in Eq.(525) to compute the diﬀerential conductance also applies in the present case except that
+now the two densities of states are the densities of states of the (Laughlin) quasiparticles (instead
+of electrons). The local quasiparticle density of states is Nqp(ω) ∼ |ω|
+2
+m −1. Then, the diﬀerential
+tunneling conductance G(V) in this case is
+G(v) ∼ V2( 1
+m −1)
+(526)
+This behavior is also non-Ohmic but, unlike the case of electron tunneling of Eq.(522), the diﬀer-
+ential tunneling conductance now diverges as V → 0. This behavior means that the Γ → 0 ﬁxed
+point is unstable and that the point contact ﬂow to a strong coupling ﬁxed point at Γ → ∞. Early
+experiments by Milliken, Umbach and Webb in 1996 [317] showed indications of CLL behavior.
+The predicted behavior was conﬁrmed in the experiments of Roddaro and coworkers [318, 319].
+In the RG language, we can deﬁne a dimensionless tunneling amplitude g by Γ = a∆−1g,
+where ∆ is the scaling dimension of the tunneling operator and a is a UV cutoﬀ. The “tree-level”
+beta function is readily found to be
+β(g) = a∂g
+∂a = (1 − ∆) g + O(g2)
+(527)
+which shows that if the scaling dimension of the operator of the tunneling particle is ∆ < 1, then
+this tunneling process is relevant, while is ∆ > 1 it is irrelevant.
+Kane and Fisher argued that there should be a crossover from the IR unstable weak-coupling
+ﬁxed point governed by quasiparticle tunneling to an IR stable strong coupling ﬁxed point gov-
+erned by electron tunneling. This crossover is reminiscent of the the crossover in quantum impu-
+rity problems such as the Kondo problem of a magnetic impurity in a metal [320, 32, 52, 53, 321].
+The main diﬀerence is that the impurity coupling is marginal and the crossover scale, the Kondo
+temperature TK, which is related to the coupling constant g as TK ∼ exp(−1/g), while in the FQH
+constriction the crossover scale is TK(g) ∼ g1/(1−∆). Kane and Fisher argued that this crossover
+can be viewed as a process that interpolated between a FQH ﬂuid with a weak constriction to a
+regime in which the Hall bar splits into two pieces with weak electron tunneling between them.
+It turns out that, after some manipulations, the model of the constriction can be mapped into
+a one-dimensional compactiﬁed boson ϕ on a semi-inﬁnite line, x ≥ 0, with a vertex operator
+acting at the boundary. The action of this system, known as boundary sine-Gordon is
+S = 1
+8π
+� ∞
+−∞
+dt
+� ∞
+0
+dx (∂µϕ)2 + Γqp
+� ∞
+−∞
+dt cos
+� �
+ν
+2ϕ(0, t)
+�
+(528)
+This theory has two ﬁxed points: a) the IR unstable ﬁxed point at Γqp = 0 where the ﬁeld
+Neumann boundary conditions at x = 0, ∂xϕ = 0, and b) an IR stable ﬁxed point at Γqp → ∞
+102
+
+where the ﬁeld has Dirichlet boundary conditions, ϕ = 2π √2/ν n (with n ∈ Z). It turns out that
+this is an integrable ﬁeld theory. Paul fendley, Andreas Ludwig and Hubert Saleur [322] used the
+thermodynamic Bethe Ansatz to calculate the diﬀerential tunneling conductance for the Laughlin
+state at ν = 1/3 as a function of voltage V and temperature T and obtained the full weak to strong
+crossover RG ﬂow. Detailed predictions of this theory have been veriﬁed experimentally by the
+work of Roddaro and coworkers [318, 319]. Point contact experiments were performed in the
+more challenging ν = 5/2 FQH state by Miller and coworkers [323] and Radu and coworkers
+[324]. They found that the MR state is the the one that best ﬁts their experimental results.
+The charge of particle can, in principle, be found directly by measuring the noise of a weak
+current. This process is known as shot noise. In the case of the constriction, the quasiparticle
+current operator is Iqp = 2eνΓqp sin
+� √ν/2φ + ω∗
+0t
+�
+, where ω∗
+0 = eνV/ℏ. The noise spectrum,
+S (ω), of the tunneling current qp is [325]
+S (ω) =
+� ∞
+−∞
+dt ⟨{Iqp(t), Iqp(0)}⟩ eiωt
+(529)
+Using the expression of the quasiparticle correlator of Eq.(514), one readily ﬁnds that, to leading
+order in Γqp, the noise spectrum is
+S (ω) = eν⟨Iqp⟩
+
+�
+1 − ω
+ω∗
+0
+�2ν−1
++
+�
+1 + ω
+ω∗
+0
+�2ν−1
+(530)
+In the limit of zero frequency the noise spectrum takes the shot noise form
+lim
+ω→0 S (ω) = 2e∗⟨Iqp⟩
+(531)
+where the expectation value of the tunneling current is
+⟨Iqp⟩ =
+2π
+Γ(2ν)eν|Γqp|2 ω∗
+0
+2ν−1
+(532)
+Chamon, Freed and Wen [326] calculated the exact noise spectrum for the case of a ν = 1/2
+bosonic FQH state and were able to investigate the crossover as well, and Fendley, Ludwig and
+Saleur [327] used the Bethe Ansatz to construct a soliton basis to compute the DC shot noise.
+These theoretical predictions were tested in tour-de-force experiments by de Picciotto and
+coworkers [253] and Samindayar and coworkers [254] who were able to measure the fractional
+charge of e/3 in the ν = 1/3 state and, with some caveats, e/5 in the ν = 2/5 state. Dolev and
+coworkers [328] went on to measure the charge from noise experiments in the nonabelian state
+at ν = 5/2 and obtained results consistent with e∗ = e/4 as predicted by theory.
+We will now consider experiments on Hall bars with two quantum point contacts created by
+two narrow gates transversal to the bar and separated at some distance d each other. Quantum
+devices of this type are Fabry-P´erot quantum interferometers. The way they operate is as follows
+[329, 298]. A current is injected in the bottom edge. At the ﬁrst quantum point contact (QPC)
+part if the current I1 tunnels to the opposite edge and the other part I2 goes on and tunnels at the
+second QPC. The FQH ﬂuid is assumed to occupy uniformly the region between the two QPC’s.
+There is some magnetic ﬂux Φ is that region which also contains a number of localized vortices
+(quasiparticles). When both currents rejoin at the top edge they interfere and the interference has
+information on the charge of the particles that tunnels through the Aharonov-Bohm eﬀect with
+the ﬂux Φ. The interference also has information on the fractional statistics of the quasiparticles
+that tunneled through their braiding with the static vortices. Chamon, Freed, Kivelson, Sondhi
+and Wen proposed a setup of this type to measure the fractional statistics of the quasiparticles for
+the abelian states [329]. They showed that the total current It = I1 + I2 is given by
+It = e∗|Γeﬀ|2 2π
+Γ(2ν) |ω0|2ν−1 sign(ω∗
+0)
+(533)
+where Γeﬀ is give by
+|Γeﬀ|2 = |Γ1|2 + |Γ2|2 + (Γ1Γ∗
+2 + Γ∗
+1Γ2) Fν
+�ω∗
+0d
+�
+�
+(534)
+Here Γ1 and Γ2 are the tunneling amplitudes at the two QPCs, ν = 1/m is the ﬁlling fraction, � is
+the velocity of the edge modes, d is the distance between the two QPCs and
+Fν(x) = √πΓ(2ν)
+Γ(ν)
+Jν−1/2(x)
+(2x)ν−1/2
+(535)
+103
+
+where Γ(z) is the Euler Gamma function and Jν−1/2(z) is the Bessel function of the ﬁrst kind. In
+the presence of Nq localized quasiparticles in the area between the two QPCs, the contribution
+of the phases of the tunneling matrix elements get shifted to
+Γ∗
+1Γ2 = ¯Γ∗
+1 ¯Γ2 exp
+�
+−2πi
+�
+ν Φ
+φ0
+− νNq
+��
+(536)
+where φ0 is the ﬂux quantum. In Eq.(536) the ﬁrst term in the phase shift is the Aharonov-
+Bohm eﬀect of the tunneling quasiparticles, while the second term in the phase shift 2πνNq is
+the contribution of the fractional statistics of the tunneling quasiparticle as its worldline braids
+with the Nq localized quasiparticles. This means that there is an interference contribution to
+the tunneling current (and also to the transmitted current) which is sensitive to both the charge
+and to the fractional statistics of the quasiparticles. This is this eﬀect that is being measured in
+experiments.
+Early attempts at doing this experiment were made by Camino and coworkers [330] but were
+diﬃcult to interpret partly due to subtle reasons related to the diﬃculty in controlling the actual
+area of the region comprised between the two QPCs [331]. Technical advances in the fabrication
+of these devices led in 2020 to the ﬁrst successful measurement of the fractional statistics of the
+quasiholes of the Laughlin state at ν = 1/3 (and also at the Jain state at ν = 2/5) by Nakamura,
+Liang, Gardner and Manfra [233].
+The nonabelian case is more subtle both theoretically and experimentally. The theory of the
+abelian interferometer of Chamon and coworkers was generalized to the nonabelian FQH states
+by Fradkin, Nayak, Tsvelik and Wilczek in 1998 [298]. The structure of the interferometer is the
+same as in the abelian case but the interference eﬀects are diﬀerent. In addition, in the case of
+the MR state, taside from the nonabelian quasiparticle σ ∼ χ exp(iφ/2
+√
+2) has charge e/4 and
+nonabelian fractional statistics, the MR state has two more abelian anyons, the charge neutral
+Majorana fermion χ and the charge e/2 Laughlin quasiparticle. The number of anyons and their
+properties are very diﬀerent in diﬀerent nonabelian FQH states.
+Ignoring for now the existence of more quasiparticles, let us focus now on the fundamen-
+tal anyon which is nonabelian. Fradkin and coworkers showed [298] considered a nonabelian
+quasihole that is injected to the bottom edge, tunnels at the ﬁrst QPC and arrives at the left end
+of the top edge in state |ψ⟩. If a second such quasiparticle is now injected but now tunnels to
+the top edge at the second QPC., arriving at the left end of the top edge in the state eiαBNq|ψ⟩,
+where α is the Aharonov-Bohm phase determined by the ﬂux Φ piercing the interferometer and
+BNq si the braiding operator os the second quasiparticle that is circling around the Nq localized
+quasiparticles in the interferometer. Then the tunneling conductance measured at the left exit of
+the top edge has an interference contribution
+σxx ∝ |Γ1|2 + |Γ2|2 + Re
+�
+Γ∗
+1Γ2eiα⟨ψ|BNq|ψ⟩
+�
+(537)
+The matrix element ⟨ψ|BNq|ψ⟩ is given in terms of the expectation value of the Wilson loop oper-
+ators of the tunneling quasiparticles braided with the Wilson loops of the Nq localized quasiparti-
+cles in the area of the interferometer. In 1989 Edward Witten [165] showed that this expectation
+value, which is computed in the nonabelian Chern-Simons gauge theory, is equal to a topolog-
+ical invariant of the braid known as the Jones polynomial VNq(e1π/4). Therefore, the oscillatory
+component of the tunneling current (and of the conductance) measures a topological invariant!
+Ref.[298] gave a general algorithm for the computation of this matrix element. However, in
+the particular case of the MR states, the non-abelian gauge theory associated with these states is
+the SU(2)2 Chern-Simons gauge theory. In this case, an explicit calculation Bonderson, Kitaev
+and Shtengel [332] and by Stern and Halperin [333] leads to the result
+σxx ∝|Γ1|2 + |Γ2|2,
+for Nq odd
+σxx ∝|Γ1|2 + |Γ2|2 + 2 |Γ1||Γ2|(−1)Nψ cos
+�
+α + arg
+�Γ2
+Γ1
+�
++ π
+4 Nq
+�
+,
+for Nq even
+(538)
+Here, Nψ = 1 when the Nq quasiparticles fuse into the state ψ and Nψ = 0 otherwise. The
+interference eﬀect is absent for Nq odd since an odd number of σ quasiparticles cannot fuse into
+the identity state I. This simple even-odd eﬀect is special for SU(1)2. In the general case and,
+in particular in the SU2)3 case which applies to the k = 3 Read-Rezayi (“Fibonacci”) state, the
+expressions are more complex.
+104
+
+At any rate, even in the MR state the situation is more complicated for two reasons. One
+is that the charge e/2 abelian Laughlin quasiparticle is always able to tunnel thus spoiling the
+even-odd eﬀect. the other complication is that the nonabelian FQH occurs in the N = 1 Landau
+level and the edge states of the nonabelian FQH state is surrounded by an abelian ν = 2 state,
+which makes accessing the interesting edge states more diﬃcult. Robert Willett has pioneered the
+fabrication and operation of the interferometer for the nonabelian state at ν = 5/2 [334, 335, 336].
+With some signiﬁcant caveats although there is solid evidence of nonabelian braiding, more work
+remains to be done on this system.
+11. Particle-Vortex Dualities in 2+1 dimensions
+11.1. Electromagnetic Duality
+The oldest form of duality in Physics is, perhaps, Dirac’s observation that in the absence of
+electric charges and currents Maxwell’s equations are invariant under the exchange of electric
+and magnetic ﬁelds, E → B and B → −E [114]. This observation led him to conjecture the
+existence of magnetic monopoles. In a relativistic invariant formulation, Maxwell’s equations in
+free space can be expressed in terms of the electromagnetic ﬁeld tensor Fµν and the dual ﬁeld
+tensor F∗
+µν
+∂µFµν = 0,
+∂µF∗
+µν = 0
+(539)
+where F∗
+µν = 1
+2ǫµνλρFλρ, and where ǫµνλρ is the fourth-rank antisymmetric Levi-Civita tensor, The
+ﬁrst Maxwell equation in Eq.(539) is just the wave equation in free Minkowski spacetime. The
+second equation in Eq.(539) is known as the Bianchi identity. The Bianchi identity is a constraint
+which implies that there are no magnetic monopoles and that the second rank antisymmetric ﬁeld
+strength tensor can be expressed in terms of the vector potential Aµ as Fµν = ∂µAν − ∂νAµ.
+This is an example of what in diﬀerential geometry is called Hodge duality, which relates a
+vector or, more generally a tensor ﬁeld to its Hodge dual. In general in D dimensions the Hodge
+dual of a (fully antisymmetric) tensor of rank p is an antisymmetric tensor of rank D − p. When
+contracted with the oriented inﬁnitesimal element of a p-dimensional hypersurface, dx1 ∧ dx2 ∧
+. . . ∧ dxp, an antisymmetric tensor of rank p, Fµ1µ2...µp, deﬁnes a p-diﬀerential form, called a p-
+form. Thus, diﬀerential forms embody the physically intuitive notions of circulation of a vector
+ﬁeld, ﬂux of a second rank tensor, etc. We will see below that duality transformations are closely
+related to these geometric notions of duality.
+11.2. Particle-Vortex Duality in 2+1 dimensions
+In this section we will extend the particle-vortex duality discussed in 2D in section 5.1.1 (see
+Eq.(75)) to 3D. In the ﬁeld theory interpretation, 2D is a 1+1-dimensional Minkowski spacetime
+and the XY model is a representation of a complex scalar ﬁeld of unit modulus. In the duality, the
+particles are the particle-like excitations of the complex scalar ﬁeld. The high temperature phase
+of the XY model is viewed as a partition function of a set of oriented loops that carry charge. the
+loops are the worldlines of the particles of the XY model.
+11.2.1. The 3D XY Model
+This picture can be seen as follows. Consider an XY model on a D-dimensional hypercubic
+lattice whose sites are labelled by {r}. At each site there is a periodic variable θ(r) ∈ [0, 2π). The
+partition function is
+ZXY =
+�
+r
+� 2π
+0
+dθ(r)
+2π
+exp
+
+1
+T
+�
+r
+�
+j=1,...,D
+�
+cos ∆jθ(r)
+�
+(540)
+where ∆jθ(r) = θ(r+e j)−θ(r), with j = 1, . . ., D, is the lattice diﬀerence, and T is the temperature
+(in the classical statistical mechanical picture). Since the interaction on each link is a periodic
+function of the phase diﬀerence ∆jθ(r), we can expand the Gibbs weight (for each link!) in a
+Fourier series or, what is the same, in the integer-valued representations of the group U(1) (which
+is the global symmetry of the XY model) to obtain
+ZXY =
+�
+r
+� 2π
+0
+dθ(r)
+2π
+∞
+�
+ℓj(r)=−∞
+exp
+−
+�
+r, j
+T
+2 ℓ2
+j(r) + i
+�
+r
+θ(r)∆jℓ j(r)
+
+(541)
+105
+
+where we used a Gaussian approximation for the modiﬁed Bessel function In(z), and ∆jℓ j(r) =
+�D
+j=1(ℓ j(r)−ℓ j(r−e j)) (where e j is the lattice unit vector along the direction j) denotes the lattice
+divergence. Integrating-out the phase variables θ(r) we obtain
+ZXY =
+�
+r
+∞
+�
+ℓj(r)=−∞
+�
+r
+δ(∆jℓ j(r)) exp
+−
+�
+r
+D
+�
+j=1
+T
+2 ℓ2
+j(r)
+
+(542)
+Therefore, the partition function is given by a sum over loops of conserved currents ℓ j(r) deﬁned
+on the links of the lattice, with a weight on each link ∝ exp(−ℓ2
+j/2β). These loops are the (lattice)
+worldlines of the particles of the complex scalar ﬁeld. In the phase where this representation
+is convergent, the complex scalar ﬁeld is massive, the XY model is gapped, and these particles
+have short range interactions. This picture is true in all dimensions, 3D included.
+On the other hand, in 2D the vortices are point-like events in Euclidean spacetime which in-
+teract with each other through long range, logarithmic, interactions. In the ﬁeld theory language,
+in 2D the point-like vortices are instantons which govern the low temperature phase of the XY
+model. In spacetime dimensions D > 2 the vortices become extended objects: vortex loops (or
+strings) in 3D, closed vortex surfaces in 4D, etc. In 3D the vortex loops are magnetic ﬂux tubes
+which interact with each other through a Biot-Savart type interaction, namely bits of vortices
+interact with each other with a 3D Coulomb interaction much in the same way as with loops of
+current in magnetostatics,
+Z3DXY =
+�
+{mj(˜r}
+�
+˜r
+δ(∆jm j(˜r)) exp
+−2π2
+T
+�
+˜r,˜r′
+3
+�
+j=1
+m j(˜r)G0(˜r − ˜r′)m j(˜r′) − α
+�
+˜r, j
+m2
+j(˜r)
+
+(543)
+where {˜r} labels the sites of the dual (cubic) lattice, the variables m j(˜r) take values on the integers,
+α is a (short-distance) vortex core energy, and G0(˜r − ˜r′) is the 3D lattice propagator (Green
+function) which at long distances has the standard form
+G0(x − x′) =
+1
+4π|x − x′|
+(544)
+In Eq.(543) the vortex loops are represented by the integer-valued conserved currents m j(˜r),
+which are naturally interpreted as the world lines of magnetic charges.
+We could have also reached the same result by following the line of reasoning that we used in
+section 5.1.1. Indeed, let θ(x) be the phase ﬁeld of a complex scalar ﬁeld φ(x) deep in its broken
+symmetry state where the amplitude of the ﬁeld φ(x) can be taken to be approximately ﬁxed. The
+partition function in this phase reduces to
+Z[aµ] =
+�
+Dθ exp
+�
+− 1
+2g
+�
+d3x
+�
+∂µθ − aµ
+�2�
+(545)
+where g is a coupling constant which in the XY model is proportional to the temperature. Here,
+much as in Eq.(70), the gauge ﬁeld aµ represents the vortices. Indeed, in 3D the vorticity is
+represented by a locally conserved vector ﬁeld ωµ
+ωµ = ǫµνλ∂νaλ = 2π
+�
+k
+mk
+µδ(x − xk)
+(546)
+where mk
+µ are the integer-valued vortex (magnetic) currents we used above. As before, the pe-
+riodic nature of the phase ﬁeld θ(x) implies that the vortex currents must be quantized and be
+integer-valued.
+Here too we can perform a Hubbard-Stratonovich transformation in terms of a vector ﬁeld bµ
+to ﬁnd a dual theory which now is
+Z(aµ) =
+�
+DbµDθ exp
+�
+−g
+2
+�
+d3x b2
+µ − i
+�
+d3x bµ(∂µθ − aµ)
+�
+(547)
+Upon integrating out the phase ﬁeld θ, which acts as a Lagrange multiplier which imposes the
+constraint ∂µbµ = 0. This constraint implies that we can write the ﬁeld bµ in terms of a dual
+gauge ﬁeld ϑµ
+bµ = ǫµνλ∂νϑλ
+(548)
+106
+
+Therefore, we can rewrite the partition function as
+Z(aµ) =
+�
+Dϑ exp
+�
+−g
+4
+�
+d3x f 2
+µν + i
+�
+d3x ωµϑµ
+�
+(549)
+where fµν = ∂µϑν − ∂νϑν is the ﬁeld strength of the dual gauge ﬁeld ϑµ. Notice that the vortex
+current ωµ is minimally coupled to the dual gauge ﬁeld ϑµ. If we now integrate-out the gauge
+ﬁeld ϑµ we obtain an expression for the weight in the path integral for a conﬁguration of vortices
+mk
+µ which is identical to the result of Eq.(543).
+What we have shown above is that in 3D the Goldstone phase of a complex scalar ﬁeld
+with coupling constant g is the dual of a Maxwell gauge theory. with coupling constant 1/g.
+Moreover, the periodicity of the phase ﬁeld implies that the dual gauge ﬁeld ϑµ is compact in
+the sense that its ﬂuxes must obey ﬂux quantization. It is straightforward to show that a charge
+operator Vn = exp(inθ) with electric charge n in the scalar ﬁeld theory is represented in the dual
+gauge theory by a magnetic monopole of magnetic charge n.
+In summary we showed that the 3D XY model (equivalent to the theory of the complex scalar
+ﬁeld) can be written in terms of two diﬀerent models of loop conﬁgurations. We saw that the
+XY model is a theory of particle (electric) loops in the high temperature phase and of vortex
+(magnetic) loops in the low temperature phase. However the particle loops have short range
+interactions while the vortex loops have long range interactions. Thus, the 3D XY model is not
+self-dual. In spite of having a representation in terms of vortex loops, the 3D XY model is in
+reality has a very diﬀerent behavior that the 2D system. The Kosterlitz-Thouless theory describes
+the phase transition in terms of a vortex-antivortex unbinding transition upon which the vortices
+proliferate. In addition, in 2D the Goldstone phase is actually a line of ﬁxed points, there is never
+true long range order but, instead, power-law correlations of the physical observables. In the
+Kosterlitz-Thouless theory the disordered phase arises the strong ﬂuctuations of the phase ﬁeld
+due to the proliferation of vortices while, at the local level, the order parameter still has a ﬁnite
+magnitude. A consequence of this behavior is that the superﬂuid density has a universal jump at
+the Kosterlitz-Thouless transition [337].
+In contrast in the 3D XY model the Goldstone phase is a true phase with long range order.
+The theory has a continuous phase transition (the thermodynamic superﬂuid transition) at which
+the superﬂuid density vanishes continuously. This means that the phase transition of the 3D XY
+is better described by the Wilson-Fisher ﬁxed point of a complex scalar ﬁeld [30, 48], rather than
+by a vortex proliferation picture of the Kosterlitz-Thouless theory [84, 85]. In particular, since
+this theory does not have magnetic monopoles, the vortex loops cannot proliferate as they do in
+2D. Instead, as the phase transition is approached, the vortex loops grow large in size but also
+become fractal-like objects whose eﬀective core size diverges as the transition is approached. In
+fact, qualitative vortex proliferation arguments readily lead to the incorrect conclusion that the
+transition should be (strongly) ﬁrst order [338].
+In spite of these diﬀerences, particle-vortex duality still plays an important role in 3D by
+relating the 3D XY model to another theory which is its dual under electromagnetic duality. Here
+electromagnetic duality is understood as the exchange of the electric worldlines (electric loops)
+and the magnetic loops (vortex loops) while exchanging strong and weak coupling, T ↔ 1/T, in
+this case between diﬀerent theories. This duality was investigated by Michael Peskin [339], by
+Paul Thomas and Michael Stone [338], and by Chandan Dasgupta and Bertrand Halperin [340].
+11.2.2. Scalar QED in 3D
+These authors considered a theory of a superconductor represented by a complex scalar ﬁeld
+ϕ(x), coupled to a ﬂuctuating electromagnetic (Maxwell) ﬁeld, also known as the abelian Higgs
+model, or scalar electrodynamics (scalar QED) aµ, with µ = 1, 2, 3 The partition function of this
+theory is
+ZSC =
+�
+DϕDϕ∗Daµ exp
+�
+−
+�
+d3x L[ϕ, ϕ∗, aµ]
+�
+(550)
+where
+L = |Dµ(a)ϕ|2 + m2|ϕ|2 + u|ϕ|4 + 1
+4e2 f 2
+µν
+(551)
+where the covariant derivative is Dµ(a) = ∂µ + iqaµ, with the integer q being the charge of the
+scalar ﬁeld (in units of the coupling constant e), and fµν = ��µaν − ∂νaµ is the ﬁeld strength. This
+theory is known as (Euclidean) scalar electrodynamics (or, the abelian Higgs model).
+107
+
+The scalar electrodynamics was extensively studied using the perturbative renormalization
+group within the epsilon expansion (near 4 dimensions) which leads to the conclusion that it
+has a weakly (ﬂuctuation-induced) ﬁrst order phase transition [109, 341]. Scalar QED of an N-
+component scalar ﬁeld coupled to a Maxwell gauge ﬁeld has a continuous phase transition with
+a non-trivial ﬁxed point for N ≥ Nc ≃ 183 (!) [342]. Since these results rely on perturbation
+theory (or in the large-N limit) it was long suspected that the physics may be diﬀerent in D = 3.
+We will see that particle-vortex duality provides the answer to this question [340].
+We begin by writing the lattice version of Eq.(550) which is obtained by coupling the XY
+model of Eq.(540) to a dynamical (Euclidean) Maxwell ﬁeld
+ZSC =
+�
+r
+� 2π
+0
+dθ(r)
+2π
+� ∞
+−∞
+daµ(r)
+2π
+exp
+�
+−S (θ, aµ)
+�
+(552)
+where the action S (θ, aµ) is
+S (θ, aµ) = − 1
+T
+�
+r,µ
+cos
+�
+∆µθ(r) − qaµ(r)
+�
++ 1
+4e2
+�
+r,µ,ν
+�
+∆µaν(r) − ∆νaµ(r)
+�2
+(553)
+In this action we assumed that the abelian gauge ﬁelds do not have monopole conﬁgurations.
+The partition function of Eq.(550) admits a representation as a sum over loops. We can then
+used the same line of reasoning that led to Eq.(542) and write the partition function as a sum over
+loops of the worldlines of the particles (charges) of the complex scalar ﬁeld
+ZSC
+Zgauge
+0
+=
+�
+r
+∞
+�
+ℓµ(r)=−∞
+δ(∆µℓµ(r)) exp
+−
+�
+r,µ
+T
+2 ℓ2
+µ(r)
+
+�
+exp
+iq
+�
+r,µ
+ℓµ(r)aµ(r)
+
+�
+a
+(554)
+where ⟨O[a]⟩a is the expectation value of the operator O[a] over the gauge ﬁelds aµ, and Zgauge
+0
+is
+the partition function of the free gauge ﬁelds. After computing this free-ﬁeld expectation value
+we obtain the result
+ZSC
+Zgauge
+0
+=
+∞
+�
+{ℓµ(r)}=−∞
+δ(∆µℓµ(r)) exp
+−
+�
+r,µ
+T
+2 ℓ2
+µ(r) − q2e2
+2
+�
+r,µ
+�
+r′,ν
+ℓµ(r) Gµν(r − r′) ℓν(r′)
+
+(555)
+where
+Gµν(r − r′) = ⟨aµ(r)aν(r′)⟩
+(556)
+is the Euclidean propagator of the free gauge ﬁelds. Since the conﬁgurations of the worldlines,
+the currents represented by ℓµ(r), are conserved and satisfy the local constraint ∆µℓµ = 0, the
+second term in the exponent of Eq.(555) is gauge invariant. Hence, we can use the propagator in
+the Feynman gauge
+Gµν(r − r′) = δµνG0(r − r′)
+(557)
+where G0(r− r′) is the 3D lattice propagator which has the same Coulomb form at long distances
+already given in Eq.(544).
+Therefore we can write the partition function of the scalar QED model (the superconductor)
+as a sum over worldline loop conﬁgurations of the particles of the scalar ﬁeld in the equivalent
+form
+ZSC
+Zgauge
+0
+=
+∞
+�
+{ℓµ(r)}=−∞
+δ(∆µℓµ(r)) exp
+−
+�
+r,µ
+T
+2 ℓ2
+µ(r) − q2e2
+2
+�
+r,r′,µ
+ℓµ(r) G0(r − r′) ℓµ(r′)
+
+(558)
+which is the same as the partition function of the 3D XY model in the broken symmetry state,
+given in Eq.(543), with the identiﬁcation of the particle loops of the abelian Higgs model with the
+vortices of the 3D XY model and a relation between the coupling constants. Hence we obtain the
+duality between the broken symmetry phase of the 3DXY model (“low T”) and the symmetric
+phase of the abelian Higgs model (“high T”)
+ℓµ ↔ mµ,
+q2e2 ↔ 2π2
+T ,
+T
+2 ↔ α
+(559)
+where the quantities on the left hand side of the identiﬁcations refer to the 3D abelian Higgs
+model and the quantities on the right hand side to the 3D XY model.
+108
+
+We can now repeat the same analysis but in the broken symmetry of the abelian Higgs model.
+In this phase the gauge ﬁeld aµ becomes massive by the Higgs mechanism (or, what is the same,
+by the Meissner eﬀect of the superconductor), and in this phase the vortex loops mµ of the
+complex scalar ﬁeld have short range interactions. This phase then is mapped onto the unbroken
+phase of the 3DXY model with the screened vortex loops of the abelian Higgs model identiﬁed
+with the particle loops of the 3D XY model, and with the same identiﬁcations between the
+coupling constants of the two theories given in Eq.(559). The identiﬁcation between the two
+partition functions implies that the phase transitions must be the same. In other words, the
+duality implies that the 3D abelian Higgs model has a continuous phase transition with the same
+ﬁxed point as that of the complex scalar ﬁeld in 3D. The only diﬀerence is that the phases are
+reversed and, for this reason, this is sometimes called an “inverted” 3D XY transition [340].
+11.2.3. The duality mapping
+We can summarize these results with the following identiﬁcation of two Lagrangians (in real
+time, Minkowski signature)[208]
+|∂µ(B)φ|2 − m2|φ|2 − u|φ|4 ↔ |∂µ(a)ϕ|2 + m2|ϕ|2 − u|ϕ|4 − 1
+4e2 f 2
+µν + 1
+2πǫµνλaµ∂νBλ
+(560)
+The left hand side of this equivalence is the Lagrangian of the 3D complex scalar ﬁeld φ and Bµ
+is a background electromagnetic gauge ﬁeld. The right hand side is the Lagrangian of the abelian
+Higgs model with a scalar ﬁeld ϕ and a U(1) gauge ﬁeld aµ. Notice that the mass terms of the
+two sides are inverted reﬂecting the reverse order of their two phases. This identiﬁcation also
+shows that the current of the scalar ﬁeld jµ = − i
+2(φ∗∂µφ − φ∂µφ∗) is identiﬁed as
+jµ ↔ 1
+2πǫµνλ∂νaλ
+(561)
+The ﬁeld operator of the complex scalar ﬁeld creates worldlines (as opposed to loops) of charged
+particles. In the dual abelian Higgs theory this operator is identiﬁed with an operator that creates
+a magnetic monopole of the U(1) gauge ﬁeld aµ with unit magnetic charge. It is easy to see that in
+the broken symmetry phase of the abelian Higgs model a monopole-antimonopole pair creation
+operator decays exponentially with distance due to the ﬂux expulsion eﬀect (the Meissner eﬀect
+of a superconductor). This means that the monopoles are conﬁned with a linear potential. This
+is the same behavior of the complex scalar ﬁeld φ in its symmetric phase where the propagator
+of the complex scalar ﬁeld decays exponentially with distance. Conversely, in the broken sym-
+metry phase of the complex scalar ﬁeld theory, the ﬁeld operator φ condenses and its propagator
+approaches the value |⟨φ⟩|2 at long distances. This is the same behavior one readily ﬁnds for the
+monopole operator of the abelian Higgs model in the symmetric phase .
+11.3. Bosonization in 2+1 dimensions
+The particle-vortex duality that we discussed in section 11.2 essentially has been understood
+since the 1980s. In section 7.4 we discussed in detail the problem of bosonization in 1+1 dimen-
+sions whic is a mapping between a massless Dirac fermion and a massless compactiﬁed scalar
+ﬁeld. As we noted, this mapping is a powerful theoretical tool to investigate non-perturbatively
+the structure of many theories in 1+1 dimensions of great physical interest. This success has
+motivated a sustained eﬀort to extend these concepts to higher dimensions where the problem in
+many ways is more subtle.
+Fermi systems in dimensions higher diﬀer from the 1+1-dimensional case in two signiﬁcant
+ways. Due to the kinematic restriction of one space dimension a massless relativistic fermion
+and a theory of fermions (relativistic or not) at ﬁnite density are mostly equivalent to each other.
+The reason is quite simple. Consider a free massless Dirac theory in 1+1 dimensions. As we
+saw, it is equivalent to a theory of two chiral fermions, the right and left moving component of
+the spinor. If we add a chemical potential µ, the right and left moving states will be ﬁlled up to
+µ, which is the Fermi energy. While in space dimensions d > 1 a ﬁnite chemical potential leads
+to a ﬁnite fermi surface, in d = 1 space dimensions the Fermi “surface” is just two points, for
+the right and left moving fermion modes respectively. The Dirac Hamiltonian at ﬁnite chemical
+potential is
+H = ψ†(−i)α∂xψ − µψ†ψ
+(562)
+109
+
+Under a chiral transformation with angle θ(x), the ﬁeld operators change as ψR → eiθψR and
+ψL → e−iθψL and the Hamiltonian becomes
+H = ψ†(−i)α∂xψ − (µ − ∂xθ) ψ†ψ
+(563)
+Clearly if we choose ∂xθ = µ (or, what is the same, θ = µx) the explicit dependence on the
+chemical potential has been cancelled and the transformed Dirac Hamiltonian is the same as the
+Hamiltonian at zero chemical potential. However under this transformation the mass operators
+transform as ¯ψψ → cos(2µx) ¯ψψ and i ¯ψγ5ψ → sin(2µx) i ¯ψγ5. In the non-relativistic context this
+change amounts to a modulation of the charge density with wave vector Q = 2pF(µ).
+In contrast, in space dimensions d > 1 the massless Dirac system and a system of fermions
+at ﬁnite density (relativistic or not) are no longer equivalent to each other as the latter system has
+a Fermi surface (of co-dimension d − 1) wile the former system does not. This diﬀerence has
+profound eﬀects on their dynamics: the system of fermions at ﬁnite density is, at least in the weak
+coupling regime, is expected to be a Fermi liquid (except for an instability to a superconducting
+state even for inﬁnitesimal attractive interactions) while the massless Dirac theory is stable as all
+local interactions are irrelevant.
+11.3.1. Bosonization of the Dirac theory in 2+1 dimensions
+We will now turn to the problem of Bose-Fermi mappings in relativistic systems in 2+1
+dimensions. This problem is of great interest both in Condensed Matter Physics and in High
+Energy Physics.
+Bosonization in 2+1 dimensions is a more subtle problem than in 1+1 dimensions that we
+discussed in section 7.4. There we saw, in 1+1 dimensions bosonization consists of a set of
+operator identities relating two free ﬁeld theories, a theory of massless Dirac fermions and a
+compactiﬁed massless free scalar ﬁeld φ(x). The success of that program is largely based on the
+fact that both theories are massless (and hence are scale and conformally invariant), on the chiral
+anomaly, and on the relation between the gauge current of the Dirac theory with the topological
+current of the compactiﬁed boson.
+The physics in 2+1 dimensions (and in general) is very diﬀerent. For instance, instead of
+the chiral anomaly in 2+1 dimensions theories of relativistic fermions have a parity anomaly,
+discussed in section 10.1.6. We will also see that although in 2+1 dimensions there are theories
+of free massless Dirac fermions and free massless compactiﬁed bosons they are no longer dual to
+each other. For these and other reasons it has been diﬃcult to extend the bosonization program
+to 2+1 dimensions.
+One of the ﬁrst bosonization constructions for a theory of massive (relativistic) Dirac ﬁelds
+was put forth by Alexander Polyakov in 1988 [168, 167]. For reasons that no longer relevant,
+Polyakov considered a theory of CP1 complex scalar ﬁelds (in its unbroken phase) coupled to
+a U(1)1 Chern-Simons gauge theory. Nevertheless, his main argument, with some corrections
+associated with the parity anomaly, still holds correct.
+Here we will follow the work of Hart Goldman and myself [343] and consider instead a
+theory of a single massive complex ﬁeld coupled to a U(1)1 Chern-Simons gauge theory whose
+Lagrangian density is
+L = |Dµ(Aµ)φ|2 − m2|φ|2 − λ|φ|4 + 1
+4πǫµνλA µ∂νA λ
+(564)
+In essence, this theory is a relativistic version of what we did in section 10.3.7 where we mapped
+non-relativistic fermions to (also non-relativistic) bosons whereas here the theory is relativistic
+and we are mapping bosons to fermions. In his work Polyakov actually considered the problem
+of the propagator of a free massive scalar ﬁeld coupled to the Chern-Simons gauge ﬁeld. In this
+limit, the propagator G(x, x′) is the transition amplitude from x to x′ (in Minkowski spacetime).
+In the case in which φ(x) is a free massive ﬁeld this propagator can be expressed in the for of a
+Feynman path-integral as a sum over all open oriented paths {Px,x′} with endpoints at x and x′
+which in Euclidean spacetime is
+Gγ(x,x′) =
+�
+{Px,x′ }
+exp(−mL(Px,x′))
+�
+exp
+�
+i
+�
+Px,x′
+dzµA µ(z)
+� �
+U(1)1
+(565)
+where the expectation value is over the Chern-Simons gauge ﬁelds Aµ of Eq.(564).
+110
+
+Let us consider now the partition function of the bosons which is a sum over all closed paths
+{P}, which we will assume to be non-intersecting (which requires a short distance repulsion)
+Z =
+�
+{P}
+exp(−mL(P))
+�
+exp(i
+�
+P
+dzµA µ(z))
+�
+U(1)1
+(566)
+The expectation value is given by
+�
+exp
+�
+i
+�
+P
+dzµA µ(z)
+� �
+U(1)1 = exp
+�
+−1
+2
+�
+P
+�
+P
+dxµdyν⟨Aµ(x)Aν(y)
+�
+≡ exp(iπW(P))
+(567)
+where W(P) is the writhe of the closed path P. The computation of the writhe requires a regu-
+larization. One possible regularization is to thicken the path which is equivalent to including a
+Maxwell term for the gauge ﬁeld in the Lagrangian. In general, the writhe of the path P is
+W(P) = S L(P) − T(P)
+(568)
+where S L(P) is an integer-valued topological invariant called the self-linking number and T(P)
+is the twist (or torsion) of the path. The twist T(P) is a Berry phase which depends on the
+coordinates of the space in which the path P is embedded, and it is not a topological invariant.
+We will now compute the Berry phase T(P). Let ˆe(s) be a unit vector tangent to the path
+P and where we parametrized the closed path P by a coordinate s ∈ [0, L]. The closed path
+P is the boundary of a surface Σ we can take to be a disk. We will write the Berry phase by
+extending ˆe(s) smoothly to the interior of Σ as ˆe(s, u), where 0 ≤ u ≤ 1 and ˆe(s, u = 1) = ˆe(s)
+and ˆe(s, u = 0) = ˆe0 ≡ constant. With these deﬁnition the Berry phase is
+T(P) ≡ W(ˆe) = 1
+2π
+� L
+0
+ds
+� 1
+0
+du ˆe · ∂sˆe × ∂uˆe
+(569)
+which is deﬁned modulo an integer.
+In this formulation, in the bosonic theory the amplitude for a path of length L with tangent
+vector ˆe(s) with endpoints at x and x′ is
+G(x − x′) =
+� ∞
+0
+dL
+�
+D ˆe(s) δ
+�
+1 − |ˆe|2�
+δ
+�
+x − x′ −
+� L
+0
+ds ˆe(s)
+�
+exp(−|m|L ± iπW(ˆe)) (570)
+In momentum space we ﬁnd
+G(p) =
+� ∞
+0
+dL
+�
+Dˆe δ(1 − |ˆe|2) exp(−|m|L ± iπ W(ˆe)) exp(ipµ
+� L
+0
+ds ˆeµ(s))
+(571)
+By inspection of Eq.(569) we see that this is the same expression that we looked in the theory of
+the path integral for spin in section 4.3.2. for spin S = 1/2 particle in a magnetic ﬁeld bµ = ±2pµ.
+The equation of motion for ˆe is
+∂sˆeµ = ±2ǫµνλˆeλ = i[H, ˆeµ]
+(572)
+where H = ∓pµˆeµ is the Hamiltonian. Upon quantization, ˆeµ satisﬁes the commutation relations
+[ˆeµ, ˆeν] = 2iǫµνλˆeλ
+(573)
+This means that we should make the identiﬁcation ˆeµ → σµ where σµ are the three 2 × 2 Pauli
+matrices. Up to rescaling of the mass m → M, upon performing the path integral of Eq.(571) we
+ﬁnd that G(P) is the (Euclidean) Dirac propagator [168]
+G(p) =
+1
+ipµσµ − M
+(574)
+Using these results the partition function of Eq.(566) becomes
+Zfermion = det[i/∂ − M] =
+�
+DJ δ(∂µJµ) exp �−|m|L[J] − isgn(M)πΦ[J]�
+(575)
+where Φ[J] = W[J], deﬁned in Eq.(568).
+111
+
+We will return to this construction shortly below when we include the eﬀects of the parity
+anomaly.
+Early attempts at deriving a mapping for a theory of Dirac fermions in 2+1 dimensions were
+based on their behavior in the presence of gauge ﬁelds and the associated parity anomaly, dis-
+cussed in section 10.1.6. These early theories are essentially a hydrodynamic description of the
+Dirac theory deep in the massive phase. With minor diﬀerences, the same results were derived
+independently (and simultaneously) by two groups, Fidel Schaposnik and myself [344] and Cliﬀ
+Burgess and Francisco Quevedo [345]. This approach was reexamined more recently by A. Chan,
+T. Hughes, S. Ryu and myself in 2016 in a derivation of a hydrodynamic eﬀective ﬁeld theory for
+topological insulators in diﬀerent dimensions [346, 347]. These derivations are similar in spirit
+to what we discussed in section 10.3.6 for the fractional quantum Hall eﬀect. We will follow the
+approach of Ref.[346] for the special case of a Chern insulator 2+1 dimenxions.
+The Dirac theory, in any dimension, is globally gauge invariant. By Noether’s theorem this
+means that it has a locally conserved current, jµ = ¯ψγµψ which thus satisﬁes ∂µ jµ = 0. Here
+too, this means that we can write jµ = ǫµνλ∂νbλ, where bµ is a gauge ﬁeld. We expect that the
+eﬀective action of the gauge ﬁeld bµ should be local, gauge invariant and, in this case, relativistic
+invariant. In 2+1 dimensions the eﬀective action should break time reversal invariance and hence
+it should have a Chern-Simons term.
+We will consider the free massive Dirac theory coupled to a background probe gauge ﬁeld Aµ.
+The Lagrangian is L = ¯ψ(i/(D) − M)ψ, where Dµ = ∂µ + iAµ. As we saw in section 10.1.6, gauge
+invariance and locality require that we have an even number of Dirac fermions. Here we will
+assume that one of the Dirac fermions is massive and acts as a regulator. The partition function
+Z[Aµ] =
+�
+D ¯ψDψ exp(iS F[ ¯ψ, ψ, Aµ])
+(576)
+By deﬁnition the expectation value of a product of current operators jµ can be obtained by func-
+tional diﬀerentiation of the partition function with respect to Aµ.
+The partition function is gauge invariant, i.e.
+Z[Aµ = Z[Aµ + aµ]
+(577)
+where the vector ﬁeld aµ is a pure gauge, aµ = ∂µφ and fµnu = ∂µa − ν − ∂µaµ = 0. Hence, aµ is
+said to be ﬂat. Therefore, up to a normalization
+Z[Aµ] =
+�
+D[aµ]pure Z[Aµ + aµ]
+=
+�
+Daµ δ(fµν = 0) Z[Aµ + aµ]
+=
+�
+DaµDbµ Z[Aµ + aµ] exp
+�
+− i
+2
+�
+d3x ǫµνλ bµ fνλ
+�
+(578)
+where in the last equality we introduced a representation of the delta function and the vector ﬁeld
+bµ plays the role of a Lagrange multiplier ﬁeld. using the invariance of the integration measure
+under aµ → aµ − Aµ we ﬁnd
+Z[Aµ] =
+�
+DaµDbµZ[aµ] exp
+�
+− i
+2
+�
+d3x ǫµνλ bµ (fνλ − Fνλ)
+�
+(579)
+where Fµν is the ﬁeld strength of the external ﬁeld Aµ. From these identities and by diﬀerentiation
+of Eq.(579) it follows that a general expectation value of products of currents is
+⟨jµ(x) jν(y) . . .⟩ =
+δ
+δAµ(x)
+δ
+δAν(y) . . .Z[Aµ] = ⟨ǫµαβ∂αbβ ǫνγδ∂γbδ . . .⟩
+(580)
+which means that, at the operator level, we can identify [344]
+jµ(x) ⇔ ǫµνλ∂νbλ
+(581)
+This result is actually general. The only diﬀerence is the nature of the ﬁeld b isn diﬀerent di-
+mensions: in 1+1 is a pseudo-scalar, in 2+1 is a vector (gauge) ﬁeld, in 3+1 is an anti-symmetric
+(Kalb-Ramond) tensor ﬁeld, etc.
+112
+
+Returning to the partition function of Eq.(579) we ﬁnd, using the result of Eq.(324) applied
+the partition function Z[aµ], that Z[Aµ] is given by
+Z[Amu] =
+�
+DaµDbµ exp(i
+�
+d3x Leﬀ[aµ, bµ, Aµ])
+(582)
+where the eﬀective Lagrangian is
+Leﬀ = −ǫµνλ bµ ∂νaλ + s
+4πǫµνλaµ∂νaλ −
+1
+4geﬀ
+fµν f µν + Aµǫµνλ∂νbλ + . . .
+(583)
+where fµν = ∂µaν − ∂νaµ. Here s = 1 in the Chern insulator, s = 0 in the trivial insulator and
+s = 1/2 at the quantum critical point, and geﬀ an eﬀective coupling constant (with dimensions of
+length−1). Here we included the eﬀects of the parity anomaly. We recognize that the ﬁrst term of
+the eﬀective Lagrangian of Eq.(583) is a BF term.
+In Ref.[346] a similar result is also derived for a 3+1-dimensional topological insulator. The
+main diﬀerences are that there is no parity anomaly but an axial anomaly and that the Lagrange
+multiplier ﬁeld is a Kalb-Ramond ﬁeld,
+Leﬀ[aµ, bµν, Aµ] = ǫµνλρ bµν ∂λaρ +
+θ
+8π2 ǫµνλρ∂µaν∂λaρ − 1
+4g2 fµν f µν + Aµǫµνλρ∂νbλρ + . . . (584)
+where θ = π in the topological insulator and θ = 0 in the trivial insulator.
+Although these results are correct, they do not give a full bosonization mapping. In particular,
+these results do not identify a ﬁxed point for the dual bosonic theory and, along with it, a full
+mapping of the observables. Progress on this problem has only been achieved in the past few
+years both in the high energy literature [348, 349, 350, 351, 352, 199, 353, 354], and in the
+condensed matter physics literature [355, 356, 357, 358, 359]. Much of that new insight was
+presented in a 2016 insightful paper by Nathaniel Seiberg, T. Senthil, Chong Wang and Edward
+Witten [208]. Many of the dualities are actually conjectures supported by strong consistency
+checks. A derivation of the basic bosonization duality based on loop models was constructed by
+Hart Goldman and myself [343].
+The basic conjectured bosonization duality is a mapping of a free Dirac fermion to a gauge
+complex scalar ﬁeld with a Chern-Simons term [352, 208]. We can think of the Dirac theory as
+either being deﬁned entirely in 2+1 dimensions or as a being deﬁned at the boundary of a non-
+trivial 3+1 dimensional topological insulator. In the ﬁrst scenario we need to take into account
+the contribution of the fermionic doublers (this is what happens in the case of a lattice theory)
+or as the Pauli-Villars heavy fermionic regulators. In both cases, the additional heavy degrees of
+freedom cancel the anomaly of the 2+1-dimensional Dirac fermion, see section 10.1.6. In the
+second scenario, there is only one Dirac fermion at the boundary and the bulk θ-term cancels the
+anomaly of the boundary theory, see section 10.2.3. With these provisos, the Lagrangian LA of
+the free massive (or massless) Dirac fermion in 2+1 dimensions coupled to the electromagnetic
+gauge ﬁeld Aµ is
+LA = ¯ψ(i /D(Aµ) − M)ψ − 1
+8πǫµνλAµ∂νAλ
+(585)
+where M is the Dirac mass. We will call this Theory A. Here /Dµ(A) = γµ(∂µ − iAµ) and Aµ
+is a background (non-dynamical) gauge ﬁeld. The last term in Eq.(585) is the Chern-Simons
+term with the 1/2-quantized coeﬃcient, the usual short-hand for the η-invariant term of Eq.(342)
+needed to cancel the parity anomaly.
+From the eﬀective ﬁeld theories we discussed above we know that the fermionic current
+maps to the curl of a gauge ﬁeld, see Eq(581). Then, the conjectured bosonic dual is given by the
+Lagrangian of Eq(564) with an extra term for the (dual) coupling to the electromagnetic gauge
+ﬁled. We will call this Theory B whose Lagrangian LB is
+LB = |Dµ(Aµ)φ|2 − m2|φ|2 − λ|φ|4 + 1
+4πǫµνλA µ∂νA λ + 1
+2πǫµνλAµ∂νA λ
+(586)
+To check the consistence we ﬁrst observe that both theories are anomaly free. By functional
+diﬀerentiation of the two partition functions we check that we get the correct mapping for the
+fermionic current, jµ ↔ ǫµνλ∂νA λ. This mapping implies that the charge density of Theory A
+maps onto the gauge ﬁeld ﬂux of Theory B, which is electromagnetic duality. We see that this
+bosonization is a relativistic version of ﬂux attachment.
+113
+
+Let us now identify the mapping of the phases of both theories. If the Dirac mass M < 0,
+Theory A describes an anomalous quantum Hall insulator. Integrating out the massive fermions
+we fund that the eﬀective action for the electromagnetic gauge ﬁeld Aµ is a U(1)1 Chern-Simons
+theory. This means that in this phase σxy = −1/(2π) (in units in which e = ℏ = c = 1). Looking
+now at Theory B we see that if m2 > 0, the scalar ﬁeld is in the unbroken phase, and it is massive.
+In this phase we set φ = 0 and ﬁnd that the low energy resulting theory if just a U(1)1 Chern-
+Simons gauge theory coupled to the curl of the electromagnetic ﬁeld. Upon integrating out the
+Aµ gauge ﬁeld we ﬁnd that the eﬀective action of Aµ is also a U(1)1 Chern-Simons term, which
+implies that σxy = −1/(2π).
+Conversely, for M > 0 the eﬀective electromagnetic action of the fermionic theory is a
+Maxwell term (the Chern-Simons term canceled). This phase is a trivial insulator. Looking now
+at Theory B we see that for m2 < 0 this theory is in its Higgs phase where ⟨φ⟩ � 0 and the gauge
+ﬁeld Aµ now has a mass term, ∝ A 2
+µ . In the low energy limit Aµ → 0, and the Hall conductivity
+vanishes, consistent to what is expected from Theory A.
+On the other hand, for M = 0 Theory A is at a quantum critical point with a very simple CFT
+structure but a non-vanishing Hall conductivity σxy = 1/(4π). It is natural to map this CFT to
+a Wilson-Fisher (WF) ﬁxed point of Theory B. At the WF one sets the (renormalized!) mass
+m2
+R = 0 (not the bare mass). In the absence of the Chern-Simons gauge ﬁeld this WF ﬁxed point
+in well understood form high quality (ﬁve loop!) epsilon expansion calculations (enhanced with
+Borel resummation) [49], and by more recent numerical Conformal Bootstrap methods [360].
+However, not much is known of the gauged version of the CFT of Theory B since it cannot
+be accessed by either methods. If the mapping to Theory A is correct it should have a Hall
+conductivity of σxy = 1/(4π). This conjectured value hast not yet been conﬁrmed.
+Consider now a monopole operator of the gauge ﬁeld Aµ of Theory B. We will denote this
+operator MA . The Chern-Simons term is not gauge invariant in a monopole background. To put
+it diﬀerently it has charge 1 under the Chern-Simons gauge ﬁeld Aµ. It also has charge 1 under
+the electromagnetic ﬁeld Aµ. Likewise the scalar ﬁeld φ has charge 1 under the Chern-Simons
+gauge ﬁeld Aµ. The operator φ†MA is gauge-invariant under the Chern-Simons gauge ﬁeld Aµ
+(i.e. it has charge 0). This composite operator has electromagnetic charge 1 (which it inherited
+from the monopole). Moreover, it has spin-1/2 required by the Wu-Yang construction [151].
+In other words, the operator φ†MA has the same quantum numbers as the Dirac fermion and
+are identiﬁed by this conjecture. Notice, however, that the free massless Dirac ﬁeld has scaling
+dimension ∆ψ = 1 in 2+1 dimensions. The conjecture implies the composite operator φ†MA
+should also have scaling dimensions 1 at the (gauged) WF ﬁxed point of Theory B. Many of these
+conjectures have been veriﬁed in non-abelian versions of Theory A and Theory B: a theory of
+free massless Dirac fermions coupled to a Chern-Simons gauge theory with gauge group SU(N)k
+and a theory of complex scalars coupled to a non-abelian Chern-Simons gauge theory with gauge
+group SU(k)N, both in the limits N → ∞ and k → ∞ with N/k ﬁxed [348, 349, 350, 351, 352].
+The particle-vortex duality discussed in section 11.2 combined with the fermion-vortex du-
+ality we sketched here provide a web of dualities. These identiﬁcations constitute a powerful
+non-perturbative tool which has led to many signiﬁcant developments. For instance, Goldman
+and myself [361] used the web of dualities to explain the experimentally observed self-duality
+at fractional quantum Hall plateau transitions, which was a long standing puzzle. It has also
+provided a powerful new tool to derive eﬀective ﬁeld theories of non-abelian fractional quantum
+Hall states [301] and even to propose novel states with a single (Fibonacci) anyon [303].
+11.3.2. Bosonization of the Fermi Surface
+A form of bosonization has been developed for the case of systems of fermions with a Fermi
+surface. Dense Fermi systems at suﬃciently weak coupling are well described by the Landau
+theory of the Fermi liquid [13, 3, 194, 260, 259]. In this regime, the system of fermions has a
+collective mode, a bound state of particles and holes, which at long wavelengths is well described
+by the random phase approximation (RPA) [362]. However, in space dimensions d > 1 there is no
+kinematical restriction and quasi-particles and quasi-holes may move in their separate ways. The
+result is that, in addition to the collective modes, there is a low-energy spectrum of renormalized
+but essentially free quasi-particles. Because of the existence of this quasi-particle spectrum the
+collective modes generally (but not always) decay into particle-hole pairs resulting in a ﬁnite
+lifetime of the collective modes.
+Superﬁcially the particle-hole collective modes are similar to the scalar ﬁeld of the bosonized
+theory in d = 1. To an extent it has been possible to “bosonize the Fermi surface” and to treat it
+as a quantum mechanical object [363, 364, 365, 366, 367, 368]. In this approach one essentially
+114
+
+regards each direction normal to the Fermi surface as a one-dimensionalchiral fermion, including
+the current algebra structure, subject to global constraints that cancels the anomalies. This point
+of view ensures that the total fermion number in the Fermi sea is conserved. Here I will only
+present a short summary of the main ideas.
+As in the one-dimensional case one deﬁnes a ﬁlled Fermi sea state |FS⟩. This is a state of non-
+interacting fermions ﬁlling up all one-particle states up to the Fermi energy EF. For concreteness
+we consider a system in two space dimensions. In this case, neglecting lattice eﬀects, the Fermi
+surface is a circumference of radius pF, the Fermi momentum. At ﬁxed fermion number, the
+excitations are particle-hole pairs. The operator nk(q) = c†(k+ q
+2) c(k− q
+2) creates a particle-hole
+pair with relative momentum q with total momentum k. In the low energy regime k is a point on
+the Fermi surface (with |k| = pF) and q is small momentum compared with pF. In what follows
+we will label the point k on the Fermi surface by the angle θ of the arc spanned by k and an
+arbitrary origin on the Fermi surface.
+We will normal-order the particle-hole creation operator with respect to the ﬁlled Fermi sea
+and deﬁne δ(q, θ) =: nk(q) := nk(q) − ⟨FS|nk(q)|FS⟩. Haldane [363], Houghton and Marston
+[364, 367], and Castro Neto and Fradkin [365, 366] showed that in real space that these normal
+ordered density operators obey the equal-time commutation relations
+[δn(x, θ), δn(x′, θ′)] = − 1
+2π kF(θ) · ▽δ2(x − x′)δ(θ − θ′)
+(587)
+where kF(θ) is a unit vector normal to the Fermi surface at angle θ. This is the algebra of the
+quantum ﬂuctuations of the Fermi surface. We can further deﬁne at each point θ on the Fermi
+surface a Bose ﬁeld ϕ(x, θ) such that
+δn(x, θ) = N(0)�F(θ) · ▽ϕ(x, θ)
+(588)
+where N(0) is the density of one-particle states at the Fermi surface and �F(θ) is the Fermi veloc-
+ity at the location θ. The scalar ﬁeld ϕ(x, θ) is a chiral boson at θ which parametrizes the quantum
+ﬂuctuations of the Fermi surface. The quantum dynamics of the chiral bosons ϕ(x, θ) is governed
+by the action
+S =1
+2N(0)
+� 2π
+0
+dθ
+2π
+�
+d2x dt
+�
+−∂tϕ(x, θ) �F(θ) · ▽ϕ(x, θ) − (�F(θ) · ▽ϕ(x, θ))2�
++1
+2N(0)
+� 2π
+0
+dθ
+2π
+� 2π
+0
+dθ′
+2π
+�
+d2x d2x′ dt F(x − x′; θ − θ′) �F(θ) · ▽ϕ(x, θ) �F(θ′) · ▽ϕ(x, θ′)
+(589)
+where x = (x, t), and F(x−x′; θ−θ′) are the Fermi liquid parameters that parametrize the eﬀective
+forward scattering interactions among the fermion quasiparticles on the Fermi surface [13]. The
+equation of motion predicted by this (quadratic) action is equivalent to the linearized quantum
+Boltzmann equation familiar from the Landau theory of the Fermi liquid. A non-linear extension
+has been introduced recently by Delacr´etaz and coworkers [368]. Recent work on the role of
+quantum anomalies in dense Fermi systems has yielded new insights on these problems [369].
+In the regime, where the Landau theory of the Fermi liquid is expected to work [13], bosoniza-
+tion of the Fermi surface approach has reproduced the previously known results. There remain
+many open problems in this approach (and others) in the vicinity of quantum phase transitions. In
+spite of intense research using many diﬀerent approaches, quantum phase transitions in metallic
+systems are not yet fully understood beyond perturbation theory [255, 256]. Non-perturbative
+ideas such as deconﬁned quantum criticality have been proposed [370] and signiﬁcant work has
+been done using large-N methods [371, 372]. A particularly important metallic quantum phase
+transition (and perhaps the simplest) occurs near the Pomeranchuk instability of the fermi liquid.
+A quantum phase transition to an electron nematic state [373] has been predicted and studied
+within the Hertz-Millis approach. It has also been studied using higher dimensional bosoniza-
+tion [373, 374, 375]. Quantum Monte Carlo simulations [376] have shown that the vicinity
+of a nematic quantum critical point can trigger a superconducting state. Nematic Fermi ﬂuids
+have been found in many physical systems of interest ranging from high temperature supercon-
+ductors (such as cuprates and iron superconductors), to electron gases in large magnetic ﬁelds
+[377, 378, 379, 380].
+115
+
+12. Conclusions
+In this chapter I have attempted to cover the role of Quantum Field Theory in modern Con-
+densed Matter Physics. Its role and inﬂuence is vast and deep. In spite of the length of this
+chapter, I have not done justice to its role in many important areas of Condensed Matter. In
+particular, I have not touched on the the theory of quantum spin liquids, fractons, and particu-
+larly topological phases in three space dimensions, among others. An area in which the ideas
+and concepts of QFT have a huge impact , both conceptually and as a tool, is on the problem of
+quantum entanglement. The concept of quantum entanglement as applied to condensed matter
+systems, both in equilibrium and out of equilibrium, is a major area of current development. It
+has become a crucial tool for characterizing systems at quantum criticality as well as topological
+phases of matter. Nevertheless, I hope that the reader will ﬁnd this chapter both insightful and a
+useful resource.
+Acknowledgments
+This work was supported in part by the US National Science Foundation through grant No.
+DMR 1725401 at the University of Illinois.
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