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+Anisotropic Quantum Hall Droplets
+Blagoje Oblak,1 Bastien Lapierre,2 Per Moosavi,3 Jean-Marie Stéphan,4 and Benoit Estienne5
+1CPHT, CNRS, École Polytechnique, IP Paris, F-91128 Palaiseau, France
+2Department of Physics, University of Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
+3Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
+4Univ Lyon, CNRS, Université Claude Bernard Lyon 1,
+Institut Camille Jordan, UMR5208, F-69622 Villeurbanne, France
+5Sorbonne Université, CNRS, Laboratoire de Physique Théorique et Hautes Energies, LPTHE, F-75005 Paris, France
+(Dated: January 4, 2023)
+We study two-dimensional (2D) anisotropic droplets of non-interacting electrons in the lowest
+Landau level, confined by trapping potentials whose level curves have an arbitrary shape at large
+distances. Using semiclassical methods, we show that energy eigenstates are localized on equipoten-
+tials of the trap, with angle-dependent local widths and heights. We exploit this one-particle insight
+to deduce explicit formulas for many-body observables in the thermodynamic limit. For instance,
+the droplet’s density falls off at the boundary with an angle-dependent width inherited from that of
+the underlying wave functions, while the many-body current is localized on the edge, to which it is
+tangent. Correlations along the edge are long-ranged, in accordance with the system’s low-energy
+edge modes which are described by a free chiral conformal field theory in terms of the angle variable
+of the trapping potential. These results are likely to be observable in solid-state systems or quantum
+simulators of 2D electron gases with a high degree of control on the confining potential.
+CONTENTS
+I. Introduction
+1
+II. Setup and main results
+2
+III. Anisotropic states from area-preserving maps
+4
+IV. Edge-deformed anisotropic traps
+6
+V. Many-body observables
+10
+VI. Conclusion and outlook
+14
+Acknowledgments
+14
+A. Isotropic droplets
+15
+B. Semiclassical expansion of P V P
+15
+C. The transport equation
+16
+References
+19
+I.
+INTRODUCTION
+Quantum Hall (QH) droplets are mesoscopic two-
+dimensional (2D) electron gases placed in a strong per-
+pendicular magnetic field and confined by some electro-
+static potential: see Fig. 1. They lie at the heart of the
+QH effect [1–3] and provide a key benchmark for topolog-
+ical phases of matter as a whole. In practice, however,
+the vast majority of detailed analytical studies of QH
+droplets are limited to highly symmetric cases, typically
+involving isotropic traps or harmonic potentials that are
+translation-invariant in one direction [4, 5]. This is espe-
+cially troubling as far as edge properties are concerned,
+since these are sensitive to the shape of the trap and
+determine the system’s low-energy excitations [6–10].
+The goal of this paper is to address this lack of analyt-
+ical results by predicting universal aspects of many-body
+observables near the edge of essentially any anisotropic
+droplet. We achieve this by providing general, explicit,
+one-line formulas for the density, current, and correla-
+tions in the regime of strong magnetic fields. We also
+study the corresponding low-energy edge modes, which
+are described by a free-fermion chiral conformal field the-
+ory (CFT) whose Fermi velocity is constant provided dis-
+tances along the boundary are measured by the canonical
+angle variable determined by the potential. These pre-
+dictions are likely to be observable thanks to direct local
+imaging techniques in condensed matter systems [11–16]
+or quantum simulators [17–24].
+This is not the first time such questions appear in the
+literature. Indeed, random potentials with no symme-
+tries are essential to model disorder, whose importance
+for the robustness of QH physics is hard to overstate [25–
+27]. An especially relevant series of works in that con-
+text is [28, 29], which study the density and current of
+QH droplets with arbitrary potentials, at finite temper-
+ature, generally including Landau level mixing, in the
+semiclassical limit of strong magnetic fields and weak
+traps [26, 27]. However, the coherent states used in these
+references do not allow for any resolution at the single-
+particle level, precluding the computation of low-energy
+dynamics and long-range correlations along the bound-
+ary. Our objective here is instead to find explicit wave
+functions that depend on the shape of the edge, and use
+that as a starting point for many-body objects.
+Regarding electronic edge correlations, similar ques-
+tions have been addressed in the context of classical 2D
+arXiv:2301.01726v1  [cond-mat.mes-hall]  4 Jan 2023
+
+2
+Energy
+Fermi
+energy
+Figure 1.
+2D electron droplet (shaded area) placed in a
+strong perpendicular magnetic field and confined by a typical
+anisotropic edge-deformed potential well (10). At leading or-
+der in the thermodynamic limit, the droplet’s boundary (thick
+black curve) coincides with the equipotential of the trap at the
+Fermi energy.
+Coulomb gases, where holomorphic methods play a key
+role [30–32]. More broadly, the results put forward here
+may be seen as microscopic, first-principles derivations
+of quantities that are normally studied within less con-
+trolled approximation schemes in the geometry of the QH
+effect [33–39]. Our hope is thus to build a bridge between
+these theoretical works and concrete observations that
+may soon be accessible in tabletop experiments with a
+high degree of control on the confining potential [23, 24].
+Here is the plan of the paper. To begin, Sec. II sum-
+marizes our methods and results, avoiding all techni-
+cal details. The next two sections are devoted to one-
+body physics in the lowest Landau level: Sec. III first
+discusses generalities on semiclassical holomorphic wave
+functions, while Sec. IV presents a detailed computation
+of the semiclassical energy spectrum in a class of ‘edge-
+deformed’ potentials of particular interest. This finally
+leads to Sec. V, where we investigate many-body densi-
+ties, currents, correlations, and low-energy edge modes in
+anisotropic traps. We conclude in Sec. VI by discussing
+several follow-ups and open questions. To streamline the
+text, non-essential details are deferred to Apps. A–C.
+II.
+SETUP AND MAIN RESULTS
+This section summarizes our methods and key results.
+To start, we describe the general setup: a semiclassi-
+cal limit (strong magnetic field, small magnetic length)
+in the lowest Landau level (LLL) [26, 27, 40]. We then
+introduce edge-deformed potentials and present their ap-
+proximate energy eigenstates, before finally giving simple
+formulas for the corresponding local many-body observ-
+ables and dynamics in the vicinity of the edge. Some of
+these findings are depicted in Fig. 2.
+A.
+Semi-classical limit in the LLL
+This work concerns spin-polarized non-interacting elec-
+trons of mass M and charge q in a 2D plane.
+Each
+electron is governed by a Landau Hamiltonian with an
+anisotropic potential V (x),
+H1-body =
+1
+2M (p − qA)2 + V (x),
+(1)
+where A is the vector potential of the magnetic field
+B = dA. (We view A as a one-form, which simplifies
+some notation but is otherwise inconsequential.) We shall
+assume that V (x) is ‘monotonous’, by which we mean
+that it has a unique global minimum away from which it
+grows monotonously, but it is otherwise general. Conse-
+quently, the level curves or ‘equipotentials’ of V (x) are
+nested and take the form shown in Fig. 2(a). We assume
+throughout that the potential is weak relative to the mag-
+netic field [26–29, 41, 42], and that it is nearly constant
+on length scales comparable to the magnetic length
+ℓ2 ≡ ℏ
+qB > 0.
+(2)
+In that regime, the potential is a small perturbation of
+the pure Landau Hamiltonian ∝ (p−qA)2 and the eigen-
+states of (1) are expected to be well-approximated by
+wave functions in the LLL. For instance, if the potential
+V (x) = V0(r2/2) is isotropic, the eigenstates of (1) have
+some definite angular momentum and reduce at strong
+B to standard LLL wave functions in symmetric gauge:
+φm(x) =
+1
+√
+2πℓ2
+zm
+√
+m!
+e−|z|2/2,
+(3)
+where m ≥ 0 is an integer angular momentum and we
+have introduced the dimensionless complex coordinate
+z ≡ x + iy
+√
+2ℓ .
+(4)
+Each wave function (3) reaches its maximum on the cir-
+cle |z| = √m, away from which it decays in a Gaussian
+manner within a magnetic length. Our goal will be to ob-
+tain similar approximate eigenstates in anisotropic traps,
+using the squared magnetic length (2) as a small expan-
+sion parameter [43]. Equivalently, we shall carry out a
+semiclassical (small ℏ), high field expansion (large B).
+In practice, the projection to the LLL is implemented
+thanks to the (one-body) operator P ≡ �∞
+m=0 |φm⟩⟨φm|,
+whose kernel can be read off from the wave functions (3):
+⟨z, ¯z|P|w, ¯w⟩ =
+1
+2πℓ2 e−(|z|2+|w|2)/2 ez ¯
+w.
+(5)
+This is manifestly Gaussian and reduces to a delta func-
+tion in the formal semiclassical limit ℓ → 0. At small but
+finite ℓ, the projection (5) makes space non-commutative
+in the sense that the LLL-projected position operators
+(x, y) satisfy the Heisenberg algebra
+[PxP, PyP] = iℓ2.
+(6)
+
+3
+���
+���
+���
+x
+x
+x
+y
+y
+y
+√
+N
+−
+√
+N
+−
+√
+N
+√
+N
+(a)
+(b)
+(c)
+x
+x
+x
+y
+y
+y
+ℓ
+√
+2N
+−ℓ
+√
+2N
+−ℓ
+√
+2N
+ℓ
+√
+2N
+Figure 2. (a) A planar plot of the many-body density (14) along with several equipotentials (dashed lines), for a droplet of
+N = 100 electrons confined by the edge-deformed trap of Fig. 1. The constancy of the bulk density and its fall at the boundary
+are manifest. (b) The current’s norm (15) for the same droplet, together with the edge (black line) on which it is localized.
+(c) The correlation function (16) for the same droplet, seen as a function of x2 = (x, y) with x1 = (ℓ
+�
+Nλ(0), 0) denoted by a
+cross and fixed at the edge. Long-range correlations along the boundary are clearly visible.
+One can thus think of the plane R2 as a ‘phase space’
+whose canonical variables are x, y. This interpretation
+pervades much of the QH literature [40, 44–53] and will
+similarly affect our discussion.
+Indeed, projecting the
+Hamiltonian (1) to the LLL and looking for its spectrum
+leads to the eigenvalue equation
+P V P|ψ⟩ = E|ψ⟩,
+(7)
+where the unknowns are the energy E and the quantum
+state |ψ⟩ ∈ LLL. Note that the kinetic term of (1) has
+disappeared in (7): the potential itself plays the role of
+an effective Hamiltonian in the non-commutative phase
+space (x, y).
+Exact solutions of Eq. (7) are generally out of reach,
+so one has to resort to approximations. The semiclassical
+one that we shall use is well known in the QH context
+[26–29, 54, 55]. Accordingly, we will look for solutions of
+Eq. (7) labeled by a large quantum number m ∈ N, seen
+as a generalization of angular momentum.
+This large
+m limit is accompanied by a small ℓ limit, such that
+the area 2πℓ2m remains finite. In that regime, the mth
+eigenstate is approximately Gaussian and localized on an
+equipotential γm of V (x), enclosing a quantized area such
+that the Bohr-Sommerfeld condition holds:
+�
+γm
+x dy = 2πℓ2 m.
+(8)
+Equivalently, the flux of the magnetic field through the
+area enclosed by γm is m times the flux quantum. The
+energy of the mth state is then
+Em = E0
+m + ℓ2E1
+m + O(ℓ4),
+(9)
+where E0
+m = V (γm) is the leading classical approxima-
+tion and the quantum correction E1
+m involves the Lapla-
+cian of the potential and the curvature of the equipo-
+tential γm [54, 55]. Note that the more familiar Wentzel-
+Kramers-Brillouin (WKB) approximation of 1D quantum
+mechanics [56] includes (topological) Maslov corrections
+on the right-hand side of (8); we will find similar correc-
+tions below, although their interpretation as topological
+invariants is prevented by a subtle distinction between
+real and Kählerian polarizations in geometric quantiza-
+tion [54, 55].
+B.
+One-body results
+The semiclassical limit just outlined applies to any
+(monotonous) weak potential. In practice, our main con-
+cern is the physics of QH droplets near the edge, where
+the details of the bulk potential are irrelevant.
+Most
+of our explicit results will therefore be given for ‘edge-
+deformed’ potentials, obtained as follows. Consider any
+monotonously increasing function V0(t) for t ≥ 0, and let
+λ(ϕ) be any strictly positive 2π-periodic function of the
+angle ϕ ∈ [0, 2π); normalize λ so that
+�
+dϕ λ(ϕ) = 4π,
+where we write
+�
+dϕ as a shorthand for
+� 2π
+0
+dϕ. Then
+adopt polar coordinates in the plane such that x + iy =
+r eiϕ and define the potential
+V (r, ϕ) = V0
+� r2
+λ(ϕ)
+�
+.
+(10)
+We refer to this as an edge-deformed trap because it re-
+sults from a deformation r2 �→ r2/λ(ϕ) that changes the
+shape of the boundary of isotropic droplets in a finite
+and smooth way, even in the thermodynamic limit where
+the droplet’s area goes to infinity [57]. In fact, the cor-
+responding infinitesimal transformations are expected to
+
+1
+1
+-
+-
+1
+1
+-
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
+-4
+be conformal transformations of the edge CFT [58–63].
+The class of potentials (10) is thus exhaustive, at least
+as far as edge effects are concerned.
+The traps (10) turn out to allow for explicit calcula-
+tions of the semiclassical energy spectrum, generalizing
+the known isotropic results reviewed in App. A. Indeed,
+we show in Sec. IV that the corresponding eigenfunc-
+tions, solving (7) in the LLL, are Gaussians localized on
+equipotentials r = ℓ
+�
+mλ(ϕ) at large quantum numbers
+m. They can be written in polar coordinates as
+ψm(x) ∼
+eiΘm(x)
+�
+2πℓ2σ(ϕ)
+e−a2/σ(ϕ)2
+(2πm)1/4 ,
+(11)
+where Θm(x) is a position-dependent phase, a ≡
+�
+r −
+ℓ
+�
+mλ(ϕ)
+��
+ℓ
+�
+λ(ϕ) is a dimensionless radial coordinate
+that measures the distance from the equipotential, and
+σ(ϕ) ≡
+�
+2
+λ(ϕ)
+�
+1 +
+� λ′(ϕ)
+2λ(ϕ)
+�2
+(12)
+is an angle-dependent width. We stress that this exhibits
+the expected ‘quantum smearing’ of wave functions in a
+strong but finite magnetic field [28, 29], which would be
+missed by the leading classical approximations ℓ2 = 0.
+As for the energy of the state (11), its expansion (9)
+up to neglected O(ℓ4) contributions is
+Em ∼ V (γm) + ℓ2
+2 Ωm
+�
+1 +
+�
+1 + Γm
+Ωm
+� � dϕ
+4π λ(ϕ)σ(ϕ)2
+�
+,
+(13)
+where V (γm) = V0
+�
+ℓ2m
+�
+is the leading term, while
+the first quantum correction involves derivatives Ωm ≡
+V ′
+0(ℓ2m) > 0 and Γm ≡ ℓ2m V ′′
+0 (ℓ2m). Each mth energy
+is thus determined by the potential and its derivatives at
+an equipotential that satisfies the quantization condition
+(8), in accordance with general theorems for holomorphic
+WKB theory [54, 55].
+C.
+Many-body results
+Now consider the ground state of a large number
+N ≫ 1 of free spin-polarized electrons, each governed by
+the single-particle Hamiltonian (1). This ground state
+is a Slater determinant of wave functions whose large m
+behavior is the Gaussian (11). As we show in Sec. V,
+the corresponding many-body density, current, correla-
+tions, and low-energy effective action can all be written
+in closed form in terms of λ(ϕ) and the number N of
+fermions. The density ρ(x) = �N−1
+m=0 |ψm(x)|2 thus sat-
+isfies the bulk behavior ρ ∼
+1
+2πℓ2 , while its form near the
+edge is given by a complementary error function:
+ρ(r, ϕ) ∼
+1
+4πℓ2 erfc
+�√
+2 a
+σ(ϕ)
+�
+,
+(14)
+where a ≡
+�
+r − ℓ
+�
+Nλ(ϕ)
+��
+ℓ
+�
+λ(ϕ) is again a dimen-
+sionless radial coordinate, now measuring the distance
+from the edge at redge = ℓ
+�
+Nλ(ϕ).
+As a result, the
+ground state forms a star-shaped droplet whose bound-
+ary has an angle-dependent width (12) inherited from
+that of one-body wave functions. Turning to the current
+J = �N−1
+m=0
+1
+2i(ψ∗
+mdψm − ψmdψ∗
+m − 2iqA|ψm|2), we write
+it as a one-form in polar coordinates to find
+J(r, ϕ) ∼ −
+exp
+�
+− 2a2
+σ(ϕ)2
+�
+(2πℓ2)3/2σ(ϕ)
+�
+ℓ
+√
+N dϕ +
+λ′(ϕ)
+2λ(ϕ)3/2 dr
+�
+.
+(15)
+This is localized on the edge and tangent to it, miss-
+ing the bulk behavior Ji ∝ εij∂jV as expected in the
+LLL [41, 42]. Finally, the two-point correlation function
+C(x1, x2) = �N−1
+m=0 ψ∗
+m(x1)ψm(x2) behaves near the edge
+as
+C(x1, x2) ∼
+eiΘN(x1,x2)
+4πℓ2�
+σ(ϕ1)σ(ϕ2)
+i exp
+�
+−
+a2
+σ(ϕ1)2 −
+b2
+σ(ϕ2)2
+�
+√
+2πN sin
+�� ϕ1
+ϕ2
+dθ
+4 λ(θ)
+�
+(16)
+with a ≡
+�
+|x1| − ℓ
+�
+Nλ(ϕ1)
+��
+ℓ
+�
+λ(ϕ1) and similarly for
+b in polar coordinates (|x1|, ϕ1) and (|x2|, ϕ2), respec-
+tively, while ΘN(x1, x2) is a complicated overall phase.
+Note again the Gaussian localization at the edge, as well
+as the long-range correlator ∝ sin(...)−1 typical of gapless
+fermions. Indeed, we eventually confirm that the under-
+lying low-energy edge modes are described by a chiral
+CFT of free fermions: see the action functional (68) be-
+low. The corresponding Fermi velocity is constant along
+the boundary when measured in terms of the angle vari-
+able of the potential (10), namely θ(ϕ) ≡
+� ϕ
+0 dϕ′ λ(ϕ′)/2.
+III.
+ANISOTROPIC STATES FROM
+AREA-PRESERVING MAPS
+This section presents the WKB ansatz [see (20)] that
+forms the basis of all our later considerations. The struc-
+ture is ultimately quite simple: given a monotonous po-
+tential V (x), we pick one of its equipotentials, γm, with
+quantized area (8). We then build a wave function with
+winding m, perfectly localized on γm, and finally project
+it to the LLL using the operator (5). General theorems
+[54, 55] ensure that LLL-projected eigenstates satisfying
+(7) can indeed be built in this way. The detailed applica-
+tion of this method to edge-deformed traps (10) is given
+in Sec. IV.
+A note: what follows relies on the mathematics of area-
+preserving diffeomorphisms, which is not reviewed in de-
+tail. We refer instead to [57] for an introduction whose
+language is similar to that adopted here. For more gen-
+eral discussions in the symplectic context, see [64, 65].
+A.
+Potentials in action-angle variables
+Let us be more precise about the geometry of the
+setup, remaining at the classical level for now. We pick
+
+5
+a smooth potential V (x) and assume as in Sec. II that it
+is monotonous. Its unique global minimum is thus sur-
+rounded by nested level curves, and one can always find
+an area-preserving deformation of the plane that sends
+each equipotential on a circle [64]. In other words, one
+can find an invertible smooth map F : R2 → R2 with
+unit Jacobian such that
+V
+�
+F(x)
+�
+= V0(r2/2),
+(17)
+where the trap on the right-hand side is isotropic (it only
+depends on r = |x|). If F is the identity (or a rotation
+around the origin), then V was isotropic to begin with
+and its eigenstates satisfying (7) are the standard wave
+functions (3) with definite angular momentum. In the
+more general case of arbitrary V , Eq. (17) suggests using
+F to map the eigenstates (3) on those corresponding to
+our general V (x).
+The existence of F in (17) is guaranteed by the mono-
+tonicity of V , and is equivalent to the existence of glob-
+ally well-defined action-angle variables. In fact, we can
+use this to write F in a more explicit form that will be
+useful below. Let therefore (ℓ2K, θ) be action-angle co-
+ordinates for the potential V (x) [66], where K ≥ 0 is
+dimensionless and θ ∈ [0, 2π) is a genuine angle. They
+are normalized so that ℓ2dK ∧ dθ = dx ∧ dy, which is
+to say that their Poisson bracket reads {ℓ2K, θ} = ℓ2 in
+terms of the phase space (x, y) with bracket (6). Then
+the map (x, y) �→ (ℓ2K, θ) is an area-preserving diffeo-
+morphism in terms of which V (x) = V0(ℓ2K(x)) is in-
+variant under rotations of θ. To be specific, write these
+coordinates as functions K(x, y) and θ(x, y) and let the
+inverse be x = F(K, θ) and y = G(K, θ) for some func-
+tions (F, G); this inverse is nothing but the deformation
+F in (17). In other words, knowing the action-angle vari-
+ables of a potential V allows us to map it on its (unique)
+isotropic cousin V0, which in turn can be used to relate
+the corresponding anisotropic eigenstates to those in (3).
+It should be clear that these considerations apply to
+any monotonous anisotropic trap, in which case one
+generally encounters intricate area-preserving maps with
+complicated action-angle variables. In Sec. IV, we will
+argue that most of these difficulties wash away when
+focusing on edge physics, whereupon the only relevant
+maps are the ‘edge deformations’ mentioned below Eq.
+(10). For now, we remain general and turn to quantum
+considerations.
+B.
+Anisotropic eigenstates
+Using the action-angle variables (ℓ2K, θ) for V (x),
+one can concretize the statements around Eqs. (8)–(9)
+into formulas and eventually obtain anisotropic eigen-
+functions that satisfy (7). Indeed, the Bohr-Sommerfeld
+quantization condition (8) implies that the equipotential
+γm is the set of points in R2 where K = m. Now consider
+the following quantum state, perfectly localized on γm:
+|Ψm⟩ ≡ 2πℓ2
+�
+dθ n(θ) eimθ��F(m, θ), G(m, θ)
+�
+,
+(18)
+where the normalization 2πℓ2 is included for later conve-
+nience, the ‘wave function’ ⟨x|F(m, θ), G(m, θ)⟩ = δ2�
+x−
+F(m, θ)
+�
+is a delta function, and n(θ) is some complex
+periodic function. The latter does not wind upon com-
+pleting one turn in the plane along the equipotential,
+meaning that all the winding of (18) is encoded in the
+phase eimθ.
+We stress that (18) is analogous to the standard WKB
+ansatz ψ(x) ∼ eiS0(x)/ℏeiS1(x) in 1D. Indeed, the phase
+eimθ is the leading classical contribution eiS0/ℏ for m ≫ 1,
+corresponding to the ‘geometrical optics’ approximation
+of the wave function, while n(θ) is the ‘physical optics’
+quantum correction eiS1 that eventually needs to satisfy a
+transport equation in order for the Schrödinger equation
+to hold [56]. The only difference lies in the interpretation
+of areas in the plane as values of an ‘action’, which ul-
+timately stems from the non-commutative geometry (6)
+of LLL physics. Note that n(θ) is the only unknown in
+(18). Indeed, most of the WKB method below will con-
+cern the derivation of a transport equation for n(θ) from
+the requirement that (7) be satisfied.
+Starting from Eq. (18), it is straightforward to build a
+state in the LLL thanks to the projector (5): denoting
+ψm(z, ¯z) ≡ ⟨z, ¯z|P|Ψm⟩,
+(19)
+one finds the wave function
+ψm(z, ¯z) = e−|z|2/2
+�
+dθ n(θ) eimθ
+× e−[F (m,θ)2+G(m,θ)2]/4ℓ2 ez[F (m,θ)−iG(m,θ)]/
+√
+2ℓ.
+(20)
+This is manifestly of the form e−|z|2/2 times a holomor-
+phic function that depends on the action variable ℓ2m
+and the uniformizing map F of Eq. (17). It will be our
+starting point for the semiclassical solution of the eigen-
+value equation (7).
+As a consistency check, note that (20) simplifies for
+isotropic potentials. In that case, action-angle variables
+are just polar coordinates ℓ2K = r2/2 and θ = ϕ, and
+the map in (17) is F(x) = x, merely implementing a
+change from polar to Cartesian coordinates: F(m, θ) =
+ℓ
+√
+2m cos(θ) and G(m, θ) = ℓ
+√
+2m sin(θ). One can then
+verify that (20) with n(θ) = const coincides (up to nor-
+malization) with the standard LLL wave function (3).
+Similarly to that case, any projected wave function (20)
+reaches its maximum on the equipotential γm and is ap-
+proximately Gaussian close to it, as ensured by the kernel
+(5). This will be confirmed explicitly in Sec. IV for edge
+deformations.
+
+6
+C.
+Expanding the eigenvalue equation
+None of what we wrote so far involves a manifest semi-
+classical expansion: it is hidden in the eigenvalue equa-
+tion (7) and the function n(θ) in (20), since n(θ) should
+be expanded as a power series n(θ) = n0(θ) + ℓ2n1(θ) +
+O(ℓ4) (as before, there are no odd powers of ℓ since ℓ2 ∝ ℏ
+is really the semiclassical parameter).
+It is therefore
+worth anticipating the first few terms of the semiclas-
+sical approximation of (7). We stress that the expansion
+below will eventually be limited to the leading order of
+the transport equation, so that only n0(θ) will be calcu-
+lated in the end. In principle, one could of course push
+the expansion to higher orders for more detailed results.
+The semiclassical expansion of the right-hand side of
+(7) is clear: it is given by the large m, small ℓ2 expan-
+sion of the projected wave function (20), including an
+expansion of n(θ). As for the energy, its expansion was
+written in (9). The left-hand side of (7) is more subtle, as
+its semiclassical expansion involves that of the operator
+P V P. The latter is a ‘Berezin-Toeplitz operator’ [54, 55]
+that will play an important role for edge-deformed po-
+tentials, so we now explain its expansion in some detail.
+First, given Cartesian coordinates (x, y), express the po-
+tential in complex coordinates (4) as V (x, y) ≡ V(z, ¯z)
+for some function V(z, ¯w) which is holomorphic in its first
+argument and anti-holomorphic in the second. Then re-
+calling that P is the LLL projector with kernel (5), one
+finds
+⟨z, ¯z|P V P|w, ¯w⟩ =
+1
+2πℓ2 e− 1
+2 (|z|2+|w|2)
+×
+�
+R2 du dv V (u, v) e−|X|2+z ¯
+X+ ¯
+wX
+(21)
+with X ≡ (u + iv)/
+√
+2ℓ similarly to (4). Our task is to
+expand the integral on the right-hand side in the semi-
+classical limit. The key is to assume that the potential
+varies slowly on the scale of the magnetic length [26–29],
+i.e. we choose once and for all a smooth potential V (x),
+independent of ℓ, and let ℓ be small. In that regime, the
+integrals in (21) are approximately Gaussian, which gives
+(see App. A 2)
+⟨z, ¯z|P V P|w, ¯w⟩
+ℓ≪1
+∼
+1
+2πℓ2 e− |z−w|2
+2
+e
+z ¯
+w−¯zw
+2
+×
+�
+V(z, ¯w) + ℓ2
+2 (∇2V )(z, ¯w)
+�
+(22)
+where
+(∇2V )(z, ¯w)
+is
+the
+bicomplex
+function
+that
+corresponds to the Laplacian of the potential,
+i.e.
+(∇2V )(z, ¯w) =
+4
+2ℓ2 ∂z∂ ¯
+wV. This is the standard semiclas-
+sical expansion of a Berezin-Toeplitz operator [54, 55].
+Note the general structure: the entire P V P operator
+boils down to P itself, with kernel (5), multiplied by a
+function that coincides with V at leading order but also
+includes quantum corrections. In the ‘zoomed-out’ limit
+where the kernel of P is a delta function, the first term of
+(22) becomes V(z, ¯z)δ2(z − w, ¯z − ¯w) as expected. More-
+over, for harmonic potentials, the truncated expression
+(22) is actually exact since the next term ∇4V and all
+subsequent ones vanish. This agrees with the common
+lore that ‘WKB is exact for quadratic Hamiltonians’.
+IV.
+EDGE-DEFORMED ANISOTROPIC TRAPS
+Here we apply the WKB ansatz of Sec. III to poten-
+tials (10) with scale-invariant level curves, obtained by
+acting with edge deformations [57] on an isotropic trap.
+As we explain below, these are the most general traps
+one expects to find close to the edge of star-shaped QH
+droplets.
+The plan is as follows.
+First, we introduce
+edge deformations and give a few examples for later ref-
+erence. Second, we apply Eq. (7) to edge-deformed traps
+and expand it in the classical limit (large m, small ℓ2
+with ℓ2m = O(1) kept fixed). We keep track of all terms
+up to order O(ℓ2), so as to capture the leading part of
+the transport equation for the function n(θ) in (18)–(20).
+This eventually yields an explicit energy spectrum [see
+(37)] along with approximately Gaussian eigenfunctions
+[see (43)]. Lastly, we conclude with a consistency check
+by showing that our wave functions reproduce the asymp-
+totic (large m) form of the known LLL-projected spec-
+trum for anisotropic harmonic traps [67–71].
+A.
+Edge deformations
+We have seen in Sec. III that area-preserving defor-
+mations play a key role for the semiclassical solution of
+the eigenvalue equation (7). The group of all such de-
+formations is obviously huge, so it is essential to identify
+the subset of transformations that are likely to be im-
+portant for low-energy physics. In fact, part of this work
+has already been carried out, at least implicitly, in the
+seminal series of papers [58–63], which we now use as a
+basis for the definition of edge deformations. (A similar
+motivation was put forward in [57].)
+Label points on the plane by their polar coordinates
+(r, ϕ), defined as usual by x + iy = r eiϕ.
+Then, the
+boundary of any isotropic QH droplet is located at some
+fixed radius redge = O(ℓ
+√
+N). What is the most general
+area-preserving deformation that preserves this order of
+magnitude? The answer is readily found by realizing that
+the constraint of preserving redge = O(ℓ
+√
+N) is equiva-
+lent, at leading order in 1/N, to the condition that the de-
+formation commutes with overall dilations r �→ const×r.
+The most general deformation satisfying this criterion is
+an edge deformation
+�r2
+2 , ϕ
+�
+�→
+�
+r2
+2f ′(ϕ), f(ϕ)
+�
+,
+(23)
+where f(ϕ) is an (orientation-preserving) deformation of
+the circle, i.e. any smooth map satisfying f(ϕ + 2π) =
+
+7
+f(ϕ) + 2π and f ′(ϕ) > 0. The angle-dependent rescaling
+of r on the right-hand side ensures that the map preserves
+area, and reproduces the argument of the potential (10)
+with λ = 2f ′. Note that the set of maps (23) is isomor-
+phic to the group of diffeomorphisms of the circle, whose
+central extension famously leads to the Virasoro algebra
+encountered in CFT. Indeed, this motivates the state-
+ment in [60, 61] that generators of maps (23) in the QH
+effect produce conformal transformations of edge modes.
+We stress that the subset of transformations (23) orig-
+inates from an asymptotic analysis of the relevant or-
+ders of magnitude. One can undoubtedly consider other
+families of deformations, motivated by different consid-
+erations, but those are irrelevant for our purposes. For
+instance, the transformations r2 �→ r2 + α(ϕ) are crucial
+for the effective low-energy description of QH droplets
+[6, 10, 61], but they are subleading compared to (23) since
+they deform the radius redge = O(ℓ
+√
+N) by terms of order
+O(1/N) instead of O(1). Conversely, one might consider
+‘higher-spin transformations’ [58, 60, 61] that change the
+radius in a dramatic way such as r2 �→ β(ϕ)r4[1+O(1/r)],
+but these stretch QH droplets to an infinite extent in the
+thermodynamic limit, which is why we discard them.
+Let us provide a few examples of edge deformations
+for future reference. First, (23) includes rigid rotations
+around the origin given by f(ϕ) = ϕ + const. A richer
+class is obtained by fixing some positive integer k and
+considering all maps of the form
+eikf(ϕ) = α eikϕ + β
+¯β eikϕ + ¯α
+(24)
+where α, β are complex and satisfy |α|2 − |β|2 = 1. For
+fixed k, such maps span a group locally isomorphic to
+SL(2, R), always containing a subgroup of rigid rotations.
+We will return to these deformations below, since they
+can be seen as Fourier modes for circle diffeomorphisms.
+In particular, setting α = cosh λ and β = sinh λ for some
+real parameter λ turns the map (24) into an analogue of
+a Lorentz boost with rapidity λ. In terms of the bulk
+action (23), any deformation (24) turns a circle into a
+‘flower’ with k petals: see Fig. 5 for k = 3. For k = 2,
+this maps the circle on an ellipse [57], which will be useful
+for anisotropic harmonic traps in Sec. IV E.
+B.
+Edge-deformed potentials
+Given an isotropic potential V0(r2/2), how is it affected
+by an edge deformation (23)? The answer is provided by
+the anisotropic trap (10) with λ(ϕ) = 2f ′(ϕ):
+V (r, ϕ) ≡ V0
+�
+r2
+2f ′(ϕ)
+�
+.
+(25)
+In what follows, we exclusively consider this class of po-
+tentials and refer to them as ‘edge-deformed traps’, for
+the reasons stated above. Having fixed once and for all
+some circle deformation f(ϕ), our goal is to solve the cor-
+responding eigenvalue equation (7) in the classical limit
+of high quantum numbers and small magnetic length.
+We begin by listing the key classical data of the prob-
+lem. The action-angle variables corresponding to (25) are
+(ℓ2K, θ) =
+�
+r2�
+(2f ′(ϕ)), f(ϕ)
+�
+with an inverse given by
+(r2/2, ϕ) =
+�
+ℓ2K/(f −1)′(θ), f −1(θ)
+�
+, where f −1 denotes
+the 1D inverse of f. Points at constant K are equipoten-
+tials, i.e. level curves of (25), each of which is a set of
+points such that
+r2
+2f ′(ϕ) = ℓ2K
+(26)
+with constant K ≥ 0.
+In Cartesian coordinates, this
+is the set of points x =
+�
+2ℓ2Kf ′(ϕ) cos(ϕ), y
+=
+�
+2ℓ2Kf ′(ϕ) sin(ϕ) for ϕ ∈ [0, 2π].
+Equivalently, in
+terms of the angle variable θ = f(ϕ) ∈ [0, 2π], the equipo-
+tential is
+x =
+�
+2ℓ2K
+(f −1)′(θ) cos(f −1(θ)) ≡ F(K, θ),
+y =
+�
+2ℓ2K
+(f −1)′(θ) sin(f −1(θ)) ≡ G(K, θ),
+(27)
+where the notation (F, G) was introduced in Sec. III A.
+Note that we will eventually focus on the regime where
+K ≫ 1 is very large in such a way that the dimensionful
+area 2πℓ2K be an O(1) quantity as ℓ → 0.
+Moving just slightly away from the classical regime, we
+have seen in Sec. III that the expansion of the operator
+P V P involves a bicomplex potential function V(z, ¯w). In
+the case of edge-deformed potentials (25), with the con-
+ventions used there and above for complex coordinates,
+one finds
+V(z, ¯w) = V0
+�
+ℓ2
+z ¯w
+f ′� 1
+2i log[z/ ¯w]
+�
+�
+.
+(28)
+Note that this only makes sense for z and w close to each
+other, otherwise taking z → e2πiz affects the argument
+of f ′ on the right-hand side. By contrast, when z and w
+remain close, taking z → e2πiz also requires w → e2πiw,
+and this time the angle
+1
+2i log[z/ ¯w] is indeed invariant.
+Finally, the expansion (22) also involves the complexi-
+fied Laplacian of the potential, but only its real value will
+be relevant at the order studied here. Let us therefore
+express the Laplacian of (25) in polar coordinates:
+∇2V = 1
+f ′
+�
+2 − 1
+2
+f ′′′
+f ′ + f ′′2
+f ′2
+�
+V ′
+0
+�
+r2/2f ′�
++ r2
+f ′2
+�
+1 + f ′′2
+4f ′2
+�
+V ′′
+0
+�
+r2/2f ′�
+.
+(29)
+Here the prime means differentiation with respect to the
+argument, namely ϕ for f(ϕ) and r2/2 for V0(r2/2). We
+shall rely on (28) and (29) below, since they directly affect
+the eigenvalue equation (7).
+
+8
+C.
+Eigenvalue equation and energy
+Having studied the potential (25), let us turn to the
+quantum state meant to solve the eigenvalue equation
+(7). As in Sec. III B, we begin by building a state (18)
+that is perfectly localized on the equipotential, project
+to the LLL using the operator (5), and obtain the wave
+function (20) that now reads
+ψm(z, ¯z) = e−|z|2/2
+�
+dϕ f ′(ϕ) n(f(ϕ))
+× exp
+�
+imf(ϕ) − 1
+2mf ′(ϕ) + z
+�
+mf ′(ϕ) e−iϕ�
+,
+(30)
+where we changed variables using θ = f(ϕ). It remains to
+show that this solves the eigenvalue equation (7) for edge-
+deformed traps (25) in the semiclassical regime, provided
+the function n(θ) satisfies a suitable transport equation.
+The latter is derived by expanding the energy (9) and
+the potential (22) to get
+0 =
+�
+dϕ f ′(ϕ) n(f(ϕ))
+× exp
+�
+imf(ϕ) − 1
+2mf ′(ϕ) + z
+�
+mf ′(ϕ) e−iϕ�
+×
+�
+V
+�
+z,
+�
+mf ′(ϕ) e−iϕ�
++ ℓ2
+2 ∇2V − E0
+m − ℓ2E1
+m
+�
+(31)
+where V(z, ¯w) is the bicomplex function (28) and the
+equation holds up to neglected O(ℓ4) corrections.
+At
+leading order in the classical limit, the potential expan-
+sion (22) boils down to ⟨z|P V P|w⟩ ∼ V(z, ¯z)δ2(z − w),
+so (31) merely states that E0
+m = V0(ℓ2m) = V (γm). The
+issue is to find the two remaining unknowns: the function
+n(f(ϕ)) and the first-order energy correction E1
+m.
+To determine these, the crucial step is to evaluate (31)
+along the equipotential (26) labeled by K = m, i.e. for
+z =
+�
+mf ′(α) eiα with α ∈ [0, 2π), where as usual we
+assume m ≫ 1. Indeed, if (31) holds on a level curve,
+then it holds for all z by holomorphicity. This is writ-
+ten in more detail in App. C, where we show that the
+integrand of (31) has a saddle point at ϕ = α, eventu-
+ally resulting in a transport equation for the unknown
+function n. Here we skip the computation and analyse
+separately the real and imaginary parts of the transport
+equation. We start with the real part, which will allow
+us to deduce the LLL-projected energy spectrum. The
+imaginary part is postponed to Sec. IV D, where we also
+display the resulting nearly Gaussian wave functions.
+Let Φ(ϕ) denote the phase of n(f(ϕ)) ≡ N(ϕ) eiΦ(ϕ).
+Then the real part of the transport equation [see (C17)]
+yields
+Φ′(ϕ) = E1
+m
+Ωm
+f ′(ϕ) − 1
+2
+� Γm
+Ωm
++ 1
+��
+1 + f ′′(ϕ)2
+4f ′(ϕ)2
+�
+− 1
+2 + ∂ϕ
+� f ′′(ϕ)
+8f ′(ϕ)
+�
++ 1
+2
+∂ϕ[f ′′(ϕ)/2f ′(ϕ)]
+1 + f ′′(ϕ)2/4f ′(ϕ)2 ,
+(32)
+where E1
+m is the first order correction to the energy (9)
+and we introduced the parameters
+Ωm ≡ V ′
+0(ℓ2m) > 0,
+Γm ≡ ℓ2m V ′′
+0 (ℓ2m).
+(33)
+In many-body droplets with N electrons, these will re-
+spectively measure the Fermi velocity and the curvature
+of the potential at the Fermi surface when m = N. Note
+that all terms in (32) are total derivatives save for the
+factor 1 + [f ′′/2f ′]2, so the solution is
+Φ(ϕ) = E1
+m
+Ωm
+f(ϕ) − 1
+2
+� Γm
+Ωm
++ 1
+� � ϕ
+0
+dθ
+�
+1 + f ′′(θ)2
+4f ′(θ)2
+�
+− ϕ
+2 + f ′′(ϕ)
+8f ′(ϕ) + 1
+2 arctan
+� f ′′(ϕ)
+2f ′(ϕ)
+�
++ const.
+(34)
+This turns out to imply a quantization condition for en-
+ergy. Indeed, when we initially introduced the function
+n(θ) in (18), we mentioned that it must have a vanishing
+winding number along the equipotential, so that all the
+winding of the wave function is contained in the expo-
+nential factor eimθ. The phase Φ(ϕ) must therefore be
+strictly 2π-periodic, i.e. Φ(2π) = Φ(0). Using (34), this
+fixes the first quantum correction of the energy (9):
+E1
+m
+Ωm
+= 1
+2 +
+� Γm
+Ωm
++ 1
+� � dϕ
+4π
+�
+1 + f ′′(ϕ)2
+4f ′(ϕ)2
+�
+.
+(35)
+The latter generally generally depends on m through Γm
+and Ωm in (33).
+A simplification only occurs in ‘har-
+monic’ setups where Γm = 0 and the right-hand side of
+(35) is an f-dependent constant, for all m [72]. In any
+case, the full mth energy (9) in the semiclassical limit can
+be written as
+Em ∼ V0
+�
+ℓ2m
+�
++ ℓ2
+2
+�
+Ωm +
+�
+Γm + Ωm
+� � dϕ
+2π
+�
+1 + f ′′(ϕ)2
+4f ′(ϕ)2
+��
+,
+(36)
+reproducing the result announced in (12)–(13) with λ =
+2f ′ and generalizing the isotropic value obtained for
+f ′ = 1 [see (A3)]. The leading-order Bohr-Sommerfeld
+quantization condition (8) is manifestly satisfied, while
+the first quantum correction can be expressed in terms
+of a Maslov-like shift and an integral of the Laplacian,
+confirming the general result in [55]:
+Em = V0
+�
+ℓ2�
+m + 1
+2
+��
++ ℓ2
+4
+� dϕ
+2π f ′(ϕ)∇2V
+��
+r2=2ℓ2(m+1/2)f ′(ϕ) + O(ℓ4).
+(37)
+(In the language of [55], our ‘Maslov-like’ term actually
+stems from an integral of the curvature of γm.) It now
+remains to write the corresponding wave functions.
+D.
+Gaussian wave functions
+As above, write n(f(ϕ)) = N(ϕ) eiΦ(ϕ) for the un-
+known function of the WKB ansatz, with a norm N(ϕ) =
+
+9
+|n(f(ϕ))|.
+Then the imaginary part of the transport
+equation [see (C18)] can be recast into
+N ′(ϕ)
+N(ϕ) = 1
+4∂ϕ log
+�
+1
+f ′(ϕ)
+�
+1 + f ′′(ϕ)2
+4f ′(ϕ)2
+��
+,
+(38)
+which remarkably has the form of an overall logarithmic
+derivative. The general solution is thus
+��n
+�
+f(ϕ)
+��� = N0
+�
+1
+f ′(ϕ)
+�
+1 + f ′′(ϕ)2
+4f ′(ϕ)2
+��1/4
+,
+(39)
+where the normalization N0 will soon be fixed. Note the
+exponent 1/4, typical of WKB approximations [56].
+We can now use (39) to evaluate approximate eigen-
+functions (30) near their maximum, i.e. close to the
+equipotential (C2). To see this, zoom in on the equipo-
+tential by writing
+z ≡
+�√m + a
+��
+f ′(α) eiα
+(40)
+for m ≫ 1 and some finite a. The integral (30) then has
+a unique saddle point at ϕ = α−iδ1/√m+O(1/m), with
+δ1 = a
+�
+1 − i f ′′(α)
+2f ′(α)
+�−1
+. The saddle-point approximation
+of the wave function (30) thus yields
+ψm(z, ¯z) ∼
+1
+√
+2πℓ2
+1
+(2πm)1/4 eimf(α)+iΦ(α)
+×
+1
+�
+σ(α)
+exp
+�
+�− f ′(α) a2
+1 − i f ′′(α)
+2f ′(α)
+�
+� ,
+(41)
+where we used Eqs. (34)–(39) for the phase and norm of
+n(f(α)), fixed the integration constant in (39) to N0 =
+1
+2πℓ( m
+2π)1/4, and introduced the ubiquitous width
+σ(ϕ)2 ≡
+1
+f ′(ϕ)
+�
+1 + f ′′(ϕ)2
+4f ′(ϕ)2
+�
+.
+(42)
+This is the angle-dependent variance of the probability
+density of (41), written in (12) in terms of λ = 2f ′. In-
+deed, one has
+|ψm(z, ¯z)|2 ∼
+1
+2πℓ2
+e−2a2/σ2(α)
+√
+2πm σ(α),
+(43)
+which
+satisfies
+the
+desired
+normalization
+condition
+�
+d2x |ψ|2 = 1. Note that Eq. (41) reproduces the wave
+function stated in (11), once more with λ = 2f ′ and now
+involving the phase
+Θm(x) = mf(ϕ) + Φ(ϕ) −
+a2f ′′(ϕ)
+2f ′(ϕ)σ(ϕ)2 .
+(44)
+The Gaussian behavior of LLL-projected is thus mani-
+fest, as anticipated at the end of Sec. III B for the gen-
+eral WKB ansatz (20). It is illustrated in Fig. 3 for two
+choices of the confining potential (25). Finally, (41) gen-
+eralizes the behavior of isotropic states (3) [see (A1)], in-
+cluding the O(1/√m) contribution that we did not state
+here but that can be computed by incorporating the next
+order term δ2/m for the saddle point and repeating the
+analysis [73].
+Out[� ]=
+0
+0.2
+0.4
+0.6
+0
+1
+|ψm|2
+Figure 3. The density of a wave function (41) at m = 30 in an
+edge-deformed trap (25). The Gaussian behavior is manifest,
+as is the angle-dependent ‘roller coaster’ predicted by (43).
+Left: Elliptic potential given by (25) with f of the form (24)
+and k = 2. Right: Same edge-deformed trap as in Figs. 1–2.
+E.
+Comparison with elliptic wave functions
+To conclude this section, we now focus on the ‘flower’
+deformations (24) and show that the resulting transport
+equation is integrable: both the phase (34) and the norm
+(39) can be expressed in terms of elementary functions.
+These results are valuable in themselves since flower maps
+are the simplest edge deformations—they are analogues
+of Fourier modes for circle diffeomorphisms—, but also
+because their special case k = 2 reproduces known wave
+functions in elliptic harmonic traps [68, 74], providing an
+important benchmark for our WKB approach.
+Consider first the deformation (24) with α = cosh λ
+and β = sinh λ for an arbitrary integer k and a real pa-
+rameter λ. Then the energy quantization condition (35)
+can be integrated exactly, yielding
+E1
+m
+Ωm
+= 1
+2 + 1
+2
+�
+1 + Γm
+Ωm
+��
+1 + k2
+4
+�
+cosh(2λ) − 1
+��
+. (45)
+As for the solution of the transport equation, consisting
+of the phase (34) and norm (39), it is found to be
+n(θ) = N0 e
+i
+8
+Γm
+Ωm k sin(kθ) sinh(2λ)e
+i θ
+2
+�
+(
+Γm
+Ωm +1)
+�
+1− k2
+4
+�
++1− k
+2
+�
+×
+�cosh λ + eikθ sinh λ
+eikθ cosh λ + sinh λ
+� k2−4
+8k (1+ Γm
+Ωm )− 1
+2k + 1
+4
+×
+�
+−4eikθ + (e2ikθ − 1)k sinh(2λ)
+eikθ cosh λ + sinh λ
+(46)
+up to an overall constant phase.
+Note that (46) depends in a non-trivial way on the po-
+tential derivatives (33), with some simplification in the
+‘harmonic’ regime Γm = 0. Let us therefore apply Eqs.
+(45)–(46) to the case of an elliptic potential k = 2 with
+constant stiffness Ωm = Ω > 0 (hence Γm = 0). The cor-
+responding edge deformation (23) maps the isotropic har-
+monic potential V0(x) = Ω r2/2 on its anisotropic cousin,
+V (x) = Ω e−2λx2 + e2λy2
+2 cosh(2λ)
+,
+(47)
+
+10
+whose equipotentials are ellipses rather than circles. The
+energy correction (45) then becomes E1
+m =
+1
+2Ω[1 +
+cosh(2λ)] and the solution (46) of the transport equation
+can be written as
+n(θ) = N0
+�
+cosh λ − e2iθ sinh λ.
+(48)
+It is straightforward to use this data to obtain the elliptic
+version of the normalized Gaussian wave function (41):
+ψm(z, ¯z) ∼
+�2π
+m
+�1/4
+1
+2πℓ
+eimθ
+√
+cosh λ − e−2iθ sinh λ
+× exp
+�
+−e2iθ + tanh(λ)
+e2iθ − tanh(λ)a2
+�
+.
+(49)
+Crucially, the latter coincides with the large m approxi-
+mation of the exact LLL-projected eigenstates of the har-
+monic potential (47), as can be verified thanks to known
+asymptotic formulas for Hermite polynomials [68]. This
+is actually true even at subleading order in 1/√m, which
+we omit here for brevity.
+V.
+MANY-BODY OBSERVABLES
+This section finally applies the results of Secs. III–IV
+to fully-fledged QH droplets consisting of a large num-
+ber N ≫ 1 of electrons.
+Specifically, we exploit our
+insights on near-Gaussian single-particle wave functions
+(41) to compute many-body observables and read off the
+universal shape-dependent effects implied by the width
+(42). The density is treated first: it equals
+1
+2πℓ2 in the
+bulk, then drops to zero as an error function at the edge
+redge = ℓ
+�
+2Nf ′(ϕ). We then turn to the current and
+show that it is localized as a Gaussian on the edge, to
+which it is tangent.
+Third, correlations near the edge
+are found to display the usual power-law behavior of free
+fermions, dressed by radial Gaussian factors.
+This re-
+duces to known expressions in isotropic traps [59] and in
+the case of flower deformations (24) with k = 2, where
+one recovers the elliptic results of [68, 74]. Finally, the
+radial behavior of correlations is shown to be consistent
+with the effective low-energy field theory of edge modes:
+we derive it microscopically and obtain a chiral CFT in
+terms of the angle variable on the boundary.
+A.
+Density
+Consider a QH droplet of N ≫ 1 non-interacting 2D
+electrons subject to the Hamiltonian (1), with a very
+strong magnetic field B = dA and a weak edge-deformed
+potential (25). The ground state of this many-body sys-
+tem is a Slater determinant of the wave functions ψm for
+occupied states m = 0, 1, . . . , N −1, where we recall that
+m is a quantized action variable generalizing angular mo-
+mentum (see the red dots in Fig. 4). Each ψm yields a
+m
+Energy
+◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦••••
+N
+Fermi energy
+••••••••◦◦◦◦◦◦· · ·
+◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦· · ·
+2Λ
+Figure 4. The one-body spectrum (37), where the states that
+are occupied in the many-body ground state are highlighted in
+red and those that contribute to the low-energy Hamiltonian
+(64) are filled (black for ‘particles’, red for ‘holes’). The cutoff
+Λ is large but much smaller than N in the sense that Λ = O(1)
+in the thermodynamic limit.
+single-particle probability density |ψm(x)|2, the sum of
+which gives the many-body density
+ρ(x) =
+N−1
+�
+m=0
+|ψm(x)|2.
+(50)
+While WKB theory does not give access to the form of ψm
+at low m, large values of m should be correctly captured
+by the analysis of Sec. IV, in which case the one-body
+density is approximately Gaussian and given by (43). We
+now exploit this Gaussian form to evaluate the many-
+body density, both in the bulk and close to the edge.
+(Some technical details are highlighted along the way, as
+the same method will later allow us to study the many-
+body current and correlations.)
+The key point is that each wave function (43) is local-
+ized on an equipotential of V (x) with area 2πℓ2m, so the
+density close to some equipotential |z| = const ×
+�
+f ′(ϕ)
+only receives sizeable contributions from wave functions
+whose quantum number is close to |z|2/f ′(ϕ). Accord-
+ingly, the bulk density for 1 ≪ |z| ≪
+√
+N is obtained by
+letting the upper summation bound of (50) go to infinity
+and writing the approximate density as
+ρ(x) ∼
+1
+2πℓ2
+∞
+�
+m=m0
+e
+−
+2
+σ2(ϕ)
+�
+|z|
+√
+f′(ϕ) −√m
+�2
+√
+2πm σ(ϕ)
+,
+(51)
+where the lower summation bound m0 ≫ 1 is irrele-
+vant as long as it is much smaller than |z|2, and σ(ϕ) is
+the angle-dependent width (42). At large |z|, the Euler-
+Maclaurin formula allows us to approximate the sum over
+m by a (Gaussian) integral over √m. This yields the ex-
+pected density
+ρ(x) ∼
+1
+2πℓ2
+(52)
+in the bulk of a QH droplet with filling fraction ν = 1.
+An analogous argument can be carried out close to
+the droplet’s edge, with one key difference: the upper
+summation bound of (50) is now crucial. Thus, letting
+|z| =
+�√
+N + a
+�
+f ′(ϕ) with finite a in the large N limit
+
+11
+and using once more the approximate Gaussian behavior
+(43), the density (50) near the edge behaves as
+ρ(x) ∼
+1
+2πℓ2
+∞
+�
+k=1
+e
+−
+2
+σ2(ϕ)
+�
+a+
+k
+2
+√
+N
+�2
+√
+2πN σ(ϕ)
+,
+(53)
+where we changed variables as m ≡ N − k with k =
+O(
+√
+N) at large N and only kept track of leading-order
+terms. For N ≫ 1, the sum over k can once more be
+converted into an integral, now over k/2
+√
+N. This yields
+the asymptotic behavior
+ρ(r, ϕ) ∼
+1
+4πℓ2 erfc
+�
+1
+σ(ϕ)
+r − ℓ
+�
+2Nf ′(ϕ)
+ℓ
+�
+f ′(ϕ)
+�
+(54)
+where erfc denotes the complementary error function and
+the width (42) is inherited from that of LLL wave func-
+tions. This is a remarkably explicit result, announced in
+(14) with λ = 2f ′. It confirms that the density is roughly
+constant and equal to (52) in the bulk, then drops to zero
+within a distance of the order of the magnetic length (2)
+around the edge at r = ℓ
+�
+2N f ′(ϕ). See Figs. 2(a)–5(a).
+We stress that, in contrast to wave functions, the den-
+sity (54) only depends on the potential near the edge of
+the droplet: bulk deformations of the potential do not af-
+fect the quantized bulk density (52) in the limit of strong
+magnetic fields. In this sense, (54) is a universal formula
+for the density of any QH droplet of LLL states whose
+edge traces an equipotential of the form r2 = 2ℓ2N f ′(ϕ).
+It would be instructive to probe this local density in mod-
+ern quantum simulators [21–24].
+Note that the leading-order formula (54) receives a
+number of subleading corrections that can be systemati-
+cally computed in our formalism; these are omitted here
+for brevity. A related comment is that the bulk value
+density (52) is only valid at extremely strong magnetic
+fields, which stems from the simplification provided by
+the LLL projection. The actual density profile depends
+on the gradient of the potential, but this involves higher
+Landau levels that are beyond our scope [28, 29].
+B.
+Current
+The current of a droplet of N ≫ 1 electrons can simi-
+larly be evaluated as a sum over single-particle currents.
+To this end, recall that the gauge-invariant one-body
+probability current of a charged wave function ψ with
+mass M is a one-form ℏ j/M given by
+j = 1
+2i
+�
+ψ∗dψ − ψdψ∗ − 2iq
+ℏA|ψ|2�
+,
+(55)
+where the first term is only sensitive to the phase of ψ
+and A = 1
+2Br2 dϕ = ℏ
+q |z|2 dϕ in symmetric gauge. Thus,
+the many-body current of a Slater determinant of the
+occupied states ψm with m = 0, . . . , N − 1 is
+J =
+N−1
+�
+m=0
+jm,
+(56)
+where jm is the single-particle current (55) of each ψm.
+As before, the WKB approximation does not give ac-
+cess to wave functions for small m, but this is unim-
+portant close to the edge. In that regime, we have al-
+ready gathered all ingredients needed to evaluate the
+currents (55) up to small quantum corrections: the one-
+body density is given by (43), while the derivative of
+the phase is contained in (41) and the phase transport
+equation (32).
+In practice, the WKB phase Φ turns
+out to be negligible at leading order, and the only rel-
+evant parts of the phase are those already visible in (41):
+the (fast) phase eimf(ϕ) together with the contribution
+from A =
+�
+ℏ|z|2/q
+�
+dϕ eventually gives rise to the lead-
+ing ϕ component of the current, while the (slow) phase
+e−i[f ′′(ϕ)/2f ′(ϕ)]a2/σ2 yields its radial component that is
+non-zero whenever f ′′(ϕ) ̸= 0.
+Starting from these facts, it is straightforward to adapt
+the method of Sec. V A to the many-body current (56).
+Writing |z| =
+�√
+N +a
+��
+f ′(ϕ), the sum over m ≡ N −k
+becomes an integral over k/(2
+√
+N) = O(1), which yields
+the leading order result quoted in (15) with λ = 2f ′:
+J(r, ϕ) ∼ −
+e
+−
+2a2
+σ(ϕ)2
+(2πℓ2)3/2σ(ϕ)
+ℓ
+�
+2Nf ′(ϕ) dϕ + f ′′(ϕ)
+2f ′(ϕ) dr
+�
+2f ′(ϕ)
+(57)
+where a =
+�
+r−ℓ
+�
+2Nf ′(ϕ)
+��
+ℓ
+�
+2f ′(ϕ) and σ(ϕ) is given
+by (42). Both components in (57) receive subleading cor-
+rections that are omitted here. In particular, there is an
+O(1) term in Jϕ that is non-zero on the edge, even in the
+isotropic case f ′ = 1. The computation of that term re-
+quires the O(1/√m) correction that was omitted in (43).
+Using the metric ds2 = dr2 + r2dϕ2, one verifies that
+the one-form ℓ√2N f ′ dϕ+(f ′′/2f ′) dr in (57) is the dual
+of a vector tangent to the equipotential at the droplet’s
+edge [75]. Moreover, the norm squared of (57) is
+∥J(r, ϕ)∥2 ∼
+1
+2(2πℓ2)3 exp
+�
+−2
+�
+r − ℓ
+�
+2Nf ′(ϕ)
+�2
+ℓ2σ(ϕ)2f ′(ϕ)
+�
+,
+(58)
+showing that the current has a constant maximum along
+the edge but a varying width: see Fig. 5.
+Similarly to the density, it is important to remember
+that the LLL projection misses some important physics.
+Indeed, the actual bulk current is the symplectic gra-
+dient of the confining potential multiplied by the Hall
+conductance [28, 29, 41, 42].
+No such effect occurs in
+(57) because it requires higher Landau levels, which are
+beyond our scope.
+C.
+Correlations
+The methods that we have applied to density and cur-
+rent can also be used to compute electronic correlations
+near the edge. Indeed, consider as before an anisotropic
+droplet whose occupied one-body states have quantum
+
+12
+���
+���
+���
+x
+x
+x
+y
+y
+y
+√
+N
+−
+√
+N
+−
+√
+N
+√
+N
+(a)
+(b)
+(c)
+x
+x
+x
+y
+y
+y
+ℓ
+√
+2N
+−ℓ
+√
+2N
+−ℓ
+√
+2N
+ℓ
+√
+2N
+Figure 5. (a) The density (54) for N = 100 electrons and a ‘flower’ edge deformation (24) of order k = 3. The constancy of
+density in the bulk and its sharp decay at the boundary are manifest. (b) The current’s norm (58) for the same flower-shaped
+droplet. The localization on the edge equipotential (black line) is clearly visible, as is the position-dependent width of the
+Gaussian jump. (c) The correlation function C(x1, x2) (61) for the same flower-shaped droplet, seen as a function of x2 when
+x1 = (ℓ
+�
+Nλ(0), 0) is fixed close to the edge of the droplet.
+numbers m = 0, 1, . . . , N − 1. Then the correlation func-
+tion between the points x1 and x2 is
+C(x1, x2) =
+N−1
+�
+m=0
+ψ∗
+m(x1) ψm(x2),
+(59)
+which reduces to the density (50) when x1 = x2.
+As
+before, we rename m ≡ N − k and let the complex coor-
+dinates z, w corresponding to x1, x2 be such that
+z =
+�√
+N + a
+��
+f ′(ϕ1) eiϕ1,
+w =
+�√
+N + b
+��
+f ′(ϕ2) eiϕ2,
+(60)
+where a, b are finite at large N and ϕ1, ϕ2 are the po-
+lar angles of x1, x2.
+One can then plug the Gaussian
+wave functions (41) into (59), this time assuming k fi-
+nite, and performing the sum over k. The gradient ex-
+pansion of the potential implies that the ratio Γm/Ωm ∼
+ΓN/ΩN + O(ℓ2) is nearly constant in this regime, so Eq.
+(59) becomes a geometric sum over k that reproduces the
+result stated in (16) with λ = 2f ′:
+C(x1, x2) ∼
+eiΘN(x1,x2)
+(2π)3/2ℓ2√
+N
+1
+�
+σ(ϕ1)σ(ϕ2)
+×
+i e
+−
+a2
+σ(ϕ1)2 −
+b2
+σ(ϕ2)2
+2 sin
+�
+[f(ϕ1) − f(ϕ2)]/2
+�,
+(61)
+where σ(ϕ) is the angle-dependent width (42). The over-
+all phase ΘN(x1, x2) = ΘN(x2) − ΘN(x1) is given by
+(44), and involves in particular the WKB phase (34).
+Some features of (61) are worth emphasizing. First,
+note
+the
+Gaussian
+jump
+of
+power-law
+correlations
+near the edge, involving a free fermion correlation ∝
+sin([f(ϕ1)−f(ϕ2)]/2)−1 expressed in terms of f(ϕ1) and
+f(ϕ2); this is the standard static diagnostic of the pres-
+ence of edge modes [10, 59, 76]. A second striking aspect
+is the apparent lack of translation-invariance along the
+edge, caused not only by the argument f(ϕ1) − f(ϕ2) =
+� ϕ1
+ϕ2 dθ f ′(θ) but also by the widths σ(ϕ1) and σ(ϕ2). In
+particular, the product σ(ϕ1)−1/2σ(ϕ2)−1/2 is reminis-
+cent of prefactors picked up by primary fields in CFT
+under conformal maps.
+Since the boundary correlator (61) holds in any edge-
+deformed trap (25), it also applies to special cases of in-
+terest such as the anisotropic harmonic potential (47).
+The corresponding correlations were actually computed
+long ago in [68], and they coincide with our result (61)
+upon using the map (24) with k = 2, α = cosh λ,
+β = sinh λ. In fact, this specific setup is also well known
+in the context of the Coulomb gas, since edge correla-
+tions can then be related by a conformal map to the
+standard Euclidean correlation function (z1 − z2)−1 of a
+free fermion CFT [77].
+Finally,
+it is a simple matter to include time-
+dependence in the correlator (61). Indeed, the occupied
+one-particle states in (59) have definite energies Em given
+by (37) at large m. This spectrum is approximately linear
+close to the Fermi energy: changing variables according
+to m = N + k with k finite at large N, one has
+EN+k − EN ∼ ℏω k
+(62)
+where ω ≡ ℓ2ΩN/ℏ is the angular Fermi velocity given
+by the derivative (33) at m = N.
+Note that ω re-
+ceives a number of subleading quantum corrections in-
+volving e.g. the curvature ΓN in (33) [78–80]; we ne-
+glect those.
+In the linear regime (62), one can repeat
+the asymptotic computation of the correlator and find
+
+-
+1
+1
+1
+1
+-
+4
+r
+-
+1
+1
+1
+1
+1
+1
+-
+1
+-
+1
+1
+-
+1
+1
+1
+113
+once more a result of the form (61), now with a time-
+dependent overall phase and a time-dependent denom-
+inator 2 sin
+�
+[f(ϕ1) − f(ϕ2) − ω(t1 − t2)]/2
+�
+.
+This ex-
+hibits the standard ballistic propagation of correlations
+in a CFT, which we confirm below by deriving the effec-
+tive low-energy dynamics of our droplet.
+D.
+Edge modes
+The effective low-energy description of anisotropic QH
+droplets can be derived similarly to the isotropic case
+[59] inspired by Luttinger-liquid theory [81].
+This has
+the advantage of circumventing topological field theory,
+at the cost of failing to apply in fractional QH states [6–
+10]. We now provide such a first-principles calculation,
+eventually concluding that edge modes span a free chiral
+CFT expressed in terms of the angle coordinate f(ϕ)
+along the boundary.
+Aside from its intrinsic interest,
+this computation provides an independent check of the
+validity of the correlation (61).
+Our starting point is the second-quantized Hamilto-
+nian in a Fock space of free fermions,
+H =
+�
+m,n
+(Em,n − µ)a†
+m,nam,n.
+(63)
+Here the sum runs over eigenstates of the one-body
+Hamiltonian (1), labeled by the Landau level n ∈ N
+and the ‘action variable’ quantum number m ∈ N within
+each level. Their energies are denoted Em,n, and µ is
+some chemical potential to be fixed below.
+For each
+pair (m, n), the operator a†
+m,n creates the corresponding
+eigenstate. The exact energy spectrum is unknown, but
+this is not an issue since low-energy excitations all be-
+long to the LLL, with an approximately linear dispersion
+(62) within a window [−Λ, Λ] around the Fermi momen-
+tum. Then choosing the chemical potential of (63) as
+µ = const − 1
+2ℏω, the low-energy approximation of the
+many-body Hamiltonian (63) becomes
+H ∼
+�
+k∈[−Λ,Λ]
+ℏω (k + 1/2) a†
+N+kaN+k,
+(64)
+where the sum is over all integers k in the specified win-
+dow. Since aN+k are fermionic Fock space operators, the
+Λ → ∞ limit of (64) yields a gapless Hamiltonian for free
+fermions written in Fourier modes, and the chemical po-
+tential enforces antiperiodic (Neveu-Schwarz) boundary
+conditions. It now only remains to relate the edge CFT
+to bulk wave functions.
+To this end, note that the Fock space operators in (64)
+create states in the LLL that are given by the Gaussian
+wave function (41). One can therefore express them in
+terms of 2D creation operators c†(x):
+a†
+N+k ∼
+� f ′(ϕ) dϕ
+√
+2π
+ei(k+1/2)f(ϕ) Ψ†(f(ϕ)),
+(65)
+where the 1D fermionic field Ψ is defined as a radial in-
+tegral involving the wave function (41),
+Ψ†(f(ϕ)) ≡ ei(N−1/2)f(ϕ) eiΦ(ϕ)
+f ′(ϕ)
+�
+σ(ϕ)
+×
+� ∞
+0
+r dr
+ℓ(2πN)1/4 c†(x) e
+−
+1
+σ2(ϕ)
+�
+r
+√
+2ℓ2f′(ϕ) −
+√
+N
+�2
+(66)
+with σ the width (42) and Φ the WKB phase (34). Note
+that in writing (66) we assumed that the momentum in-
+dex k and the cutoff Λ are of order O(1) in the thermo-
+dynamic limit. It is then clear that the operator Ψ†(ϕ)
+creates an electron at the position ϕ on the edge.
+From this point onward, the derivation of the low-
+energy effective field theory is essentially done. Indeed,
+the Hamiltonian (64) expressed in terms of the edge field
+(66) reads
+H = ℏω
+�
+dθ Ψ†(θ) (−i∂θ)Ψ(θ).
+(67)
+Furthermore, the normalization of the field Ψ defined by
+(66) is canonical in angle variables: using the standard
+anticommutator {c(x1), c†(x2)} = δ(2)(x1 −x2), one sim-
+ilarly finds {Ψ(f(ϕ1)), Ψ†(f(ϕ2))} = δ(f(ϕ1) − f(ϕ2)) in
+terms of the Dirac delta function on a circle. This de-
+termines the kinetic term of the action functional for Ψ,
+which is thus a canonical fermionic expression ∼ Ψ†∂tΨ.
+It can be combined with the Hamiltonian (64) to write
+down the fermionic action functional of edge modes,
+S[Ψ, Ψ†] = ℏ
+�
+dt dθ iΨ†(θ)
+�
+∂t + ω∂θ
+�
+Ψ(θ),
+(68)
+where we recall that the angular Fermi velocity is ω =
+ℓ2ΩN/ℏ, given by (33). This is manifestly a (1+1)D free
+chiral CFT in terms of the angle variable θ = f(ϕ).
+The low-energy effective theory (68) is universal: for
+any trapping potential, edge modes are described by a
+chiral fermionic CFT expressed in terms of the angle co-
+ordinate of the trap at the boundary, as could have been
+guessed from the dynamics of electronic guiding centers
+induced by the potential V in the non-commutative plane
+(6). In the present case, the angle coordinate was just
+θ = f(ϕ); more general cases involve more complicated
+action-angle variables. We stress that the angle variable
+generally has nothing to do with other obvious position
+coordinates, such as the polar angle ϕ or the arc length
+s(ϕ) = ℓ
+√
+2N
+� ϕ
+0
+dα
+�
+f ′(α) + f ′′(α)2
+4f ′(α) .
+(69)
+Any such ‘wrong’ coordinate makes the apparent Fermi
+velocity of edge modes position-dependent. This is rem-
+iniscent of inhomogeneous CFTs whose light-cones are
+curved owing to the presence of a non-zero space-time
+curvature [82–89].
+However, one should keep in mind
+that our edge modes sense a flat metric ω2dt2 − dθ2 =
+
+14
+ω2dt2 − f ′(ϕ)2dϕ2 whose light-cones are straight lines in
+terms of the angle variable θ = f(ϕ).
+Let us conclude this section by showing that the action
+(68) is consistent with the seemingly complicated corre-
+lator (61). We start from the definition (66) to write the
+1D correlation function ⟨Ψ†(θ1)Ψ(θ2)⟩ as a double radial
+integral of the 2D quantity ⟨c†(x1)c(x2)⟩. Now using the
+asymptotic relation (61), one finds that all normaliza-
+tions simplify and the correlator of the edge field boils
+down to
+⟨Ψ†(θ1)Ψ(θ2)⟩ = 1
+2π
+i
+2 sin
+�
+[θ1 − θ2]/2
+�.
+(70)
+The same result would have been obtained directly from
+the low-energy action (68): it is a correlation function
+of free gapless fermions written in terms of the angle
+coordinates θ1 = f(ϕ1) and θ2 = f(ϕ2). As a bonus,
+time-dependent correlations automatically satisfy the be-
+haviour ∝ sin
+�
+[θ1 − θ2 − ω(t1 − t2)]/2
+�−1 stated at the
+end of Sec. V C.
+VI.
+CONCLUSION AND OUTLOOK
+This work was devoted to a detailed study of meso-
+scopic droplets of non-interacting planar electrons placed
+in a perpendicular magnetic field and confined by star-
+shaped anisotropic traps with scale-invariant level curves.
+In particular, we provided explicit formulas for the corre-
+sponding wave functions and energy spectrum, allowing
+us to compute the many-body density, current, and cor-
+relations of an entire droplet. The resulting low-energy
+edge modes were eventually shown to behave as a chiral
+CFT in terms of the angle variable along the boundary.
+This was all achieved in a regime of high magnetic fields,
+ultimately equivalent to a semiclassical limit.
+These results pave the way for a number of applications
+and follow-ups. Indeed, recent advances in quantum sim-
+ulation suggest the tantalizing possibility of probing lo-
+cal properties of QH-like droplets in the lab [21–24], both
+for static ground states and their dynamical edge excita-
+tions. The density (54) or the current (57) then predict
+observable shape-dependent widths, while the low-energy
+theory (68) predicts the ballistic propagation of local
+boundary disturbances with a position-dependent veloc-
+ity. More generally, the geometry of the QH effect [33–
+39] could soon become relevant for experiments involv-
+ing ultracold atoms or photonics. Our framework pro-
+vides a bridge between this field of mathematical physics
+and concrete observables in mesoscopic quantum physics.
+Verifying the predictions put forward here, through lin-
+ear response experiments or direct imaging, would be a
+fascinating example of many-body quantum mechanics
+at work.
+Turning now to theory, the link between our formal-
+ism and QH symmetries deserves further study: following
+the series of works [58–63], one can think of edge defor-
+mations as approximately unitary operators acting on
+many-body QH states. It is then natural to wonder how
+these operators get composed together; do they span a
+Virasoro group as in CFT? If yes, how to derive the cen-
+tral charge c = 1 in terms of microscopic wave functions?
+More broadly, what are the operators implementing area-
+preserving deformations in the sense of the WKB ansatz
+(20)? One expects these operators to provide a finite (ex-
+ponentiated) form of the operators studied in [58, 60, 61],
+with non-commutative composition laws consistent with
+the geometry (6) of LLL physics.
+Note that the discussion above was mostly focused on
+leading-order properties. For instance, one may wonder
+what are irrelevant corrections to the edge field theory
+(68), especially following the recent numerical observa-
+tion [79] that density waves on the edge satisfy a non-
+linear Korteweg-de Vries equation at late times.
+This
+regime is presumably described by small droplet defor-
+mations of the form r2 �→ r2 + α(ϕ), spanning a U(1)
+Kac-Moody algebra whose level is sensitive to the fill-
+ing fraction [58, 60, 61]. The resulting non-linear edge
+waves would then be described by an evolution equa-
+tion in an infinite-dimensional group manifold. This per-
+spective is standard in hydrodynamics [90–92], but it
+has only recently come to be appreciated in condensed
+matter physics [93]. The geometric study initiated here
+provides a basis for considerations of this kind in the
+QH effect, including the possibility of inhomogeneous
+(position-dependent) corrections in anisotropic traps.
+Another obvious extension of this work is the fractional
+QH regime. In that context, no single-particle descrip-
+tion is available, but many-body predictions such as the
+edge density (54) or the current (57) conceivably display
+universal geometric features that would remain true in
+interacting many-body ground states [42]. It would be
+thrilling to derive such predictions from the family of
+edge transformations studied here, either from a micro-
+scopic analysis of the Laughlin wave function, or thanks
+to the reformulation of fractional QH states as CFT cor-
+relation functions [94].
+ACKNOWLEDGMENTS
+We are grateful to Laurent Charles for illuminating
+discussions on semiclassical methods in Kählerian geo-
+metric quantization. B.O. also thanks Mathieu Beauvil-
+lain, Nathan Goldman, and Marios Petropoulos for col-
+laboration on closely related subjects.
+Finally, we ac-
+knowledge useful and motivating interactions with Jean
+Dalibard, Benoit Douçot, Jean-Noël Fuchs, Marc Geiller,
+Gian Michele Graf, Semyon Klevtsov, Titus Neupert
+and Nicolas Regnault. The work of B.O. is supported
+by the European Union’s Horizon 2020 research and in-
+novation programme under the Marie Skłodowska-Curie
+grant agreement No. 846244. B.L. acknowledges fund-
+ing from the European Research Council (ERC) under
+the European Union’s Horizon 2020 research and inno-
+vation program (Grant No. ERC-StG-Neupert-757867-
+
+15
+PARATOP). P.M. gratefully acknowledges financial sup-
+port from the Wenner-Gren Foundations (No. WGF2019-
+0061). B.E. is supported by the ANR grant TopO No.
+ANR-17-CE30-0013-01.
+Appendix A: Isotropic droplets
+Most of this work is concerned with anisotropic prop-
+erties, so isotropic results provide a useful benchmark.
+They are simpler than their anisotropic counterparts and
+mostly well-known in the literature, so their properties
+are concisely summarized here. We begin by recalling el-
+ementary aspects of the one-body energy spectrum based
+on the exact wave functions (3), then turn to many-body
+observables.
+1.
+One-body spectrum
+Consider a spin-polarized 2D electron governed by the
+Landau Hamiltonian (1) with an isotropic confining po-
+tential V (x) = V0(r2/2). At very strong magnetic fields,
+the resulting one-body spectrum is well approximated
+by the solution of the LLL-projected eigenvalue equa-
+tion (7). As the potential is isotropic, it commutes with
+angular momentum, so the eigenstates of P V P are wave
+functions (3) with definite angular momentum. Note that
+these confirm the general near-Gaussian behavior found
+in (43): letting |z| = √m+a with finite a, one finds that
+(3) behaves at large m as
+φm(x) =
+eimϕ
+√
+2πℓ2
+1
+(2πm)1/4 e−a2 �
+1 + O(1/√m)
+�
+. (A1)
+The energy Em of each state (3) is readily found by com-
+puting the wave function ⟨z, ¯z|PV0(r2/2)P|φm⟩, which
+yields the exact eigenvalue
+Em = ⟨φm|V |φm⟩ = 1
+m!
+� ∞
+0
+dt tm e−t V0(ℓ2t)
+(A2)
+in terms of the integration variable t ≡ |z|2. Note in pass-
+ing that this is the value one would find from first-order
+perturbation theory of the full Landau Hamiltonian (1):
+by construction, LLL-projected physics is only sensitive
+to first-order effects of the potential, while higher-orders
+ultimately involve higher Landau levels.
+Now fix an index m ≥ 0. What is the corresponding
+equipotential in the sense of (9)? To answer this in the
+classical limit, we let m → ∞ while fixing the value of
+ℓ2m = O(1), and evaluate the integral (A2) thanks to a
+saddle-point approximation. The result is
+Em = V0(ℓ2m) + ℓ2Ωm + ℓ2
+2 Γm + O(ℓ4),
+(A3)
+where Ωm and Γm were defined in (33). This is consistent
+with Eqs. (13) and (36) with λ = 2f ′ = 2.
+2.
+Many-body aspects
+The sequence followed here is the same as in Sec. V: we
+start with the density, then consider the current and cor-
+relations close to the edge. In all cases, the edge asymp-
+totics reproduce the formulas of Sec. V for the simplest
+case where f ′(ϕ) = 1.
+Density. Let N ≫ 1 non-interacting planar electrons
+be subjected to the Hamiltonian (1), with a very strong
+magnetic field B = dA and a weak isotropic poten-
+tial V (x) = V0(r2/2).
+The ground state wave func-
+tion of this many-body system is a Slater determinant of
+the occupied single-particle eigenstates φ0, φ1, . . . , φN−1
+given by (3), each of which has a one-body density
+|φm(x)|2. The resulting many-body density is thus (50),
+which can be computed in closed form in the very spe-
+cial case of states (3) with definite angular momentum:
+it is a normalized incomplete gamma function ρ(x) =
+(2πℓ2)−1Γ(N, |z|2)/Γ(N).
+Constancy of density in the
+bulk is then manifest, as is its drop to zero close to
+the edge |z| =
+√
+N, with an error function behavior
+that can be deduced from known asymptotic formulas
+for gamma functions [76] and reproduces Eqs. (14)–(54)
+with λ = 2f ′ = 2.
+Current. For the LLL states (3) with definite angular
+momentum, each one-body current (55) is purely angu-
+lar, i.e. it reads jm = (. . .)dϕ. The sum (56) can then
+be evaluated in closed form owing to an exact cancella-
+tion between the contribution of the states m and m+ 1,
+eventually leading to a current only due to the (N − 1)th
+wave function. It is then trivial to show that the current
+is localized as a Gaussian close to the edge, since this is
+also true of the underlying single-particle wave function.
+This reproduces Eqs. (15)–(57) with λ = 2f ′ = 2.
+Correlations.
+The computation of electronic correla-
+tions close to the edge is similar to that of the density.
+Indeed, since the many-body ground state wave function
+is a Slater determinant, the two-point correlation func-
+tion in the ground state can be expressed as in (59). The
+exact wave functions (3) can then be used to write the
+correlation (59) as an incomplete gamma function (this
+time with a complex argument):
+C(z, ¯z, w, ¯w) =
+1
+2πℓ2
+Γ(N, ¯zw)
+Γ(N)
+e−(|z|2+|w|2)/2 ez ¯
+w. (A4)
+It is then manifest that bulk correlations coincide with
+the kernel (5) at leading order in the thermodynamic
+limit. As for the edge behavior, it can be extracted e.g.
+from a steepest descent argument [76] and reproduces
+Eqs. (16)–(61) with λ = 2f ′ = 2.
+Appendix B: Semiclassical expansion of P V P
+In this appendix, we derive (22) starting from (21).
+To this end, think of V (x, y) as some smooth function of
+
+16
+(x, y) whose arguments can be complexified, and change
+the integration variables (x, y) of (21) to
+s ≡ x −
+ℓ
+√
+2(z + ¯w),
+t ≡ y +
+iℓ
+√
+2(z − ¯w).
+(B1)
+In terms of (s, t), the integrals in (21) are two line in-
+tegrals in the complex plane, each along a path from
+−∞ + ic to +∞ + ic, where c is some irrelevant real con-
+stant (a different one for s and t). The advantage of the
+change of variables (B1) is to make the exponential factor
+in (21) purely Gaussian:
+⟨z, ¯z|P V (x)P|w, ¯w⟩ =
+1
+(2πℓ2)2 e− |z−w|2
+2
+e
+z ¯
+w−¯zw
+2
+×
+�
+ds dt V
+�
+s+ ℓ
+√
+2(z + ¯w), t− iℓ
+√
+2(z − ¯w)
+�
+e− s2+t2
+2ℓ2 .
+(B2)
+We then complexify V , thus replacing V (x, y) by V(z, ¯z),
+where V(z, ¯w) is a function of two complex variables,
+holomorphic in the first one and anti-holomorphic in the
+second.
+We can then deform independently both inte-
+gration contours for s and t back to the real line. For
+small ℓ, the Gaussian factor of (B2) localizes everything
+to s = t = 0. We now use our assumption of slow varia-
+tion of V (x) to Taylor-expand it as
+V
+�
+s +
+ℓ
+√
+2(z + ¯w), t −
+iℓ
+√
+2(z − ¯w)
+�
+∼
+�
+V + s2
+2 ∂2
+xV + t2
+2 ∂2
+yV
+����� ℓ
+√
+2 (z+ ¯
+w),− iℓ
+√
+2 (z− ¯
+w)
+�,
+(B3)
+where we only kept terms that give non-zero contribu-
+tions to the O(ℓ2) approximation of the integral (B2).
+Note that everything is evaluated at (x, y) = ( ℓ
+√
+2(z +
+¯w), − iℓ
+√
+2(z − ¯w)); in complex coordinates, this is just
+the point (z, ¯w), so it is simpler to write the potential
+as V(z, ¯w). Plugging the expansion (B3) into (B2) then
+yields the result (22).
+Appendix C: The transport equation
+The purpose of this appendix is to derive the real and
+imaginary parts of the transport equation in (32) and
+(38), respectively, by imposing the eigenvalue equation
+(7) based on our WKB ansatz (30) in the case of edge-
+deformed droplets. The derivation relies on expanding
+the energy and the potential as in (9) and (22).
+It is
+divided in two parts. First, we use the eigenvalue equa-
+tion to derive the constraint (31), and let z belong to an
+equipotential so that the whole equation boils down to
+a 1D integral identity. Second, we show that the inte-
+gral has a sharp saddle point in the large m limit; this
+allows us to rephrase the integral constraint as a first-
+order transport equation for the unknown function n.
+1.
+Evaluation along an equipotential
+Using the wave functions (18)–(19) and the expansion
+(22) of the potential along with the projector property
+P 2 = P, the eigenvalue problem (7) reads
+0 =
+�
+R2
+d2w
+2πℓ2 e− |z−w|2
+2
++ z ¯
+w−¯zw
+2
+��
+V+ ℓ2
+2 ∇2V
+����
+(z, ¯
+w)−Em
+�
+×
+�
+dθ n(θ) eimθ δ2�
+w −
+�
+F(m, θ), G(m, θ)
+��
+(C1)
+up to O(ℓ4) corrections [95]. In the case of edge-deformed
+traps, V(z, ¯w) is the bicomplex potential given in (28)
+and the delta function localizes the whole integral over
+w to a level curve (27) with K = m. Integrating over w
+and changing the integration variable from θ = f(ϕ) to
+ϕ yields Eq. (31).
+Note that the structure of Eqs. (C1) and (31) is 0 =
+e−|z|2/2 F(z) for a holomorphic function F(z), so setting
+F(z) = 0 on a closed curve implies F(z) = 0 everywhere.
+Accordingly, we will solve (C1)–(31) along the equipo-
+tential (26) by fixing K = m and parametrizing
+z =
+�
+mf ′(α) eiα,
+α ∈ [0, 2π).
+(C2)
+This ensures that all three terms in the exponent of (31)
+are of the same order O(m). Then (31) with the choice
+(C2) and ϕ ≡ α + ε becomes
+0 =
+� π
+−π
+dε f ′(α + ε) n(f(α + ε)) exp
+�
+imf(α + ε) − 1
+2mf ′(α + ε) + m
+�
+f ′(α)f ′(α + ε) e−iε�
+×
+�
+V
+��
+mf ′(α) eiα,
+�
+mf ′(α + ε) e−i(α+ε)�
++ ℓ2
+2 ∇2V − E0
+m − ℓ2E1
+m
+�
+.
+(C3)
+This rewriting will allow us to carry out the integral thanks to the saddle-point approximation, obtained by expanding
+all terms in powers of ε and leading to a differential equation for n(θ).
+
+17
+2.
+Saddle point and transport equation
+The saddle-point expansion of the integral (C3) is cumbersome but straightforward. The strategy is to expand all
+factors in the integrand up to a suitable power of ε, then perform the resulting integrals of the form
+�
+dε ε# e−Cε2,
+where C is some f-dependent coefficient [see e.g. (C5)]. The powers of ε involved are typically small, as higher powers
+are suppressed in the classical limit [large m and ℓ2m = O(1)]. The fact that the argument of n(θ) also involves a
+factor ε eventually converts the integral into a transport equation of the form n′(θ) ∝ n(θ) [see (C16)].
+We start with (C3) and first expand the exponential, then the potential with its Laplacian, and finally the simplest
+f ′(ϕ)n(f(ϕ)) prefactor. For convenience, we introduce the notation
+A ≡ f ′′
+f ′ ,
+B ≡ f ′′′
+f ′
+(C4)
+for combinations of derivatives of f that often appear below; from now on, expressions of the form f or f ′, etc., are
+all implicitly evaluated at α unless specified otherwise (so f ≡ f(α), f ′ ≡ f ′(α), etc.). Note for future reference the
+useful relation A′ = B − A2.
+The exponential. Using the notation (C4), one has
+exp
+�
+imf(α + ε) − 1
+2mf ′(α + ε) + m
+�
+f ′(α)f ′(α + ε) e−iε�
+∼ eimf+ 1
+2 mf ′ exp
+�
+− 1
+2mf ′ �
+1 + A2
+4
+�
+ε2� �
+1 + mf ′ε3 �
+i
+6 − A
+4 − iB
+12 + iA2
+8 − AB
+8 + A3
+16
+��
+(C5)
+where the factor exp
+�
+imf + mf ′/2
+�
+is ultimately irrelevant for the eigenvalue equation (C3), so we will not include
+it in what follows. The main point of (C5) is to exhibit the leading Gaussian behavior exp
+�
+−(mf ′/2)(1 + A2/4)ε2�
+of the integrand, which will eventually allow us to convert (C3) into a differential equation for the unknown function
+n(θ). In fact, the same exponential term appears in the approximately Gaussian wave function (43).
+The potential. We now turn to the expansions of the potential and of its Laplacian. As a first step, our task is to
+expand the potential
+V
+��
+mf ′ eiα,
+�
+mf ′(α + ε) e−i(α+ε)�
+= V0
+�
+�
+�ℓ2m
+√f ′ �
+f ′(α + ε) e−iε
+f ′
+�
+1
+2i log
+� √f ′ e2iα+iε
+√
+f ′(α+ε)
+��
+�
+�
+�
+∼ V0
+�
+ℓ2m
+�
+1 − iε
+�
+1 + A2
+4
+�
++ ε2 �
+− 1
+2 + B
+8 − 3A2
+8
+− A3
+4i − A4
+16 + AB
+4i + A2B
+32
+� ��
+∼ V0
+�
+ℓ2m
+�
+− iℓ2m ε
+�
+1 + A2
+4
+�
+V ′
+0
+�
+ℓ2m
+�
+− 1
+2ℓ4m2ε2 �
+1 + A2
+4
+�2
+V ′′
+0
+�
+ℓ2m
+�
++ ℓ2m ε2 �
+− 1
+2 + B
+8 − 3A2
+8
+− A3
+4i − A4
+16 + AB
+4i + A2B
+32
+�
+V ′
+0
+�
+ℓ2m
+�
+,
+(C6)
+where we used (28) and then the notation (C4). Aside
+from the contribution of the Laplacian, these are all the
+terms of the potential needed in the eigenvalue equation
+(C3) along an equipotential. As expected, they all ul-
+timately involve the potential and its derivatives at the
+equipotential (26). For ε = 0, the whole expression boils
+down to V0(ℓ2m) alone.
+Let us now turn to the Laplacian term.
+The
+eigenvalue
+equation
+(C3)
+requires
+the
+Laplacian
+evaluated
+at
+the
+complexified
+point
+(z, ¯w)
+=
+��
+mf ′(α) eiα,
+�
+mf ′(α + ε) e−i(α+ε)�
+.
+In
+practice,
+the Laplacian term is multiplied by ℓ2 in (C3), so we
+may safely set ε = 0 when computing it; this removes
+the complexification and allows us to write the Laplacian
+contribution in (C3) as
+ℓ2
+2 ∇2V ∼ ℓ2
+f ′
+�
+1 − B
+4 + A2
+2
+�
+V ′
+0(ℓ2m)
++ ℓ4m
+f ′
+�
+1 + A2
+4
+�
+V ′′
+0 (ℓ2m),
+(C7)
+which follows from the general expression (29) evaluated
+on the equipotential (26).
+All together. Let us finally consider the very first factor
+on the right-hand side of (C3), namely
+f ′(α + ε) n(f(α + ε)) ∼ f ′n(f) + ε
+�
+f ′′n(f) + f ′2n′(f)
+�
+(C8)
+where higher powers of ε are negligible at this order. To
+see why they may be neglected, it is helpful to investigate
+
+18
+the general structure of the small ℓ expansion of (C3):
+the exponential term in (C5) has the form
+exp[imf(. . .)] ∼ const × e−mΛε2(1 + mLε3)
+(C9)
+with m ≫ 1 and Λ, L some O(1) coefficients.
+Sim-
+ilarly, the potential expansion (C6) together with the
+Laplacian correction (C7) can schematically be written as
+V0+ ℓ2
+2 ∇2V0 ∼ V0+ℓ2W0+Gε+Hε2, where V0 ≡ V0(ℓ2m)
+while W0, G, H are again some O(1) coefficients. Finally,
+the expansion (C8) of the prefactor roughly has the form
+f ′n e(...) ∼ const × (f ′n + εIn′ + εJn),
+(C10)
+where I, J are O(1) coefficients.
+Putting together the
+schematic expressions (C9)–(C10) and using the fact that
+constant (i.e. ε-independent) contributions are irrelevant,
+the eigenvalue equation (C3) becomes
+0 =
+�
+dε (f ′n + εIn′ + εJn)e−mΛε2(1 + mLε3)
+×
+�
+V0 + ℓ2W0 + Gε + Hε2 − E0
+m − ℓ2E1
+m
+�
+.
+(C11)
+Here the right-hand side is a sum of integrals whose in-
+tegrand has the form εn e−mΛε2. For odd n, each such
+integral vanishes; for even n, it is non-zero and scales as
+m−n/2. This is why only the first order in ε is needed in
+the expansion (C8): higher powers of ε would yield sub-
+leading corrections to (C11), which can be consistently
+taken into account only by expanding the exponential,
+potential and Laplacian terms up to orders in ε higher
+than what we did above. Here we content ourselves with
+the zeroth and first order terms in ℓ2 (i.e. in 1/m). At
+that level of approximation, (C11) yields the zeroth order
+statement
+V0 − E0
+m = 0
+(C12)
+and the first-order result
+f ′n
+�
+Λℓ2m[W0 −E1
+m]+ H
+2 + 3LG
+4Λ
+�
++ G
+2
+�
+In′ +Jn
+�
+= 0,
+(C13)
+where ℓ2m = O(1) as before. Eq. (C12) confirms that the
+eigenvalue equation holds if E0
+m = V0(ℓ2m), i.e. if the en-
+ergy of the eigenstate |ψm⟩ is that of its equipotential at
+leading order [recall (9)]. More important, (C13) yields
+a transport equation for n, whose schematic form is
+GI
+2
+n′
+n +f ′�
+Λℓ2m(W0−E1
+m)+ H
+2 + 3LG
+4Λ
+�
++ GJ
+2 = 0. (C14)
+We now rely on the expansions (C5)–(C8) to write this
+transport equation explicitly: using the notation (33) and
+plugging (C5)–(C8) into (C3) yields the condition
+0 =
+�
+dε e
+− Kf ′
+2
+�
+1+ A2
+4
+�
+ε2 �
+1 + ε
+�
+A + f ′ n′(f)
+n(f)
+�� �
+1 + Kf ′ε3 �
+i
+6 − A
+4 − iB
+12 + iA2
+8 − AB
+8 + A3
+16
+��
+×
+�
+−iℓ2Kε
+�
+1 + A2
+4
+�
+Ωm − ℓ2Kε2
+2
+�
+1 + A2
+4
+�2
+Γm + ℓ2Kε2 �
+− 1
+2 + B
+8 − 3A2
+8
+− A3
+4i − A4
+16 + AB
+4i + A2B
+32
+�
+Ωm
++ ℓ2
+f ′
+�
+1 − B
+4 + A2
+2
+�
+Ωm + ℓ2
+f ′
+�
+1 + A2
+4
+�
+Γm − ℓ2E1
+m
+�
+,
+(C15)
+whose structure is that announced in (C11), as had to
+be the case. What remains is to multiply all the factors
+in the integrand, keep track of powers of ε and integrate
+over ε, which leads to
+iR′/R =
+�
+1 + A2
+4
+�
+Γm
+2Ωm + 1 − B
+4 + A2
+2 − f ′ E1
+m
+Ωm
+−
+1
+1 + A2
+4
+��
+B
+8 + A4
+16 − A2B
+32
+�
++ i
+�
+A
+4 + 3A3
+16 − AB
+8
+��
+,
+(C16)
+where we introduced R ≡ R(α) ≡ n(f(α)) for simplicity.
+This is the transport equation for the O(1) multiplica-
+tive factor of the WKB ansatz (30). Its real and imag-
+inary parts, respectively, govern the phase and norm of
+n(f(ϕ)) ≡ N(ϕ) eiΦ(ϕ):
+−Φ′ =
+�
+1 + A2
+4
+�
+Γm
+2Ωm − f ′ E1
+m
+Ωm
++
+1
+1+ A2
+4
+�
+1 + 3A2
+4
++ A4
+16 − 3B
+8 − A2B
+32
+�
+, (C17)
+N ′/N = −
+1
+1 + A2
+4
+�
+A
+4 + 3A3
+16 − AB
+8
+�
+.
+(C18)
+The identity B = A′+A2 then reduces these two relations
+to Eqs. (32) and (38) in the main text.
+
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