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+Kondo Resonance, Pomeranchuk Effect, and Heavy Fermi Liquid in Twisted Bilayer
+Graphene - A Numerical Renormalization Group Study
+Geng-Dong Zhou1 and Zhi-Da Song1, ∗
+1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
+(Dated: January 13, 2023)
+Low energy electron Hamiltonian in the magic-angle twisted bilayer graphene can be equivalently
+reformulated as a topological heavy fermion model [Phys. Rev. Lett. 129, 047601 (2022)]. It consists
+of effective localized f-electrons at AA-stacking regions and itinerant Dirac c-electrons. In this work,
+we applied systematic analytical and numerical renormalization group analyses to a single-impurity
+version of this model.
+We obtained a phase diagram consisting of a Fermi liquid phase in the
+Kondo regime, a Fermi liquid phase in the frozen impurity regime, and various local moment phases
+with different spin momenta. Remarkably, this single-impurity phase diagram explains a series of
+experimental discoveries reported recently: (i) the zero-energy peak at fillings 1 ≲ |ν| < 2 observed
+in STM at low temperatures (T < 1K) [Nature 588, 610 (2020), Nature Physics 17, 1375 (2021),
+Nature 600, 240 (2021), Nature 589, 536 (2021)], (ii) the cascade of transitions observed in STM at
+higher temperatures [Nature 582, 198 (2020), Nature Physics 17, 1375 (2021)], (iii) the Pomeranchuk
+effect at ν ≈ ±1 observed in transport and compressibility measurements [Nature 592, 214 (2021),
+Nature 592, 220 (2021)], which show that the Fermi liquid ground state develops local moments
+upon heating, and (iv) various transport experiments showing resistance peaks but no gaps around
+ν ≈ ±1. For the first time, we point out that all these phenomena result from a simple unified
+mechanism - the Kondo effect. The Fermi liquid state at ν ≈ ±1 exhibiting the zero-energy peak
+is stabilized by the Kondo screening with a Kondo temperature TK ≈ 1.5K. A higher temperature
+will suppress the Kondo screening and favor a local moment phase that obeys Curie’s law and
+contributes to an entropy of the order of Boltzmann’s constant (per moir´e cell).
+We computed
+the spectral densities, entropies, and spin susceptibilities at various fillings and temperatures, and
+obtained results quantitatively comparable to experiments. We also predict the heavy Fermi liquid
+as the ground state in a wide range of fractional fillings and conjecture that it is the parent state
+for the observed unconventional superconductivity.
+I.
+INTRODUCTION
+Since the first discovery of the superconductivity [1]
+and correlated insulators [2] in magic-angle twisted bi-
+layer graphene (MATBG) [3], MATBG has become a new
+platform to study novel correlation effects in flat-band
+systems and has attracted extensive attentions. Remark-
+ably rich physics, including interplay between supercon-
+ductivity [4–11] and strong correlation [4, 6–8, 12–19], in-
+teraction driven Chern insulators [20–26], strange metal
+behaviors [27–29], and the Pomeranchuk effect [30, 31],
+etc., have been observed in MATBG. Several theoretical
+understandings of the correlated states have also been
+achieved recently. The strong correlation arises from the
+two topological flat bands [32–37], each of which is four-
+fold degenerate due to the spin and valley d.o.f. A large
+U(4) symmetry group [38–42] emerges in the flat-band
+limit, where the actual bandwidth is counted as negligi-
+ble. Then the observed correlated states at integer fillings
+ν = 0, ±1, ±2, ±3 can be understood as flavor polarized
+states [38–40, 42–58] that spontaneously break the U(4)
+symmetry. Here |ν| is the number of electrons (ν > 0)
+or holes (ν < 0) per moir´e cell counted from the charge
+neutrality point (CNP). The continuous U(4) degeneracy
+leads to Goldstone mode fluctuations [59, 60] that may
+∗ songzd@pku.edu.cn
+destroy the long-range order due to the Mermin-Wagner
+theorem.
+Less theoretical understandings have been achieved for
+the gapless states, which are observed at both fractional
+and integer fillings. For example, at ν ≈ ±1, depend-
+ing on the experimental setup, both gapped correlated
+insulators [4, 6, 7] and gapless Fermi liquid states [6, 25–
+27, 29–31] have been observed in transport experiments,
+suggesting that they are competing ground states with
+close energies. Interestingly, the observed gapless Fermi
+liquid states around ν = ±1 usually exhibit resistivity
+peaks [25–27, 29, 31] above a few kelvins, and the peaks
+could become increasingly pronounced as temperature
+rises.
+Scanning tunneling microscope (STM) measure-
+ments [11, 19, 21, 22] have constantly seen that, at low
+temperatures of about a few hundred millikelvins, the
+conduction (valence) band will be pinned at the Fermi
+level and form a sharp zero-energy peak for the fillings
+1 ≲ ν < 2 (−2 < ν ≲ −1). The sharp peak does not fit
+the intuition of Stoner instability given that the interac-
+tion is indeed strong. When the temperature increases to
+a few or ten kelvins, these peaks develop into a cascade
+of transitions like a quantum dot model [17, 19].
+In this work, based on the recently developed topolog-
+ical heavy fermion (THF) model [61, 62], we find that
+the zero-energy peak, as well as the gapless Fermi liq-
+uid states at ν ≈ ±1, are results of the Kondo resonance
+[63–80]. At a higher temperature exceeding the Kondo
+arXiv:2301.04661v1  [cond-mat.str-el]  11 Jan 2023
+
+2
+energy scale, the local moments (LMs) formed by local-
+ized electrons give rise to the transition cascades, and
+the resistance peaks around ±1. We have numerically
+reproduced the temperature-dependent features of the
+spectral density. Our theory is also fully consistent with
+the Pomeranchuk effect observed around fillings ν = ±1
+[30, 31], which show that local moments appear upon
+heating. We have calculated LM entropies as functions
+of the temperature, filling, and an external field, and
+obtained curves comparable to the experimentally mea-
+sured data in Ref. [30, 31].
+Our theory shows that the observed gapless states at
+the fillings 1 ≲ |ν| < 2 are the strongly correlated heavy
+Fermi liquid state. Since this filling range overlaps with
+the superconductivity [1, 4–11] and the strange metal
+[27–29] around ν = −2+δ (for small δ), the heavy Fermi
+liquid state could be the parent state for the unconven-
+tional superconductivity, as it is in the heavy fermion
+materials with 4f or 5f electrons.
+This opens a new
+perspective - with a solid theoretical and experimental
+basis at the same time - to study the superconductivity
+in MATBG.
+This work is organized as the followings. For this work
+to be self-contained, in Sec. II we will review the THF
+model and its symmetry shortly. In Sec. III, based on a
+poor man’s scaling analysis and experimental facts, we
+argue that the Kondo screening effect is irrelevant at
+CNP, and hence the ground state at CNP is the pre-
+viously identified symmetry-broken correlated insulator
+[38–40, 42–44]. Then we derive a simpler effective peri-
+odic Anderson model describing active excitations upon
+the correlated ground state.
+We further simplify the
+model to a single-impurity version. In Sec. IV, by ap-
+plying poor man’s scaling and Wilson’s numerical renor-
+malization group (NRG) method [81–83] to the single-
+impurity problem, we obtain a phase diagram charac-
+terized by strong coupling fixed points and various LM
+fixed points. The strong coupling phase is divided into
+a Kondo regime and a frozen impurity (FI) regime. We
+also present a detailed analysis of the RG flows at these
+fixed points. The gapless 1 ≲ |ν| < 2 states are found
+to be in the Kondo regime. In Secs. V and VI we cal-
+culate the spectral densities, spin susceptibilities, and
+entropies as functions of the filling ν and the temper-
+ature T. The spectral densities feature sharp Kondo res-
+onances at low temperatures smaller than the Kondo en-
+ergy scale. Whereas at higher temperatures, the Kondo
+resonances are suppressed and the Hubbard bands be-
+come clearer, which periodically cross the Fermi level as
+ν changes from 0 to 4 as that of a quantum dot model,
+matching the STM experiments. The spin susceptibili-
+ties obey Curie’s law at high temperatures, suggesting
+the existence of LM, and approach constants at lower
+temperatures, suggesting the Fermi liquid behavior. The
+Kondo temperature TK can be estimated as the turning
+temperature of the two behaviors of the spin susceptibil-
+ity. For 1 ≲ |ν| < 2, we find kBTK ranges from 0.129meV
+to 0.675meV. (Here kB is Boltzmann’s constant.)
+At
+c
+c
+Energy (meV)
+0
+40
+-20
+(b)
+ΓM
+KM
+MM
+2|M|
+(a)
+-40
+-60
+20
+60
+KM
+ΓM
+MM
+c
+-80
+80
+f
+f
+(c)
+G
+ΓM
+KM
+MM
+c
+L=0,0
+L=1,-1
+(d)
+G
+15
+10
+0
+∆(ω) (meV)
+ω (meV)
+5
+FIG. 1.
+The THF model.
+(a) Top: red spheres represent
+the effective f-electrons located at AA-stacking regions of
+MATBG, and blue spheres represent the itinerant c-electrons.
+Bottom: the moir´e Brillouin zone. (b) Black bands are given
+by the THF model. The red and blue bands are the decou-
+pled f- and c-bands, respectively.
+M is a parameter that
+determines the bandwidth of the flat bands. We focus on the
+M → 0 limit in this work. (c) Excitation spectrum of the ac-
+tive c-electrons upon the symmetry-breaking parent state for
+ν > 0, where M and µc are set to zero. (d) The hybridization
+function ∆(ω) contributed by the c-bands in (c). A nonzero
+M will not change the asymptotic behavior of ∆(ω) around
+the gap.
+|ν| = 1 and kBTK ≈ 1meV, the entropy contributed by
+the LM is around 2 ln 2 · kB ≈ 1.39kB. In the presence of
+a strong in-plane magnetic field, the entropy is quenched
+to about ln 2·kB ≈ 0.69kB. These values will be appreci-
+ated if one notices that the measured entropies at ν ≈ 1
+and T ≈ 10K in the absence and presence of a strong in-
+plane field are about 1.2kB and 0.6kB, respectively [30].
+In the Sec. VII, we discuss the heavy Fermi liquid states
+at 1 ≲ |ν| < 2 and propose experiments to confirm them.
+We estimate the heavy fermion bands and their quasi-
+particle weights using spectral information provided by
+the NRG calculation. The possible effects of the RKKY
+interactions are also discussed.
+II.
+THE THF MODEL
+One theoretical challenge in studying correlation
+physics in MATBG is the lack of a fully symmetric lat-
+tice model for low energy physics, which is forbidden
+by the band topology protected by a C2zT symmetry
+[32–34] and an emergent particle-hole symmetry P [37]
+- even though extended Hubbard models [84–88] can be
+constructed at the sacrifice of either symmetry or local-
+ity. The band topology was thought as fragile [32–34]
+but was later shown to be a stable symmetry anomaly
+jointly protected by C2zT and P [37]. The THF model
+[61, 62] resolved this problem by ascribing the strong
+correlation to effective f-orbitals at the AA-stacking re-
+gions, which form a triangular lattice, and leaving the
+remaining low energy states to continuous c-bands de-
+scribed by a topological Dirac Hamiltonian (Fig. 1). The
+THF model faithfully reproduces the symmetry, topol-
+ogy, dispersion, and Coulomb interaction of the continu-
+ous Bistritzer-MacDonald model [3]. Its free part is given
+
+3
+by
+ˆH0 = −µ ˆ
+N +
+�
+ηs
+�
+aa′
+�
+|k|<Λc
+H(c,η)
+aa′ (k)c†
+kaηsckaηs
++
+�
+ηsαa
+�
+|k|<Λc
+�
+e− |k|2λ2
+2
+H(cf,η)
+aα
+(k)c†
+kaηsfkαηs + h.c.
+�
+.
+(1)
+Here µ is the chemical potential, ˆN is the particle-number
+operator, ckaηs is the fermion operator for the c-electron
+of the momentum k, orbital a (= 1, 2, 3, 4), valley η (=
+±), and spin s (=↑, ↓), fkαηs is the corresponding fermion
+operator for the f-electron of the orbital α (=1,2). The
+summation over k for the c-bands is in principle limited
+within the cutoff Λc. But the theory is well-defined and
+yields the same low energy physics if we take the Λc → ∞
+limit. H(c,η)(k) = v⋆(ησx⊗σ0kx−σy⊗σzky)+02×2⊕Mσx
+is the Dirac Hamiltonian of the c-bands. When M ̸= 0,
+c-bands have a quadratic band touching at the zero en-
+ergy, whereas when M = 0, c-bands become linear. The
+two-by-two block of H(cf,η)
+aα
+(k) for a = 1, 2 is given by
+γσ0 + v′
+⋆(ησxkx + σyky), and the two-by-two block of
+H(cf,η)
+aα
+(k) for a = 3, 4 vanishes.
+The parameter λ in
+the second line of Eq. (1) is the spread of the Wan-
+nier functions of f-electrons, and it truncates the hy-
+bridization at |k| ≫ λ−1.
+In this work we adopt the
+w0/w1 = 0.8 parameters of Ref. [61]: γ = −24.75meV,
+v⋆ = −4.303eV · ˚A, v′
+⋆ = 1.623eV · ˚A, λ = 1.4131/kθ,
+kθ = 1.703˚A−1 · 2 sin θm
+2 with θm = 1.05◦ being the first
+magic angle. (w0 and w1 are the interlayer couplings of
+MATBG at the AA-stacking and AB-stacking regions,
+respectively. Due to the corrugation effect [89–92], w0
+is usually smaller than w1. Ref. [85] estimates w0/w1 as
+0.817.) The resulting band structure with a nonzero M
+(3.697meV) is shown in Fig. 1(b). One can see that the
+topological flat bands result from the hybridization be-
+tween c- and f-bands. As explained in detail in Ref. [61]
+and shown in Fig. 1(b), the parameter M determines the
+bandwidth of the flat-bands.
+In each valley, the Hamiltonian ˆH0 respects a magnetic
+space group P6′2′2 [32] (#177.151 in the BNS setting
+[93]), generated by C3z, C2x, C2zT, and translation sym-
+metries. The two f-orbitals and the four c-bands have
+effective angular momenta L = −η, η and L = −η, η,
+0, 0, respectively.
+As shown in detail in [61], the six-
+by-six representations of C3z, C2x, C2zT symmetries on
+these orbitals are eiη 2π
+3 σz ⊕ eiη 2π
+3 σz ⊕ σ0, I3×3 ⊗ σx, and
+I3×3 ⊗ σxK, respectively, where σ0,x,y,z are Pauli matri-
+ces and K the complex conjugation. One can verify that
+the Hamiltonian matrices H(c,η) and H(cf,η) given in the
+last paragraph respect these crystalline symmetries. The
+interaction Hamiltonian given in the following paragraph
+also respects these crystalline symmetries.
+The interaction Hamiltonian is given by
+ˆHI =U1
+2
+�
+R
+δnf
+Rδnf
+R + U2
+2
+�
+⟨RR′⟩
+δnf
+Rδnf
+R′ +
+1
+2NM
+�
+qaa′
+V (q)δnc
+−qa′δnc
+qa +
+1
+NM
+�
+Rqa
+Wae−iq·Rδnf
+Rδnc
+qa −
+J
+2NM
+�
+ηη′αα′
+ss′
+�
+|k|,|k′|<Λc
+R
+�
+(ηη′ + (−1)α+α′)e−i(k−k′)·R− λ2(k2+k′2)
+2
+(f †
+Rα′η′s′fRαηs − 1
+2δηη′δαα′δss′)(c†
+k,α+2ηsck′,α′+2η′s′ − 1
+2δkk′δηη′δαα′δss′)
+�
+,
+(2)
+where NM is the number of moir´e cells, fRαηs is the real
+space fermion operator for the f-electrons, R are the tri-
+angular lattice shown in Fig. 1(a), ⟨RR′⟩ represents near-
+est neighbor pairs (ordered), δnf
+R = �
+αηs(f †
+RαηsfRαηs −
+1
+2) is the density operator of f-electrons,
+δnc
+qa
+=
+�
+ηsk(c†
+k+qaηsckaηs − 1
+2δq0) is the density operator for c-
+electrons of the orbital a with k and k + q being limited
+within the cutoff Λc. U1,2, V (q), Wa are the density-
+density interaction between ff, cc, cf electrons, respec-
+tively, and J is an exchange interaction between cf elec-
+trons. We adopt the w0/w1 = 0.8 parameters of Ref. [61]:
+U1 = 57.95meV, U2 = 2.329meV, W1 = W2 = 44.03meV,
+W3 = W4 = 50.20meV, J = 16.38meV. We choose
+V (q) as the double-gate-screened Coulomb interaction,
+πξ2Uξ
+Ω0
+tanh(ξ|q|/2)
+ξ|q|/2
+, with ξ = 10nm being distance between
+MATBG and the gates, Uξ = 24meV the Coulomb inter-
+action at the distance ξ, and Ω0 ≈ 156nm2 the area of
+the moir´e cell.
+Hereafter, we mainly focus on the flat-band limit, i.e.,
+M = 0. In this limit, an exact U(4) symmetry of ˆH0+ ˆHI
+between the spin, valley, and orbital flavors emerge, as
+previously recognized in the projected Coulomb Hamil-
+tonian of the continuous model [38–42]. We emphasize
+that this U(4) symmetry is not related to the so-called
+chiral limit [35, 94], i.e., w0 = 0, which leads a distinct
+U(4) symmetry [41, 39]. The U(4) symmetry in the flat-
+band limit has been shown as a good approximation for
+realistic parameters such as w0/w1 = 0.8 [41, 43]. The
+sixteen U(4) generators acting on fRαηs, ckaηs (a = 1, 2),
+and ckaηs (a = 3, 4) are
+Σf
+µν = {σ0τ0ςν, σyτxςν, σyτyςν, σ0τzςν} ,
+(3)
+Σc12
+µν = {σ0τ0ςν, σyτxςν, σyτyςν, σ0τzςν} ,
+(4)
+and
+Σc34
+µν = {σ0τ0ςν, −σyτxςν, −σyτyςν, σ0τzςν} ,
+(5)
+
+4
+respectively, where ςν (ν = 0, x, y, z) are Pauli matrices
+acting in the spin subspace, τµ (µ = 0, x, y, z) are Pauli
+matrices acting in the valley subspace, and σ0,x,y,z are
+Pauli matrices acting in the orbital subspace. With the
+help of U(4) symmetry, the J term in Eq. (2) can be writ-
+ten as a ferromagnetic coupling between the U(4) LM of
+f-electrons and the U(4) LM of c-electrons [61]. When
+M ̸= 0, only the µ = 0, z U(4) generators commute with
+the Hamiltonian, leading to a lower U(2)×U(2) symme-
+try group. The rotation generated by µ = z, ν = 0 is
+referred to as the valley-U(1) symmetry.
+Consistent with previous results [38–40, 42–44], a
+Hartree-Fock treatment of the THF model has predicted
+the ground state at CNP as a U(4) LM state that re-
+spects a U(2)×U(2) subgroup [61]. The LM forms a 20-
+fold multiplet belonging to the [2, 2]4 representation [43]
+of the U(4) group. These states can be approximately
+written as
+|Ψ0⟩ = e−iθµν ˆΣµν �
+R
+f †
+R1+↑f †
+R1+↓f †
+R2+↑f †
+R2+↓|FS⟩ ,
+(6)
+where the |FS⟩ is the Fermi sea state with the half-filled
+c-bands, ˆΣµν are the U(4) generator operators defined
+by the matrices in Eqs. (3) to (5), and θµν are the ro-
+tation parameters, and an implicit summation over re-
+peated µ, ν indices is assumed. When θµν’s are zero, |Ψ0⟩
+is the valley-polarized state because all the occupied f-
+electrons are in the η = + valley, and the U(2)×U(2)
+subgroup is generated by Σ0,ν and Σz,ν (ν = 0, x, y, z).
+For nonzero θµν’s, |Ψ0⟩ respects an equivalent U(2)×U(2)
+subgroup. The Kramers inter-valley coherent states can
+be obtained by choosing nonzero θx0 and θy0 satisfying
+θ2
+x0 + θ2
+y0 = (π/4)2. When M ̸= 0, the Kramers inter-
+valley coherent states are the ground states, while the
+valley polarized states have higher energy (∼ 0.1meV)
+[43, 61].
+III.
+EFFECTIVE ANDERSON MODEL FOR
+ν > 0 STATES
+A.
+Irrelevance of Kondo screening at CNP
+Here we argue that the Kondo screening effect is ir-
+relevant at CNP; hence, the U(4) LM state in Eq. (6)
+is valid as an approximate ground state. We first exam-
+ine the energy scale of a fully symmetric Kondo state
+at CNP. Since the f-sites are almost decoupled from
+each other, a reasonable approximation is to view each
+f-site as a single Anderson impurity coupled to a bath
+of c-electrons. If we only consider the on-site U1 inter-
+action and the single-particle hybridization between f-
+and c-electrons (H(cf,η)(k) in Eq. (1)), then it is almost
+a standard Anderson model with eight flavors. The ef-
+fect of c-bath is described by the hybridization function
+∆(ω), defined as the imaginary part of the self-energy of
+a free f-electron (in the absence of U1) coupled to the c-
+bath, i.e., Im[Σ(f)
+αηs,α′η′s′(ω)] = δα,α′δηη′δss′sgn(ω)∆(ω).
+The identity matrix form of Im[Σαηs,α′η′s′(ω)] is guar-
+anteed by the spin-SU(2) (δss′), the valley-U(1) and the
+time-reversal (δηη′), and crystalline (δαα′) symmetries.
+Low energy c-bands (k → 0) are coupled to the impu-
+rity through the constant coupling γ in H(cf,η)(k). Then
+∆(ω) would be proportional to the density of states ρ(ω)
+of c-bands. In the flat-band limit (M = 0), c-bands have
+a linear dispersion (Fig. 1(b)) and hence the density of
+states, as well as the hybridization function, linear in en-
+ergy, i.e., ρ(ω) ∼ |ω|, ∆(ω) ∼ |ω|. As a consequence,
+low-lying states of the impurity will see vanishing bath
+electrons when the energy scale is small enough. Both nu-
+merical [95–97] and analytical [98] RG studies have shown
+that Anderson impurity models with such a ∆(ω) ∼ |ω|
+hybridization function do not have the strong coupling
+fixed point that exhibits Kondo screening. Instead, the
+only stable fixed point is the LM phase.
+With a finite M, the c-bands given by H(c,η)(k) in
+Eq. (1) have a quadratic touching at the zero energy,
+i.e., ±(−M/2 +
+�
+M 2/4 + v2⋆k2), leading to a finite den-
+sity of states at the zero energy and hence a finite ∆(0).
+Nevertheless, as explained in the following, the Kondo
+energy scale resulting from realistic parameters is neg-
+ligible.
+In Appendix B 2 we derived an analytical ex-
+pression of ∆(ω) for the symmetric state at CNP. For
+|ω| > M, ∆(ω) is almost linear in |ω|, i.e., ∆(ω) ≈ b·|ω|.
+For |ω| < M, ∆(ω) is a constant plus a linear term:
+∆(ω) ≈ ∆(0)(1 + |ω|/M). Using the parameters given
+in Sec. II and M = 3.697meV, we have b ≈ 0.129,
+∆0 ≈ 0.239meV. A rough estimation of the Kondo en-
+ergy scale can be made by replacing the ω-dependent
+∆(ω) with the constant ∆(0) at ω = 0. Then naively
+applying the large-N formula at second order [99], i.e.,
+kBTK ≈ De−
+πU1
+4N ∆(0) with N = 8 being the number of
+flavors and D = U1/2 the energy scale up to which the
+perturbation theory applies, leads to an extremely small
+kBTK ≈ U1 · 2 × 10−11. A better estimation is given by
+a poor man’s scaling that considers the ω-dependence of
+∆(ω). Readers may refer to Appendix B 2 for more de-
+tails. Here we only present the main results. There are
+two stages in the RG process: (i) a stage with energy
+scale from U1/2 - scale up to which the perturbation the-
+ory applies - to M.
+(ii) a stage with an energy scale
+below M. RG in the first stage effectively enhances ∆(0)
+to g1∆(0) with g1 ≈ 2.34. Then, RG in the second stage
+gives
+kBTK ≈Meye−
+πU1
+4N g1∆(0)
+≈3.8 × 10−4meV
+(2M = 7.39meV).
+(7)
+where y ≈ 1 is a factor contributed by the ω-dependence
+of ∆(ω) in the second stage. This value is still much lower
+than the energy gain of the symmetry-broken correlated
+state [43, 59]. The bandwidth of the Goldstone modes
+at CNP from ΓM to MM is about 8meV. (See Fig. 2 of
+Ref. [59]).
+If we understand this spectrum as a tight-
+binding band of the Holstein–Primakoff bosons on the
+f-sites, which form a triangular lattice, then the nearest
+
+5
+neighbor hopping is about 8meV/8=1meV. This hopping
+indicates an RKKY interaction much larger than kBTK
+evaluated in Eq. (7).
+The actual TK can even be much smaller than the value
+in Eq. (7). First, as ∆(0) → 0 when M → 0, TK decays
+exponentially when M decreases. For example, a band-
+width 2M = 5meV corresponds to
+kBTK ≈ 5.1 × 10−6meV
+(2M = 5meV) .
+(8)
+Second, because we only considered the U1 interaction
+and the cf hybridization that gives all flavors of f-
+electrons the same ∆(ω) (guaranteed by symmetries), the
+single-impurity model has a U(8) symmetry. This U(8)
+symmetry must be broken when other interaction terms,
+e.g., J in Eq. (2), are taken into account, leading to a
+multiplet splitting.
+When the energy scale in the RG
+is smaller than the multiplet splitting, the degeneracy
+factor N should be reduced accordingly, and TK will be
+further suppressed. (One can see section 17.2 of Ref. [99]
+and Appendix B 3 for examples of how multiplet splitting
+suppresses TK.)
+The U(4) LM state at CNP is also supported by var-
+ious experiments. In contrast to the Kondo resonance,
+STM measurements have shown strong suppression of the
+density of states at the zero energy at CNP [11, 12, 14–
+17, 19, 21, 22]. Some transport experiments [4, 6, 7, 27]
+also exhibits a gap behavior at CNP. Although there are
+also transport experiments showing semimetal behavior,
+the gaplessness can be explained if there are fluctuations
+of the local moments from site to site, which is possi-
+ble due to the Goldstone mode fluctuations [59, 60] and
+possible inhomogeneity of the sample.
+B.
+Periodic Anderson model for ν > 0 states
+We aim for an effective model describing the active ex-
+citations upon the ground state |Ψ0⟩ (Eq. (6)) at CNP.
+Let us first assume the valley-polarized state, where θµν’s
+in Eq. (6) are all zero such that all the occupied f-
+electrons are in the η = + valley.
+As detailed in the
+supplementary material of Ref. [61] and in Ref. [59], the
+lowest particle and hole excitations are in the η = −
+and η = + valleys, respectively. Thus, for ν > 0 states,
+only particle excitations in the η = − valley will be in-
+volved, and the electrons in the η = + valley can be
+viewed as a static background.
+The effective Hamil-
+tonian can be obtained by replacing operators in the
+η = + valley by their expectation values on |Ψ0⟩, which
+are ⟨f †
+Rα+sfR′α′+s′⟩ = δRR′δαα′δss′, ⟨c†
+ka+sck′a′+s′⟩ ≈
+1
+2δkk′δaa′δss′, ⟨c†
+ka+sfRα+s′⟩ = 0. Substituting these ex-
+pectation values into ˆH0 + ˆHI, we obtain the effective
+free Hamiltonian
+ˆHeff
+0
+=
+�
+|k|<Λc
+aa′s
+(H(c)
+aa′(k) − µδaa′)c†
+kascka′s − µ
+�
+Rαs
+nf
+Rαs
++
+�
+|k|<Λc
+aα
+�
+e− 1
+2 λ2k2H(cf)
+aα (k)c†
+kasfkαs + h.c.
+�
+,
+(9)
+where nRαs = f †
+RαsfRαs is the density operator of f-
+electrons. Here we have dropped the valley index η as
+they are limited to η = −. The H(c)(k) and H(cf)(k)
+matrices are given by the H(c,−)(k) and H(cf,−)(k) ma-
+trices defined after Eq. (1). We also obtain the effective
+interaction Hamiltonian
+ˆHeff
+I
+= U1
+2
+�
+R
+nf
+Rnf
+R + U2
+2
+�
+⟨RR′⟩
+nf
+Rnf
+R′
++
+1
+2NM
+�
+qaa′
+V (q)δnc
+−q,a′δnc
+q,a +
+1
+NM
+�
+Rqa
+Wae−iq·Rnf
+Rδnc
+qa
+−
+J
+NM
+�
+Rss′
+�
+α
+�
+|k|,|k′|<Λc
+e−i(k−k′)·R− λ2(k2+k′2)
+2
+× (f †
+Rαs′fRαs − 1
+2δss′)(c†
+k,α+2,sck′,α+2,s′ − 1
+2δss′) ,
+(10)
+where nf
+R = �
+αs nf
+Rαs, δnc
+qa = �
+sk(c†
+k+qasckas − 1
+2δq0)
+with |k| and |k + q| being limited within the cutoff Λc.
+The δnf
+R operator in Eq. (2), which represents the density
+deviation from the charge background at CNP, is now
+replaced by nf
+R, the total density in the η = − valley,
+because the charge background is compensated by the
+occupied η = + electrons. The J term in Eq. (2) is also
+simplified: As active excitations are limited to η = −,
+the factor ηη′ + (−1)α+α′ becomes 2δα,α′.
+In the flat-band limit (M
+= 0),
+ˆHeff
+0
++ ˆHeff
+I
+ap-
+plies to arbitrary U(4) partners of the valley-polarized
+state, including the so-called Krammers intervalley co-
+herent state. To be specific, for a generic |Ψ0⟩ given in
+Eq. (6), we can always define rotated operators fRαs =
+UfRα−sU †, ckas = Ucka−sU †, where U = e−iθµν ˆΣµν is
+the U(4) rotation defining |Ψ0⟩, such that the effective
+Hamiltonian on the rotated basis is same as Eqs. (9)
+and (10).
+The effective Hamiltonian ˆHeff
+0
++ ˆHeff
+I
+respects all the
+crystalline symmetries discussed in Sec. II. The effective
+angular momenta of the active two f-orbitals and four
+c-bands are L = 1, −1 and L = 1, −1, 0, 0, respectively.
+And, the six-by-six representations of C3z, C2x, C2zT
+on these orbitals are e−i 2π
+3 σz ⊕ e−i 2π
+3 σz ⊕ σ0, 13×3 ⊗ σx,
+and 13×3 ⊗ σxK, respectively, with K being the complex
+conjugation.
+In the flat-band limit (M = 0), as |Ψ0⟩
+respects a U(2)×U(2) subgroup of the U(4) group, e.g.,
+independent spin-charge rotations in the two valleys for
+the valley polarized |Ψ0⟩, one may expect a U(2)×U(2)
+symmetry of ˆHeff
+0
++ ˆHeff
+I .
+However, since the effective
+Hamiltonian only involves half of the d.o.f., e.g., the ac-
+tive η = − valley for the valley polarized |Ψ0⟩, only a
+single U(2) factor is meaningful for ˆHeff
+0
++ ˆHeff
+I . There-
+fore, hereafter we will say that ˆHeff
+0 + ˆHeff
+I
+respects a U(2)
+symmetry group.
+When M ̸= 0, the U(4) symmetry is broken, and the
+effective Hamiltonian will have an additional term. In
+Appendix A we show that to the order of M 2, the cor-
+
+6
+ˆH0 + ˆHI
+ˆHeff
+0
++ ˆHeff
+I
+ˆHSI
+M = 0
+U(4)
+U(2)
+U(2)×U(2)
+M ̸= 0 U(2)×U(2)
+U(2)
+U(2)×U(2)
+TABLE I. Continuous symmetries of the Hamiltonians. ˆH0 +
+ˆHI is the original THF model. For ν > 0 (ν < 0), ˆHeff
+0 + ˆHeff
+I
+is the effective periodic Anderson model for the active particle
+(hole) excitations upon the symmetry broken state at CNP.
+ˆHSI is a single-impurity version of ˆHeff
+0
++ ˆHeff
+I .
+rection is simply an energy shift
+M 2
+J
+�
+|k|<Λc
+�
+a=3,4
+�
+s
+c†
+kasckas + O(M 4) .
+(11)
+Thus, the M-term breaks neither the crystalline symme-
+try nor the U(2) symmetry, and it will play a minor role
+in the effective theory. To avoid confusion, in Table I we
+summarize the continuous symmetries of different Hamil-
+tonians with M = 0 or M ̸= 0 discussed in this work.
+The effective model for ν < 0 states, which only in-
+volve hole excitations, can be obtained by applying the
+particle-hole operation Pc [41, 61] to ˆHeff
+0
++ ˆHeff
+I .
+C.
+Single impurity model for ν > 0 states
+Hereafter we mainly focus on a single-impurity version
+of ˆHeff
+0
++ ˆHeff
+I , where only the correlation at the R = 0
+f-site is considered. The interactions involving other f-
+sites will be treated at the mean-field level. The STM
+spectra [11, 19, 21, 22] that show the zero-energy peaks
+also clearly show a continuity between the gapped CNP
+state and the gapless states at 1 ≲ |ν| < 2, implying
+that they have the same symmetries. Therefore, in this
+work, we assume no additional symmetry breaking. For
+the completeness of the discussion, we also extend our
+symmetric assumption to |ν| ≥ 2 states. One should be
+aware that additional symmetry breaking may happen
+at low temperatures in |ν| ≥ 2 states due to the effec-
+tive RKKY interactions neglected in this work. Thus the
+symmetric assumption is invalid for |ν| ≥ 2 states at low
+temperatures. Nevertheless, the |ν| ≥ 2 states may re-
+cover the symmetries at higher temperatures, where our
+symmetric theory applies.
+At a given filling ν, the symmetric mean field is char-
+acterized by only a few parameters: νf = ⟨nf
+R⟩, νc,a =
+1
+NM ⟨δnc
+q=0,a⟩, where νc,1 = νc,2, νc,3 = νc,4 due to the
+crystalline symmetries. The actual values of νf and νc,a
+can be determined self-consistently. The considered cor-
+related site at R = 0 is described by the Hamiltonian
+ˆHf = −µf
+�
+αs
+nf
+αs + U1
+�
+(αs)<(α′s′)
+nf
+αsnf
+α′s′ ,
+(12)
+where the lattice index R is omitted for simplicity, µf =
+−6νfU2−�
+a νc,aWa− 1
+2U1+Jνc,3+µ is an effective chem-
+ical potential for the f-site, and µ is the global chemical
+potential determined by the total filling ν. The U2, Wa, J
+terms in µf are contributed by the Hartree mean fields of
+the U2, Wa, and J interactions in Eq. (10), respectively.
+The U1 term in µf is from the diagonal U1 interactions
+in Eq. (10), i.e.,
+1
+2U1
+�
+αs nf
+αsnf
+αs. The U1 interaction
+in ˆHf only contains the off-diagonal U1 interactions of
+Eq. (10).
+The effective Hamiltonian of c-electrons is given by
+ˆHc =
+�
+|k|<Λc
+�
+aa′s
+(H(c)
+aa′(k) + ∆H(c)
+aa′ − µcδaa′)c†
+kascka′s , (13)
+where H(c)(k) = −v⋆(σx ⊗ σ0kx + σy ⊗ σzky) is the free
+Dirac Hamiltonian, ∆H(c)
+aa′ = G·δaa′(δa3+δa4) is a mean-
+field term that split the a = 1, 2 and a = 3, 4 c-electrons,
+µc = −νfW1 −νc V
+Ω0 +µ is an effective chemical potential
+of c-electrons. The W1 and V terms in µc are contributed
+by the W and V interactions in Eq. (10), respectively.
+The mean field term G arises from two interaction terms:
+(i) A mean field treatment of the J interaction in Eq. (10)
+yields an energy shift J( 1
+2 − 1
+4νf) for a = 3, 4 c-electrons.
+(ii) The Hartree mean fields of the Wa interactions in
+Eq. (10) are νfW1 and νfW3 for a = 1, 2 and a = 3, 4
+c-electrons, respectively. As we have absorbed νfW1 to
+µc, a = 3, 4 c-electrons have an extra energy shift (W3 −
+W1)νf. Combining the two effects, the parameter G is
+given by G = J
+2 +(W3−W1− J
+4 )νf. Since the interaction
+V (q) of c-electrons is completely treated at the mean-
+field level, ˆHc is an effective free-fermion system.
+As
+detailed in Appendix A, the band structure of Eq. (13) is
+given by G/2±
+�
+G2/4 + v2⋆k2. In Fig. 1(c) we plot the c-
+bands with µc = 0 and G = 10meV. It has a gap opened
+by G, where, according to the symmetry representations
+given in Sec. III B, the lowest conduction and highest
+valence band states have angular momenta 0, 0 and 1, −1,
+respectively.
+The f-site is coupled to c-electrons via two terms. One
+is the hybridization
+ˆHhyb =
+1
+√NM
+�
+|k|<Λc
+�
+aαs
+�
+e− λ2k2
+2
+H(cf)
+aα (k)c†
+kasfαs + h.c.
+�
+.
+(14)
+The other coupling term is the remaining ferromagnetic
+exchange interaction
+ˆHJ = −
+J
+NM
+�
+|k|,|k′|<Λc
+�
+αss′
+e− 1
+2 λ2(k2+k′2)(f †
+αsfαs′ − 1
+4δss′νf)
+× (c†
+k′α+2s′ckα+2s − 1
+2δk,k′δss′νc,α+2) .
+(15)
+By “remaining” we mean that the mean field back-
+grounds
+1
+4νf and
+1
+2νc,a are deducted in
+ˆHJ.
+As ex-
+plained below, ˆHJ leads to an effective Hund’s coupling
+that changes the symmetry of the single-impurity model.
+There are also remaining density-density interactions be-
+tween c- and f-electrons, i.e., Wa(nf − νf)(nc
+q,a − νc,a).
+However, these remaining density-density interactions do
+not change the essence of the single-impurity problem as
+ˆHJ does.
+
+7
+Thanks to the C3z symmetry,
+ˆHhyb and ˆHJ couple
+the f-electrons to two independent baths belonging to
+different angular momenta. This allows us to treat the
+two terms separately. In a polar coordinate, ˆHhyb only
+couples f-electrons to
+�ckαs = 1
+A
+�
+a
+ˆ
+dφ · H(cf)
+αa (k)ckas
+(α = 1, 2) ,
+(16)
+where k = k(cos φ, sin φ), and A is a normalization fac-
+tor. Explicitly, there are �ck1s ∼
+´
+dφ·(γck1s−v′
+⋆keiφck2s)
+and �ck2s ∼
+´
+dφ · (γck2s − v′
+⋆ke−iφck1s). Under the C3z
+operation (defined in Sec. III B), �ck1s and �ck2s have effec-
+tive angular momenta 1, -1, respectively. On the other
+hand, ˆHJ only couples f-electrons to
+�ckas = 1
+A′
+ˆ
+dφ · ckas
+(a = 3, 4) .
+(17)
+Both ckas (a = 3, 4) have the effective angular momen-
+tum 0 under C3z.
+Because �ckas (a = 1, 2) and �ckas
+(a = 3, 4) form different representations of C3z, they do
+not couple to each other, hence the ˆHhyb-bath and the
+ˆHJ-bath are indeed independent.
+As a ferromagnetic coupling always flows to zero and
+becomes irrelevant in low energy physics, we can inte-
+grate out the ˆHJ-bath in a single attempt. This leads to
+an effective Hund’s coupling (Appendix A)
+ˆHH = JH
+�
+α
+nα↑nα↓ ,
+(18)
+where JH, estimated as 0.3meV, is the additional energy
+that two electrons will acquire if they occupy the same
+orbital. A nonzero M does not change the form of ˆHH.
+Integrating out the ˆHhyb-bath leads to a self-energy
+correction Σ(f)
+αs,α′s′(ω) to the f-electrons, the imaginary
+part of which defines the hybridization function ∆(ω),
+i.e., Im[Σ(f)
+αs,α′s′(ω)] = δαα′δss′sgn(ω)∆(ω). The identity
+matrix structure of the self-energy is guaranteed by SU(2)
+spin rotation symmetry and crystalline symmetries. In
+Appendix A we derived an analytical expression of ∆(ω).
+As shown in Fig. 1(d), where ∆(ω) for µc = 0 and G =
+10meV is plotted, ∆(ω) has an abnormal ω-dependence
+compared to those in usual metals. First, ∆(ω) = 0 for
+ω in the gap of c-bands (Fig. 1(c)). Second, because f-
+electrons (L = 1, −1) have different angular momenta
+as the lowest conduction c-bands (L = 0), hybridization
+between them vanishes as k → 0.
+As a result, ∆(ω)
+linearly approaches zero at the conduction band edge.
+Third, as f-electrons have the same angular momenta as
+the highest valence c-bands, ∆(ω) approaches a constant
+at the valence band edge. In summary, around the gap
+∆(ω) has the asymptotic behaviors
+∆(ω) ∼
+�
+�
+�
+�
+�
+|ω + µc − G|,
+ω + µc → G + 0+
+0,
+0 ≤ ω + µc ≤ G
+const.,
+ω + µc → −0+
+.
+(19)
+A nonzero M does not change the asymptotic behav-
+iors as these behaviors are guaranteed by the C3z sym-
+metry that is also respected by M. Due to the damp-
+ing factor e− 1
+2 λ2k2 in ˆHhyb, c-electrons with momenta
+|k| ≫ 1/λ will not contribute to ∆(ω). Thus, ∆(ω) de-
+cays exponentially when ω exceeds v⋆/λ ∼ 95meV. In
+the rest of this work, we will restrict the hybridization to
+|ω| < D = 100meV.
+Baths giving rise to the same ∆(ω) are physically
+equivalent. Therefore, we can choose a bath that is as
+simple as possible. We introduce the following effective
+single-impurity Hamiltonian
+ˆHSI = ˆHf + ˆHH +
+�
+αs
+ˆ D
+−D
+dϵ · ϵ · d†
+αs(ϵ)dαs(ϵ)
++
+�
+αs
+ˆ D
+−D
+dϵ ·
+�
+∆(ϵ)
+π
+(f †
+αsdαs(ϵ) + h.c.) ,
+(20)
+where ˆHf and ˆHH are given by Eqs. (12) and (18), re-
+spectively, and dαs(ϵ) are the auxiliary bath fermions in-
+troduced to reproduce the hybridization function. dαs(ϵ)
+satisfy {dα′s′(ϵ′), d†
+αs(ϵ)} = δα′αδs′sδ(ϵ′ − ϵ). ˆHSI is com-
+pletely defined by the following parameters: µf the chem-
+ical potential of f-electrons, U1 the Coulomb repulsion,
+JH the Hund’s coupling, and ∆(ω) the hybridization
+function, which further depends on µc the chemical po-
+tential of c-electrons and G the gap of c-bands (Fig. 1(c)).
+As explained at the beginning of this subsection, the ac-
+tual values of µf, µc, and G depend on the occupations
+νf, νc,a, which are further determined by self-consistent
+calculations at given total fillings ν. We plot µf, µc, and
+G as functions of ν in Fig. 2(a). For ν changing from 0
+to 4, µc changes from 0 to 64.70meV, µf changes from
+-28.98meV to 124.13meV, and G changes from 8.19meV
+to 14.21meV. We can now regard µc, µf, G as given pa-
+rameters that define the single-impurity problem.
+It is worth mentioning that Eq. (20) has an emergent
+U(2)×U(2) symmetry - independent spin-charge rota-
+tions in the α = 1, 2 orbitals - that is higher than the
+U(2) symmetry of Eqs. (9) and (10). It is not surpris-
+ing that a single-impurity model has a higher symmetry
+than its lattice version.
+For example, if J = 0, there
+would be no Hund’s coupling JH, and ˆHSI would have
+a U(4) symmetry, as expected in a four-flavor Anderson
+impurity model without multiplet splitting.
+IV.
+PHASE DIAGRAM OF THE
+SINGLE-IMPURITY MODEL
+A.
+Poor man’s scaling
+Before going to numerical calculations, we first apply a
+poor man’s scaling to the single impurity model Eq. (20).
+The scaling theory helps us understand several features
+of the data obtained by NRG, as will be discussed in
+the following subsections.
+And, it predicts the Kondo
+
+8
+temperatures in the same order as those predicted by
+NRG.
+We assume that the ground state of the isolated impu-
+rity has nf (integer) occupied f-electrons. One should
+not confuse nf with νf - the expectation value of f-
+occupation after the impurity is coupled to the bath. The
+chemical potential µf must be in the range (nf − 1)U1 <
+µf < nfU1.
+We apply a Schrieffer-Wolff transforma-
+tion to Eq. (20) to obtain an effective Coqblin–Schrieff
+model where the local Hilbert space of f-electrons is re-
+stricted to nf particles. The transformation involves vir-
+tual particle and hole excitations, the energies of which
+are ∆E+ = nfU1 − µf and ∆E− = µf − (nf − 1)U1, re-
+spectively. (We have ignored the JH term in ∆E± as it
+is small compared to U1.) Adding the two contributions,
+we have
+ˆH = ˆHH +
+�
+αs
+ˆ D
+−D
+dϵϵd†
+αs(ϵ)dαs(ϵ) + 4g
+πU1
+�
+αα′ss′
+ˆ D
+−D
+dϵdϵ′
+�
+×
+�
+∆(ϵ)∆(ϵ′)(f †
+αsfα′s′ − xδαα′δss′)d†
+α′s′(ϵ′)dαs(ϵ)
+�
+.
+(21)
+The parameters g, x are given by
+g = U1
+4
+�
+1
+∆E+
++
+1
+∆E−
+�
+,
+x = ∆E−
+U1
+.
+(22)
+g is a dimensionless parameter characterizing the anti-
+ferromagnetic coupling strength between the impurity
+and the bath. x appears as a “charge background” of
+the f-electrons. For µf = (νf − 1
+2)U1, there is g = 1,
+x = 1
+2. For a generic (nf − 1)U1 < µf < nfU1, g ≥ 1
+and 0 < x < 1. Flow equations of g, x are derived in Ap-
+pendix B 3, where the divergence of g indicates the strong
+coupling fixed point that exhibits the Kondo screening.
+We notice that x always flows to nf/4, i.e., the occupa-
+tion fraction of f-electrons.
+One should be careful about the cutoff D in Eq. (21)
+First, it must be smaller than ∆E+ and ∆E− for the
+Schrieffer-Wollf transformation to be valid. Second, for
+analytical conveniences, we only keep the positive branch
+of ∆(ω) (Eq. (19)) at ω > G − µc because when ν >
+0 the negative branch is far away from the Fermi level
+(Fig. 1(d)). Hence, we also require D < µc − G. We can
+choose D = min(µc − G, ∆E+, ∆E−).
+We first consider the case nf = 1. The flow equation
+of g(t) as the cutoff is reduced to De−t is given by
+dg
+dt = 4∆(0)
+πU
+Ng2 + O(e−t) ,
+(23)
+and the initial condition g(0) is given by Eq. (22). Here
+N = 4 is the number of flavors. The local Hilbert space
+for nf = 1 is four-fold. The Hund’s coupling JH does not
+split the four-fold degeneracy and hence does not appear
+in the flow equation. The O(e−t) terms are irrelevant
+at small energy scales but they may affect the coupling
+constant at an early stage of the RG process. Using a
+linear approximation of ∆(ω), i.e., ∆(ω) ≈ ∆(0)(1 +
+ω/(µc − G)), we obtain (Appendix B 3)
+kBT (1)
+K
+= Dey1 · e−
+πU1
+4N ∆(0)g(0)
+(24)
+where y1 ≈ −0.75
+D
+µc−G < 0 is factor contributed by the
+irrelevant O(e−t) terms and will slightly suppress the
+Kondo energy scale. The suppression factor ey1 appears
+because, for the nf = 1 states, the virtual processes that
+contribute to the RG equation involve more hole exci-
+tations than particle excitations in the bath, such that
+the smaller ∆(ω < 0) contributes more than the larger
+∆(ω > 0). As a result, the resulting kBTK is smaller
+than the standard case (y1 = 0) where the coupling is a
+constant, i.e., ∆(ω) = ∆(0).
+We then study the RG equations at nf = 2. Unlike
+the nf = 1 case, Hund’s coupling JH splits the six-
+dimensional local Hilbert space. According to Eq. (18),
+the four states with (nf
+1↑, nf
+1↓; nf
+2↑, nf
+2↓) =(10;10), (10;01),
+(01;10), (01;01) do not feel JH and have the energy
+−2µf + U1, whereas the two states (11;00), (00;11) have
+the energy −2µf + U1 + JH. We divide the RG process
+into two stages: (i) a stage with an energy scale from D
+to JH, (ii) a stage with an energy scale below JH. In the
+first stage, JH plays a minor role; hence, a flow equation
+similar to Eq. (23) applies. If g diverges before the energy
+scale reaches JH, then the Kondo scale is given by
+kBT (2)′
+K
+= D · e−
+πU1
+4N ∆(0)g(0)
+(25)
+there is no y-factor as in Eq. (24) because, for the nf = 2
+states, the virtual processes that contribute to the RG
+equations evolve equal holes and particles in the bath.
+Otherwise, g will be renormalized to
+g1 =
+g(0)
+1 − g(0) 4∆(0)
+πU1 N ln D
+JH
+(26)
+at the energy scale of JH. As detailed in Appendix B, RG
+in the second stage is similar to Eq. (23) except that, due
+to the multiplet splitting, the factor N = 4 is replaced
+by 2. This leads to the Kondo energy scale
+kBT (2)′′
+K
+= JH · e
+−
+πU1
+8∆(0)g1 = D
+� D
+JH
+� N
+2 −1
+e
+−
+πU1
+8∆(0)g(0) . (27)
+Considering the g may diverge in either the first or the
+second stage, the physical Kondo energy scale can be
+written as
+kBT (2)
+K
+=
+�
+kBT (2)′
+K ,
+kBT (2)′
+K
+> JH
+kBT (2)′′
+K
+,
+otherwise
+.
+(28)
+The theory at nf = 3 is almost the same as the theory
+at nf = 1 except that the four single-particle states are
+now replaced by the four single-hole states. Thus, the
+local Hilbert space is also four-dimensional and will not
+be split by Hund’s coupling JH. The Kondo energy scale
+is given by
+kBT (3)
+K
+= Dey3 · e−
+πU1
+4N ∆(0)g(0)
+(29)
+where y3 ≈ 0.75
+D
+µc−G > 0 is a factor contributed by
+the irrelevant O(e−t) terms and will slightly enhance the
+
+9
+ν
+kBT (P )
+K
+(meV) δK(meV) kBT (χ)
+K
+(meV)
+0.75
+0.280
+0.165
+0.151
+1.00
+0.250
+0.158
+0.129
+1.25
+0.366
+0.305
+0.190
+1.50
+0.780
+0.558
+0.344
+1.75
+1.620
+1.674
+0.482
+2.00
+-
+1.926
+0.675
+2.25
+2.73
+1.463
+0.670
+2.50
+3.22
+1.934
+0.736
+2.75
+4.18
+2.284
+0.872
+3.00
+4.12
+2.305
+1.024
+3.25
+4.05
+3.400
+1.271
+3.50
+2.86
+2.663
+1.391
+TABLE II. Kondo energy scales at various fillings ν. T (P )
+K ,
+δK, and T (χ)
+K
+are Kondo energy scales estimated by the poor
+man’s scaling, the NRG spectral density, and the NRG spin
+susceptibility, respectively. δK is defined as the half width at
+half maximum of the spectral peak. T (χ)
+K
+is defined as the
+turning temperature where χ(T) transitions from the Curie-
+Weiss behavior to the Fermi liquid behavior. There is no data
+of T (P )
+K
+at ν = 2 because ν = 2 is close a mixed valence case
+where µf ≈ U1 and the Schrieffer-Wolff transformation does
+not apply.
+Kondo temperature. The enhancement arises from the
+fact that, in contrast to the case of nf = 1, for nf = 3
+states the virtual processes contributing to the RG equa-
+tion evolve more particle excitations than hole excita-
+tions in the bath, and particles have larger couplings than
+holes.
+In Table II we tabulate the Kondo energy scales ob-
+tained by the poor man’s scaling at various fillings us-
+ing the filling-dependent µc, µf, G parameters given in
+Fig. 2(a), and compare them to the values obtained by
+the NRG method.
+B.
+The NRG method
+In the NRG method [81–83], the bath is alternatively
+realized by a Wilson chain constructed in the way de-
+scribed below.
+First, the energy window [−D, D] is
+discretized on a logarithmic scale, i.e., ωn = D/Λn−1
+(n = 1, 2 · · · ), where Λ > 1 is a scaling factor (cho-
+sen as 3 in this work).
+Then for each energy shell
+ωn ≤ |ω| < ωn+1 two auxiliary bath electrons corre-
+sponding to the positive and negative part of it are in-
+troduced to reproduce the corresponding ∆(ω). These
+auxiliary electrons are further recombined into a Wilson
+chain, dnαs, such that (i) only the first site d1αs couples
+to the impurity, (ii) the chain is a tight-binding model
+with only on-site and nearest neighbor hopping terms.
+The Hamiltonian Eq. (20) is now mapped to an impurity
+plus a Wilson chain
+ˆHN = ˆHf + ˆHH +
+�
+αs
+t0(f †
+αsd1αs + h.c.)
++
+N
+�
+n=1
+�
+αs
+ϵnd†
+nαsdnαs +
+N−1
+�
+n=1
+�
+αs
+(tnd†
+n+1αsdnαs + h.c.) ,
+(30)
+where N is a large number. The parameters ϵn and tn
+can be computed from ∆(ω) using a standard iterative
+algorithm [83]. For n → ∞, ϵn ∼ Λ−n and tn ∼ Λ− 1
+2 n.
+Thus, the right-most sites represent the lowest-lying bath
+states.
+One can define the Nth scaled Hamiltonian as �HN =
+(Λ)
+1
+2 N−1 ˆHN. They can be constructed iteratively
+�
+HN+1 =Λ
+1
+2 �
+HN + Λ
+1
+2 (N−1) �
+αs
+�
+ϵN+1d†
+N+1,αsdN+1,αs
++ tNd†
+N+1,αsdN,αs + tNd†
+N,αsdN+1,αs
+�
+.
+(31)
+The Hilbert space dimension increases exponentially in
+this iterative process. The NRG algorithm truncates the
+Hilbert space by keeping a fixed number (chosen to be
+∼1200 in this work) of the lowest-lying states at each
+step. In order to keep the symmetry in the truncated
+Hilbert space, in practice we keep all the states up to a
+gap above the 1200th state.
+C.
+Phase diagram and fixed points
+Two successive transformations (Eq. (31)) that take
+�HN to �HN+2 can be thought as a renormalization group
+operation [81, 82]. The system is said to achieve a fixed
+point when �HN and �HN+2 have the same low-lying many-
+body spectrum. We can obtain a zero temperature phase
+diagram in the parameter space of µc, µf, G by analyz-
+ing the fixed points. For the completeness of discussions,
+here we let µf take value in [−0.5U1, 3.5U1] such that
+the corresponding impurity occupation (in the decoupled
+limit) nf = µf/U1 +1/2 takes value in [0, 4]. In Fig. 2(b)
+and (c) we show the obtained phase diagrams in the pa-
+rameter space of µc, µf for G = 8meV and G = 14meV,
+respectively. 8meV and 14meV are chosen to be close to
+the minimal (8.19meV) and maximal (14.21meV) values
+of G (Fig. 2(a)), respectively.
+Due to the U(2)×U(2) symmetry, all the many-body
+levels can be classified into symmetry sectors labeled by
+the good quantum numbers (Q1, Q2; S1, S2), where Qi
+and Si are the charge and spin of the ith U(2) sym-
+metry, respectively.
+Here we take the convention that
+Q1 + Q2 = 0 corresponds to a total occupation 2N + 2
+(2N) for odd (even) N. A fixed point is characterized by
+low energy many-body levels and the associated quantum
+numbers. In the whole phase diagram, we find two dis-
+tinct types of stable fixed points: (i) the strong coupling
+fixed point exhibiting a Fermi liquid behavior and (ii) the
+LM fixed points exhibiting nonzero spin momenta. At a
+strong coupling fixed point, as exampled in Fig. 2(d),
+(e), for either even or odd N, the ground state is a sin-
+glet and has (Q1, Q2; S1, S2) = (2k, 2k; 0, 0) for some or-
+der one integer k, which in most cases equals to 0. The
+
+10
+(f)
+(g)
+(d)
+(h)
+EN  (meV)
+EN  (meV)
+(c)
+(a)
+(e)
+200
+μf/U
+μc  (meV)
+0
+1
+1
+2
+3
+60
+20
+0
+40
+LM1
+LM2
+LM3
+Kondo
+FI
+FI
+δK (meV)
+G=8meV
+(b)
+μf/U
+60
+20
+0
+40
+LM1
+LM2
+LM3
+FI
+Kondo
+FI
+μc  (meV)
+G=14meV
+N
+11
+21
+31
+41
+1
+(1, 1;1/2, 1/2)
+(1, 1;1/2, 1/2)
+(0, 0;0, 0)
+(1, 0;1/2, 0)
+(1, 0;1/2, 0)
+(2, 0;0, 0)
+-40
+40
+ω (meV)
+N
+11
+21
+31
+41
+1
+(2, 1;0, 1/2)
+(2, 1;0, 1/2)
+(1, 1;1/2, 1/2)
+(1, 1;1/2, 1/2)
+(2, 2;0, 0)
+(1, 0;1/2, 0)
+-40
+40
+ω (meV)
+0
+100
+200
+150
+50
+11
+21
+31
+41
+N
+(0, 0;0, 0)
+(1, 0;1/2, 0)
+(-1, 0;1/2, 0)
+(-2, -1;0, 1/2)
+(1, 0;1/2, 0)
+(-1, 0;1/2, 0)
+(0, 0;0, 0)
+1
+-40
+40
+ω (meV)
+E  (meV)
+0
+100
+150
+50
+11
+21
+31
+41
+1
+(-1, 0;1/2, 0)
+(-1, 0;1/2, 0)
+(0, 0;0, 0)
+(0, 0;0, 0)
+(1, 0;1/2, 0)
+(1, 0;1/2, 0)
+-40
+40
+ω (meV)
+N
+11
+21
+31
+41
+N
+(0, 0;0, 0)
+(1, 0;1/2, 0)
+(1, 0;1/2, 0)
+(-1, 0;1/2, 0)
+1
+(1, 1;1/2, 1/2)
+(1, 1;1/2, 1/2)
+-40
+40
+ω (meV)
+FIG. 2. Phase diagram and fixed points. (a) The self-consistent mean-field values of µc, µf, G as functions of the total filling ν
+from ν = 0 to 4, where we have enforced the symmetries of the correlated insulator state at CNP. The left y-axis represents µc
+and µf while the right y-axis represents G with a different range. (b) The phase diagram in the parameter space of µc, µf for
+G = 8meV. The white lines are phase boundaries between the local moment (LM) phases and the strong coupling phase. The
+dashed black lines are crossover boundaries between the frozen impurity (FI) and Kondo regimes of the strong coupling phase.
+The color maps the half-width of the spectral density peak, reflecting the Kondo energy scale if in the Kondo regime. The
+solid black line indicates the trajectory of µc and µf determined from a self-consistent calculation as ν changes from 0 to 4,
+where the five arrows from left to right represent ν = 0, 1, 2, 3, 4, respectively. (c) is the same as (b) but a different parameter
+G = 14meV is used. (d)(e) The RG flow of the many-body spectrum of the scaled Hamiltonian �
+HN (N ∈ odd) in the Kondo
+regime, where µc = 30.7meV, µf/U1 = 0.367, G = 9.49meV is the mean field value at ν = 1.25 for (d) and µc = 49.9meV,
+µf/U1 = 1.286, G = 11.83meV is the mean field value at ν = 2.5 for (e). The spectral lines’ colors represent the symmetry
+sectors labeled by good quantum numbers (Q1, Q2; S1, S2). Since the levels in sectors (Q1, Q2; S1, S2) and (Q2, Q1; S2, S1) are
+identical, only |Q1| ≥ |Q2| sectors are shown for simplicity. The insets are the resulting single-particle spectral densities that
+exhibit Kondo resonances. (f)(g)(h) The RG flow of many-body spectrum of the scaled Hamiltonian �
+HN (N ∈ even) in the
+LM1,2,3 phase, where µc = 5meV, µf/U1 = 0.5, 1.5, 2.5, G = 8meV respectively. The insets are the resulting single-particle
+spectral densities that exhibit Hubbard bands.
+low-lying many-body spectrum is identical to the one of
+a free-fermion chain defined by ϵn and tn with an ad-
+ditional chemical potential term.
+In other words, the
+impurity acts as if it was nonexistent. The underlying
+mechanism is either the Kondo screening, where the im-
+purity is an effective LM screened by the bath, or the
+impurity freezing, where the impurity occupation νf is
+effectively empty or full. We refer to the two cases as the
+Kondo regime and the frozen impurity (FI) regime, re-
+spectively, which are adiabatically connected. The fixed
+points shown in Fig. 2(d), (e) are in the Kondo regime
+because, if we continuously change µc to 0, they evolve
+to the LM1 and LM2 states (discussed in the next para-
+graph), respectively. There is a crossover between Kondo
+and FI regimes as one changes µf, as indicated by the
+dashed lines in Fig. 2(b), (c). Later we will determine
+the crossover boundary using the spectral density.
+At an LM fixed point, as exampled in Fig. 2(f), the low-
+lying many-body spectrum is identical to a free-fermion
+chain plus a detached LM. Depending on the represen-
+tation of the ground state, the LM fixed points can be
+further classified into LMn, where n = 1, 2, 3 is the
+effective impurity occupation.
+The flows of the spec-
+tra towards these fixed points are shown in Fig. 2(f),
+(g), (h), respectively.
+LMn ground states have the
+same SU(2)×SU(2) representations as ground states of
+ˆHf + ˆHH (Eqs. (12) and (18)) with n impurity elec-
+trons, where the Hubbard interaction freezes charge exci-
+tations and the Hund’s coupling prefers states with elec-
+trons lying in different orbitals. For n = 1, the ground
+states are four-fold degenerate and belong to the sym-
+metry sectors (Q1, Q2; S1, S2) = (2k + 1, 2k; 1
+2, 0) and
+(2k, 2k + 1; 0, 1
+2), corresponding to the spin- 1
+2 states of
+the two U(2)’s, respectively. The SU(2) representations
+are the same as those of the four single-particle states of
+ˆHf + ˆHH: (nf
+1↑, nf
+1↓; nf
+2↑, nf
+2↓) = (10;00), (01;00), (00;10),
+(00;01).
+For n = 2, the ground states are also four-
+fold degenerate but belong to a different symmetry sec-
+tor (2k + 1, 2k + 1; 1
+2, 1
+2).
+One can understand them
+as the product states of two spin- 1
+2 states of the two
+U(2)’s. They have the same SU(2) representations as the
+four two-particle states of ˆHf + ˆHH: (10;10), (10;01),
+(01;10), (01;01). Without Hund’s coupling JH, the LM2
+ground states would be
+�4
+2
+�
+= 6-fold degenerate. A fi-
+
+11
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+-40
+0
+40
+(a) kBT=0
+-40
+0
+40
+0
+1
+2
+3
+4
+5
+(b) kBT=0
+(c) kBT=2meV
+(d) kBT=5meV
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+-40
+0
+40
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+-40
+40
+-5 5
+A(𝜔,T)  (meV-1)
+𝜔  (meV)
+𝜔  (meV)
+𝜔  (meV)
+𝜔  (meV)
+𝜈=0
+𝜈=1
+𝜈=2
+𝜈=3
+𝜈=4
+FIG. 3. Spectral densities A(ω, T) at various fillings ν and
+temperatures. (a) Spectral densities at the zero temperature
+for ν = 0, 0.1, 0.2 · · · 4. The curves are offset by 0.1ν (meV−1)
+for clarity. (b) is the same as (a) but is shown with a smaller
+vertical scale for clarity of Hubbard bands, marked by in-
+verted triangles. The curves are offset by 0.01ν (meV−1). (c)
+and (d) are spectral densities at finite temperatures, where
+the curves are offset by 0.01ν (meV−1).
+nite JH raises the energy of the two many-body states
+with both electrons occupying the same orbital, i.e.,
+(11;00), (00;11). For n = 3, the ground states are still
+four-fold degenerate but belong to the symmetry sec-
+tors (2k − 1, 2k; 1
+2, 0) and (2k, 2k − 1; 0, 1
+2). Their SU(2)
+representations are same as the four single-hole states
+ˆHf + ˆHH: (01;11), (10;11), (11;01), (11;10).
+It is also helpful to look at the global U(2) symme-
+try representations, where the two U(2) rotations are the
+same. The total charge and spin of LM1,2,3 are 1, 2, 3
+(mod 4) and 1
+2, 1
+2 ± 1
+2, 1
+2, respectively.
+Phase boundaries between different LM phases and the
+strong coupling phase are described by unstable fixed
+points where different ground states cross with each
+other.
+The phase boundaries are shown by the white
+lines in Fig. 2(b), (c). Starting from an LMn phase, in-
+creasing µc will eventually drive it into a strong coupling
+phase due to the enhancement of hybridization. The crit-
+ical µc, as expected, is close to G, the conduction band
+edge (Fig. 1(c), (d)).
+V.
+SPECTRAL DENSITY
+We calculate the spectral density of the f-electrons,
+A(ω, T) = − 1
+π
+�
+αs Gαs(ω, T), with Gαs(ω, T) being the
+retarded Green’s function of fαs at the temperature T.
+We use the method described in Ref. [100] to collect
+the many-body levels at different RG steps to compute
+A(ω, T).
+The fixed points in Fig. 2(d), (e) are in the
+Kondo regime and hence have sharp zero-energy peaks
+due to the Kondo resonance, as shown in the insets of
+Fig. 2(d), (e).
+The fixed points in Fig. 2(f), (g), (h)
+are in the LM1,2,3 phases, respectively, thus, their spec-
+tral density are dominated by the upper and lower Hub-
+bard bands. We compute the spectral densities for all
+the points in the phase diagrams in Fig. 2(b), (c). We
+identify a central peak for every calculation and measure
+its half-width δK at half maximum. (If there is no cen-
+tral peak, e.g., Fig. 2(f), δK = 0.) δK is indicated by the
+color in Fig. 2(b), (c). We can distinguish the Kondo and
+FI regimes in the strong coupling phase through spectral
+density. Intuitively, a state in the Kondo regime should
+have a Kondo resonance. By contrast, a state in the FI
+regime should have its main spectral weight away from
+zero energy because the impurity occupation is either
+empty or full.
+Thus, we identify a phase point in the
+Kondo regime if δK covers the zero energy and otherwise
+in the FI regime. The crossover between the two regimes
+is indicated by dashed lines in Fig. 2(b), (c).
+Several features of δK in Fig. 2(b), (c) can be un-
+derstood using the poor man’s scaling developed in
+Sec. IV A. First, there are three domes around µf =
+1
+2U1, 3
+2U1, 5
+2U1 where δK is relatively small. They cor-
+respond to the nf = 1, 2, 3 cases discussed in Sec. IV A.
+From the poor man’s scaling perspective, these three µf’s
+correspond to the minimal initial value of the coupling
+constant g (Eq. (22)), which then leads to smaller TK’s.
+Second, when µc is small (≲ 30meV), δK in the middle
+dome is significantly smaller than those of the other two
+domes. The reason is that the Kondo energy scale TK
+for nf = 2 will be strongly suppressed due to the multi-
+plet splitting if TK is smaller than JH. One can see that
+the N factor in the exponential function of Eq. (27) is
+replaced by 2. Third, for the same µc, the first dome has
+lower δK than the third dome. This difference is a result
+of the suppression factor y1 for nf = 1 (Eq. (24)) and the
+enhancement factor y3 for nf = 3 (Eq. (29)) due to the
+particle-hole asymmetry of ∆(ω), as discussed in detail
+in Sec. IV A.
+In order to compare our results with STM measure-
+ments, we need to adopt physical µc, µf, G parameters.
+As discussed in Sec. III C, µc, µf, G can be determined as
+functions of the filling ν via a symmetric self-consistent
+calculation of Eqs. (10). µc, µf, G as functions of ν are
+shown in Fig. 2(a). The obtained spectral densities at the
+zero temperature are shown in Fig. 3(a), (b). For ν = 0,
+the state is in the FI regime with an (almost) zero occu-
+pation; hence the spectral weight is mainly distributed at
+positive energy. As ν increases, the spectral peak moves
+to the zero energy and is eventually pinned at the zero en-
+ergy to form a Kondo resonance. This is precisely what is
+seen in STM experiments at low temperatures (T < 1K)
+[11, 19, 21, 22]. One can also observe the evolution of
+Hubbard bands at T = 0 as ν changes (Fig. 3(b)), but
+they are relatively weak compared to the Kondo reso-
+nance peaks.
+At finite temperatures (Fig. 3(c), (d)),
+the Kondo resonance peaks are smeared by thermal fluc-
+tuations and the evolution of Hubbard bands becomes
+clearer. As ν increases from 0 to 4, the Hubbard bands
+periodically pass through the zero energy, matching the
+cascade of transitions seen in STM experiments at higher
+
+12
+(a)
+kBT (meV)
+4
+102
+101
+100
+10-1
+10-2
+10-3
+𝜈
+0
+1
+2
+3
+101
+100
+10-1
+10-2
+102
+101
+100
+10-1
+10-2
+10-3
+kBT (meV)
+102
+101
+100
+10-1
+10-2
+10-3
+kBT (meV)
+100
+10-1
+10-2
+10-3
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+𝜈
+(b)
+(d)
+(c)
+102
+101
+100
+10-1
+10-2
+10-3
+kBT (meV)
+𝜈
+0
+1
+2
+3
+4
+0
+1
+2
+3
+(e)
+𝜔 (meV)
+0
+20
+40
+-20
+-40
+0
+0.1
+0.2
+0.3
+A(ω,T) (meV-1)
+𝜈=1
+0
+0.1
+0.5
+1
+2
+5
+kBT (meV)
+kBT =0.82meV
+(f)
+𝜈
+0
+1
+2
+3
+4
+0
+0.4
+0.8
+1.2
+1.6
+Bloc(T)
+0
+5
+10
+15
+20
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+𝜈
+0
+1
+2
+3
+4
+Simp 
+×kBln2
+FIG. 4. Spin susceptibility and entropy contributed by the im-
+purity. (a) χloc(T)/χloc(0) as a function of filling ν and tem-
+perature T. (b) The local spin susceptibilities χloc(T) at fill-
+ings ν = 0, 0.5, 1 · · · 4. (c) The entropy contributed by the im-
+purity as a function of ν and T. (d) The entropy contributed
+by the impurity Simp(T)/(kB ln 2) at fillings ν = 0, 0.5, 1 · · · 4.
+(e) The spectral densities at ν = 1 at various temperatures.
+(f) The entropy contributed by the impurity as a function of
+ν at B = 0, 5, 10, 15, 20 T and temperature kBT = 0.82 meV.
+temperatures [17, 19].
+One can use δK to estimate the Kondo energy scale.
+Using the ν-dependent µc, µf, G parameters given in
+Fig. 2(a), we tabulate the δK’s at different fillings in Ta-
+ble II. Comparing it to TK estimated by the poor man’s
+scaling, denoted as T (P )
+K , we find T (P )
+K
+is about δK ∼ 2δK.
+VI.
+LOCAL MOMENTS AND THE
+POMERANCHUK EFFECT
+At a temperature exceeding the Kondo energy scale,
+the LM will become effectively decoupled from the bath
+and visible in experimental measurements. This is the
+mechanism of the Pomeranchuk effect [30, 31]. Refs. [30]
+observed a higher entropy (∼ 1kB per moir´e cell with kB
+being the Boltzmann’s constant) state at ν ≈ 1 at the
+temperature T ≈ 10K. As this entropy can be quenched
+by an in-plane magnetic field, it is ascribed to a free local
+moment. Ref. [31] observed a similar effect at ν ≈ −1
+and showed that an additional resistivity peak that is
+absent at T = 0 develops in the higher entropy state at
+T ≈ 10K. These observations can be naturally explained
+by the transition from the Fermi liquid phase to the LM
+phase as the temperature increases.
+To demonstrate the LM phase at higher temperatures,
+we calculate the local spin susceptibilities χloc(T) us-
+ing the filling-dependent µc, µf, G parameters given in
+Fig. 2(a).
+χloc(T) is defined as
+dMloc
+dBloc [101, 102], with
+Mloc being the spin momenta contributed by the impu-
+rity and Bloc a local magnetic field that only acts on the
+impurity. As shown in Fig. 4(a), (b), for ν ≥ 0.5, χloc(T)
+approaches a constant as T → 0, and obeys the Curie’s
+law, i.e., χloc(T) ∼ T −1, when T is larger than the Kondo
+energy scale. Obeying Curie’s law is a clear indication of
+a free LM. One may notice that the T-dependences of
+χloc(T)’s for ν < 0.5 are non-monotonous. The ν < 0.5
+states lie in the FI regime, thus the spin susceptibili-
+ties are extremely small when T → 0, and will start to
+increase when T is able to excite the LM1 states. For
+ν > 0.5, we can define the Kondo temperature TK as
+the turning point between Curie’s behavior and Fermi
+liquid behavior. Specifically, it can be obtained as the
+crossing of the extended T −1 line from the LM side and
+the extended horizontal line from the Fermi liquid side
+(Fig. 4(b)). We tabulate the resulting TK in Table II. As
+shown in the table, such defined TK is about 1
+3δK ∼ δK
+with δK being the half-width of the spectral density dis-
+cussed in Sec. V.
+We also calculate the impurity entropy Simp(T) for
+comparison with experiments. Simp(T) is defined as the
+difference of the entropy of �HN and that of a reference
+free-fermion chain (without the impurity site) defined
+by the same ϵn, tn parameters as in �HN. As shown in
+Fig. 4(c) and (d), Simp(T) is zero in the Fermi liquid
+phase at sufficiently low T and starts to increase when
+T reaches the Kondo energy scale. For ν = 1, Simp(T)
+climbs to about 2 ln 2 · kB at about kBT ≈ 1meV and
+(approximately) stays at this value until kBT reaches
+10meV. The entropy 2 ln 2 · kB ≈ 1.39kB is close to the
+measured value (∼ 1kB) in Refs. [30, 31] and can be un-
+derstood as contributed by the four degenerate states in
+the LM1 phase. Higher excited states will be involved
+when kBT is larger than 10meV, and the entropy contin-
+ues to increase for larger kBT. We also show the spectral
+density at ν = 1 in Fig. 4(e), one can see that the Kondo
+resonance peak becomes weak for kBT > 1meV, which
+is consistent with the entropy and spin susceptibility re-
+sults.
+An in-plane magnetic field will polarize the spin and
+hence suppress the entropy.
+However, as discussed in
+Sec. IV C, the four-fold degenerate LM1 states consist of
+two spin- 1
+2 states due to the orbital degeneracy, hence
+a strong field will not completely quench the entropy.
+Instead, due to the orbital degeneracy, the remaining en-
+tropy will be ln 2 · kB ≈ 0.69kB. This is also consistent
+with observations in Ref. [30]. In Fig. 4(f), we plot the
+
+13
+(a)  𝜈=1.2
+(b)  𝜈=1.8
+(c)  𝜈=2.4
+Band Energy (meV)
+0
+20
+40
+60
+-20
+-40
+-60
+ΓM
+KM
+MM
+ΓM
+KM
+MM
+ΓM
+KM
+MM
+0
+0.2
+0.4
+0.6
+0.8
+1
+FIG. 5. Heavy Fermi liquid bands at ν = 1.2 (a), 1.8 (b), 2.4
+(c). As explained in the text, the effective hybridization be-
+tween c- and f-bands is suppressed by a factor z
+1
+2 with z being
+the quasi-particle weight of f-electrons, which are estimated
+as 0.038, 0.27, 0.16 for (a), (b), and (c), respectively. The
+color of the bands represents the total quasi-particle weight,
+which is always larger than z.
+impurity entropy as a function of the filling at various
+magnetic fields Bloc. At ν = 1, the entropy saturates
+to a constant ∼ ln 2 · kB under a strong field. One can
+see that the entropy values, the shapes of the curves, and
+their field-dependences in Fig. 4(f) are comparable to the
+experimental results in Fig. 2(e) of Ref. [30].
+VII.
+DISCUSSIONS
+Based on the NRG calculations, the poor man’s scal-
+ing, and various experimental observations, we have
+shown that the gapless states at 1 ≲ |ν| < 2 are in the
+Kondo regime. Considering the translation invariance of
+the actual system, these states must be the heavy Fermi
+liquid states. Here we estimate the heavy Fermi liquid
+bands from the information provided by the NRG cal-
+culation. At the zero temperature, a spectral density in
+the Kondo regime possesses a Lorentz peak around the
+zero energy, i.e., A(ω) ≈ 4z
+π
+δK
+ω2+δ2
+K , where z is the quasi-
+particle weight, δK is the half-width given in Fig. 2(b),
+(c), and the factor 4 is from orbital and spin degenera-
+cies. Then the quasi-particle weight can be estimated as
+z = πA(0)δK/4. Substituting the quasi-particle part of
+Gαs(iω), i.e.,
+z
+iω, into Dyson’s equation of c-electrons
+G(c)(iω, k) = G(c,0)(iω, k)
++ G(c,0)(iω, k)H(cf)(k) z
+iω H(cf)†(k)G(c)(iω, k) ,
+(32)
+one can see that the cf hybridization is effectively sup-
+pressed by a factor of z
+1
+2 . Here ω is the Matsubara fre-
+quency, G(c,0)(iω, k) = (iω − H(c)(k))−1 is the free prop-
+agator of c-electrons, and H(c)(k), H(cf)(k) are Hamilto-
+nian matrices in Eq. (9). If z = 0 the effective hybridiza-
+tion will be zero, corresponding to the LM phase where
+the Fermi surface is solely contributed by the c-electrons.
+Using this method we obtain the bands at ν = 1.2 and
+1.8, as shown in Fig. 5(a), (b), respectively. One should
+be aware that the Hubbard band information is com-
+pletely neglected in this method. For future reference,
+we also estimate the heavy Fermi liquid bands at ν = 2.4
+(Fig. 5(c)) by assuming the ν = 2.4 state in the Kondo
+regime.
+The heavy Fermi liquid states at 1 ≲ |ν| < 2 can be
+further confirmed by future experimental research. For
+example, the Fermi surface can dramatically change as
+one tunes the filling and temperature or applies an ex-
+ternal field. The Fermi surface change will be reflected
+in spectral measurements such as quasi-particle interfer-
+ence. It is also in principle possible to directly measure
+the scattering phase shift [103].
+c-bands will induce RKKY interactions between LMs
+at different f-sites, which can lead to further symme-
+try breaking and should be crucial to stabilize the ob-
+served correlated insulator states at |ν| = 2. We have
+ignored these RKKY interactions and further symmetry
+breaking in the current work. A full self-consistent treat-
+ment including both RKKY and Kondo screening effects
+may result in more complicated ν-dependencies of µc, µf
+than those shown Fig. 2(a), (b), (c).
+For example, at
+ν = 2, µc given by a self-consistent symmetry breaking
+Hartree-Fock mean field [61] is about 26meV, which is
+significantly lower than the one (∼40meV) in Fig. 2(a).
+Thus, a possible mechanism for the correlated insulators
+to win the Kondo screening is that µc drops to a small
+value such that the Kondo energy scale becomes irrele-
+vant. (See Fig. 2(b), (c).) Observations in Refs. [18, 26]
+also suggest that the ν-dependencies of µc, µf are com-
+plicated. The competition between RKKY and Kondo
+screening is also a potential mechanism for the observed
+strange metal behaviors [27–29] and could play an impor-
+tant role in the unconventional superconductivity [1, 4–
+11]. We leave this for future studies.
+Note added.
+During the preparation of the current
+work, a related work [104] appeared. This work studied
+the symmetric Kondo state using a slave-fermion mean
+field in a Kondo lattice model derived from the THF
+model. Our theory is based on the symmetry-broken cor-
+related state at CNP. We are also aware of related works
+on the Kondo problem in MATBG by A. M. Tsvelik’s and
+B. A. Bernevig’s group [105, 106] and P. Coleman’s group
+[107] that will appear soon, and a generalization of the
+THF model to the magic-angle twisted trilayer graphene
+[108]. Ref. [105] also obtains a Kondo temperature about
+1 ∼ 2K around |ν| ≈ 1.
+ACKNOWLEDGMENTS
+We are grateful to B. Andrei Bernevig, Ning-Hua
+Tong, Xi Dai, Jia-Bin Yu, Xiao-Bo Lu, Yong-Long Xie,
+Yi-Lin Wang, and Chang-Ming Yue for helpful discus-
+sions. Z.-D. S. and G.-D. Z. were supported by National
+Natural Science Foundation of China (General Program
+No. 12274005), National Key Research and Development
+Program of China (No. 2021YFA1401900).
+
+60
+0.9
+40
+0.8
+0.7
+20
+0.6
+0
+0.5
+0.4
+-20
+0.3
+0.2
+-40
+0.1
+-60
+0
+K
+G
+M60
+0.9
+40
+0.8
+0.7
+20
+0.6
+0
+0.5
+0.4
+-20
+0.3
+0.2
+-40
+0.1
+-60
+0
+K
+G
+M60
+0.9
+40
+0.8
+0.7
+20
+0.6
+0
+0.5
+0.4
+-20
+0.3
+0.2
+-40
+0.1
+-60
+0
+K
+G
+M60
+0.9
+40
+0.8
+0.7
+20
+0.6
+0
+0.5
+0.4
+-20
+0.3
+0.2
+-40
+0.1
+-60
+0
+K
+G
+M14
+Appendix A: More details about the effective
+Hamiltonian
+1.
+Nonzero M term
+A generic trial ground state at CNP is given by
+(Eq. (6))
+|Ψ0⟩ = U
+�
+R
+f †
+R1+↑f †
+R1+↓f †
+R2+↑f †
+R2+↓|FS⟩ ,
+(A1)
+where U = exp(−iθµν ˆΣµν) is a U(4) rotation operator
+and an implicit summation over repeated µ, ν indices is
+assumed. We can always define the rotated fermion op-
+erators �ckaηs = UckaηsU †, �fRaηs = UfRaηsU † such that
+�fRα+s’s are occupied in |Ψ0⟩ and �fRα−s’s are empty in
+|Ψ0⟩. According to the discussions in the supplementary
+material section S4B of Ref. [61], in the flat-band limit
+(M = 0), the lowest particle (hole) excitations only in-
+volve �cka−s and �fRa−s (�cka+s and �fRa+s).
+Thus, the
+effective periodic Anderson model for ν > 0 derived
+in Sec. III B is written in terms of ckas = �cka−s and
+fRαs = �fRα−s. Here we give the explicit forms of the
+rotated operators
+�fRαηs =
+�
+α′η′s′
+�
+eiθµνΣf
+µν
+�
+αηs,α′η′s′ fRα′η′s′ ,
+(A2)
+and
+�ckaηs =
+�
+a′=1,2
+η′s′
+�
+eiθµνΣc12
+µν
+�
+aηs,a′η′s′ cka′η′s′
+(a = 1, 2), (A3)
+�ckaηs =
+�
+a′=3,4
+η′s′
+�
+eiθµνΣc34
+µν
+�
+aηs,a′η′s′ cka′η′s′
+(a = 3, 4), (A4)
+where the eight-by-eight matrices Σf
+µν, Σc12
+µν , Σc34
+µν
+are
+defined in Eqs. (3) to (5).
+The M-term in the original basis of the THF model
+(Eq. (1)) is
+M
+�
+aa′=3,4
+�
+|k|<Λc
+�
+ηs
+[σx]aa′c†
+kaηscka′ηs .
+(A5)
+It favors the Kramers inter-valley coherent state dis-
+cussed at the end of Sec. II, where θx0 and θy0 are nonzero
+and satisfy θ2
+x0 + θ2
+y0 = (π/4)2. Without loss of general-
+ity, we assume U = exp(−i π
+4 ˆΣx0) for the Kramers inter-
+valley coherent state. Writing this M-term in terms of
+the rotated operators, we obtain
+M
+�
+|k|<Λc
+�
+a,a′=3,4
+�
+ηη′ss′
+�c†
+kaηsOaηs,a′η′s′�cka′η′s′ ,
+(A6)
+where O = ei π
+4 Σc34
+x0 σxτ0ς0e−i π
+4 Σc34
+x0
+= −σzτxς0. The τx
+matrix in O represents couplings between the empty
+and occupied single-particle states. If we simply project
+this M-term onto the empty states, it vanishes, i.e.,
+[Oa−s,a′−s′] = 0.
+A better approximation is applying
+a Schrieffer-Wolff transformation to decouple the η = ±
+states, leading to a second-order correction to the effec-
+tive Hamiltonian. As ⟨Ψ0| �f †
+αηs �fαηs|Ψ0⟩ = (1 + η)/2, the
+J term in Eq. (2) yields the following mean field term (see
+also the supplementary material section S4B of Ref. [61])
+− J
+2
+�
+a=3,4
+�
+ηs
+η · �c†
+aηs�caηs
+(A7)
+Then, regarding the J
+2 term as the zeroth order Hamilto-
+nian and M as a perturbation, a Schrieffer-Wolff trans-
+formation leads to the correction
+− M 2
+J
+�
+|k|<Λc
+�
+a=3,4
+�
+ηs
+η · �c†
+kaηs�ckaηs + O(M 4) .
+(A8)
+The resulting energy levels ±(J/2 + M/J2) at k = 0 is
+fully consistent with a Taylor expansion of the one-shot
+energy levels ±
+�
+J2/4 + M 2 derived in Ref. [61]. Pro-
+jecting the correction to the active d.o.f., i.e., ckas =
+�cka−s, we obtain the correction to the effective Hamilto-
+nian
+M 2
+J
+�
+|k|<Λc
+�
+a=3,4
+�
+s
+c†
+kasckas + O(M 4) .
+(A9)
+2.
+Hund’s coupling
+The four-by-four Hamiltonian matrix H(c)(k)+∆H(c)
+in Eq. (13), i.e., −v⋆(σx⊗σ0kx+σy ⊗σzky)+02×2⊕Gσ0,
+can be diagonalized analytically. As discussed at the end
+of the last subsection, to O(M 2), the M term simply
+shifts the energy of a = 3, 4 electrons by M 2/J. Thus,
+all the analysis below applies to the M ̸= 0 after G is
+replaced by G + M 2/J. We find the energy eigenvalues
+and wave-functions of the H(c)(k) + ∆H(c) as
+ϵ1(k) =ϵ+(k) = G
+2 +
+�
+G2
+4 + v2⋆k2
+u1(k) =
+�
+sin θk
+2 e−iφk 0 − cos θk
+2
+0
+�T
+,
+(A10)
+ϵ2(k) =ϵ+(k) = G
+2 +
+�
+G2
+4 + v2⋆k2
+u2(k) =
+�
+0 sin θk
+2 eiφk 0 − cos θk
+2
+�T
+,
+(A11)
+ϵ3(k) =ϵ−(k) = G
+2 −
+�
+G2
+4 + v2⋆k2
+u3(k) =
+�
+cos θk
+2 e−iφk 0 sin θk
+2
+0
+�T
+,
+(A12)
+ϵ4(k) =ϵ−(k) = G
+2 −
+�
+G2
+4 + v2⋆k2
+u4(k) =
+�
+0 cos θk
+2 eiφk 0 sin θk
+2
+�T
+.
+(A13)
+
+15
+where
+θk = arccos
+G/2
+�
+G2/4 + v2⋆k2
+(A14)
+and φk = arg(kx + iky).
+We now derive the effective Hund’s coupling ˆHH. Ap-
+plying a second-order perturbation in terms of ˆHJ, we
+obtained the correction to the Hamiltonian
+∆ ˆH = − J2
+N 2
+M
+�
+I
+�
+α1α2
+s1s′
+1s2s′
+2
+�
+k1,k′
+1
+k2,k′
+2
+(f †
+α1s1fα1s′
+1 − νf
+4 δs1s′
+1)
+× (f †
+α2s′
+2fα2s2 − νf
+4 δs2s′
+2) · e− λ2
+2 (k2
+1+k′2
+1 +k2
+2+k′2
+2 )
+×
+⟨Ψ0|c†
+k′
+1α1+2s′
+1ck1α1+2s1|ΨI⟩⟨ΨI|c†
+k2α2+2s2ck′
+2α2+2s′
+2|Ψ0⟩
+EI − E0
+,
+(A15)
+where |ΨI⟩ are excited states with a single particle-hole
+pair and EI are the energies of the excited states. k1,2,
+k′
+1,2 are limited within the cutoff Λc. Due to the mo-
+mentum and spin conservation, for the matrix element
+to be nonzero, there must be k1 = k2, s1 = s2, k′
+1 = k′
+2,
+s′
+1 = s′
+2. For simplicity, we rewrite k1, k′
+1, s1, and s′
+1
+as k, k′, s, and s′, respectively. (k, s) and (k′, s′) label
+the particle and the hole excitations, respectively. Then
+the matrix element in the third line of Eq. (A15) can be
+written as
+nF (ϵ+(k′) − µc)(1 − nF (ϵ−(k) − µc))
+× ⟨Ψ0|c†
+k′α1+2s′ckα1+2sc†
+kα2+2sck′α2+2s′|Ψ0⟩
+(A16)
+According
+to
+the
+wave
+functions
+given
+in
+Eqs.
+(A10)-(A13),
+there are ⟨Ψ0|c†
+k′α1+2s′ck′α2+2s′|Ψ0⟩
+=
+δα1α2 sin2 θk′
+2 , ⟨Ψ0|ckα1+2sc†
+kα2+2s|Ψ0⟩ = δα1α2 cos2 θk
+2 .
+The excitation energy EI −E0 is given by ϵ+(k)−ϵ−(k′).
+Thus, ∆ ˆH is simplified to
+∆ ˆH = − J2
+N 2
+M
+�
+αss′
+kk′
+(f †
+αsfαs′ − νf
+4 δss′)(f †
+αs′fαs − νf
+4 δss′)
+× nF (ϵ−(k′) − µc)(1 − nF (ϵ+(k) − µc)) sin2 θk′
+2 cos2 θk
+2
+ϵ+(k) − ϵ−(k′)
+× e−λ2(k2+k′2)
+(A17)
+The s = s′ contribution is an effective chemical potential
+shift, estimated as 0.17meV at CNP, of the f-electrons.
+As it is much smaller than U1, we will omit the s = s′
+contribution. The s ̸= s′ contribution can be written as
+ˆHH = JH
+�
+α
+nf
+α↑nf
+α↓
+(A18)
+with JH given by
+JH =2J2
+�Ω0
+2π
+�2 ˆ Λc
+0
+dk′ · k′
+ˆ Λc
+k0
+dk · k · e−λ2(k2+k′2)
+× sin2 θk′
+2 cos2 θk
+2
+ϵ+(k) − ϵ−(k′) ,
+(A19)
+where k0 is determined by ϵ+(k0) = µc. Here we have
+made use of the fact that ϵ±(k) and θk only depends
+on |k| but not φk. At CNP, µc = 0 and G = J/2 =
+8.19meV, taking the limit Λc → ∞, we obtain JH ≈
+0.34meV. Using the self-consistent values of µc and G
+shown in Fig. 2(a), we find JH at ν = 1, 2, 3, 4 are given
+by 0.29meV, 0.26meV, 0.21meV, 0.19meV, respectively.
+As JH is small and does not change significantly with ν,
+in this work, we simply set JH = 0.34meV for simplicity.
+3.
+Hybridization function
+By definition, the hybridization function ∆(ω) is given
+by
+∆(ω) = π
+N
+�
+k
+�
+n
+|Vnα(k)|2δ(ω − ϵn(k))
+(A20)
+where Vnα(k) = �
+a u∗
+an(k)H(cf)
+aα (k)e− λ2k2
+2
+is the hy-
+bridization between fαs and the n-th energy band of c-
+electrons.
+∆(ω) does not depend on α because of the
+C2zT or C2x symmetry that flips the α index. Substitut-
+ing ϵn(k) and uan(k) in Eqs. (A10)-(A13) into the above
+equation, we obtain Vnα(k) for α = 1 as
+V11(k) =γ sin θk
+2 eiφk
+V21(k) =v′
+⋆(−kx + iky) sin θk
+2 e−iφk
+V31(k) =γ cos θk
+2 eiφk
+V41(k) =v′
+⋆(−kx + iky) cos θk
+2 e−iφk .
+(A21)
+Using the energy eigenvalues in Eqs. (A10)-(A13) and
+the Vnα(k) matrix elements given above, it is direct to
+obtain
+∆(ω) = Ω0
+2v2⋆
+����ω + µc − G
+2
+����
+�
+γ2 + v′2
+⋆ k2
+F
+�
+e−k2
+F λ2
+�
+θ(ω + µc − G) sin2 θkF
+2
++ θ(−ω − µc) cos2 θkF
+2
+�
+(A22)
+where kF is given by
+kF = 1
+v⋆
+�
+(ω + µc − G/2)2 − (G/2)2 .
+(A23)
+We now verify the asymptotic behaviors of ∆(ω).
+When ω + µc → G+, kF → 0 and only the first term
+in the second line of Eq. (A22) contributes to ∆(ω). Ac-
+cording to Eq. (A14), there is cos θkF =
+G/2
+ω+µc−G/2 and
+hence sin2 θkF
+2
+= 1
+2− 1
+2 cos θkF ≈ (ω+µc−G)/G. Then we
+obtain the asymptotic behavior of ∆(ω) as ω + µc → G+
+∆(ω) = Ω0
+4v2⋆
+γ2 · (ω + µc − G) + O((ω + µc − G)2) .
+(A24)
+When ω + µc → −0+, kF → 0 and only the second
+term in the second line of Eq. (A22) contributes to ∆(ω).
+
+16
+According to Eq. (A14), there is cos θkF =
+G/2
+G/2−ω−µc and
+hence cos2 θkF
+2
+= 1
+2 + 1
+2 cos θkF ≈ 1. Then we obtain the
+asymptotic behavior of ∆(ω) as ω + µc → −0+
+∆(ω) = Ω0
+4v2⋆
+Gγ2 + O((ω + µc)) .
+(A25)
+Appendix B: Poor man’s scaling of Anderson models
+with energy-dependent couplings
+1.
+Generic theory for U(N) models
+We consider the Anderson impurity model with N
+symmetric flavors
+ˆH = − µf ˆ
+Nf + U
+2
+ˆ
+Nf( ˆ
+Nf − 1) +
+N
+�
+µ=1
+ˆ D
+−D
+dϵ · ϵ · d†
+µ(ϵ)dµ(ϵ)
++
+N
+�
+µ=1
+ˆ D
+−D
+dϵ
+�
+∆(ϵ)
+π
+(f †
+µdµ(ϵ) + h.c.) ,
+(B1)
+where µ is the flavor index and Nf = �N
+µ=1 f †
+µfµ. We
+assume the ground state of the isolated impurity has nf
+f-electrons, which can take the values 1, 2 · · · (N − 1).
+(We do not consider the empty case (nf = 0), the full
+case (nf = N), and the mixed valence case where ground
+states with different nf are exactly degenerate.) We fur-
+ther assume the charge gaps to nf − 1 and nf + 1 elec-
+trons are ∆E− and ∆E+ = U − ∆E−, respectively. We
+then apply a Schrieffer-Wolff transformation to obtain an
+effective Coqblin–Schrieffer model for the Hilbert space
+restricted to ˆNf = nf
+ˆH =
+N
+�
+µ=1
+ˆ D
+−D
+dϵ · ϵ · d†
+µ(ϵ)dµ(ϵ) + 4g
+πU
+N
+�
+µ,µ′=1
+ˆ D
+−D
+dϵdϵ′
+�
+�
+∆(ϵ)∆(ϵ′)(f †
+µfµ′ − xδµµ′)d†
+µ′(ϵ′)dµ(ϵ)
+�
+.
+(B2)
+Terms that only involve ˆNf are omitted because they
+only contribute to an energy shift.
+The bandwidth D
+should be smaller than min(∆E+, ∆E−), otherwise, the
+Schrieffer-Wolff transformation is invalid. The parame-
+ters g and x are given by
+g = U
+4
+�
+1
+∆E+
++
+1
+∆E−
+�
+,
+x = ∆E−
+U
+,
+(B3)
+respectively. If µf = (nf − 1
+2)U, there is ∆E+ = ∆E− =
+1
+2U and g = 1, x = 1
+2.
+We now truncate the bandwidth at D−dD = D(1−dt)
+(dt ≪ 1) and consider second order (in g) corrections
+form the virtual particle (D − dD < ϵ < D) and hole
+(−D < ϵ < −D + dD) excitations. The particle excita-
+tion contributes to the correction
+− (4g)2
+(πU)2
+1
+D
+�
+µ1µ2µ′
+1µ′
+2
+ˆ D−dD
+−D+dD
+dϵ1dϵ2d
+ˆ D
+D−dD
+dϵ′
+1dϵ′
+2
+×
+�
+∆(ϵ1)∆(ϵ2)∆(ϵ′
+1)∆(ϵ′
+2)d†
+µ1(ϵ1)⟨dµ′
+1(ϵ′
+1)d†
+µ′
+2(ϵ′
+2)⟩dµ2(ϵ2)
+× (f †
+µ′
+1fµ1 − xδµ1µ′
+1)P(f †
+µ2fµ′
+2 − xδµ2µ′
+2) .
+(B4)
+The denominator D in the factor is the excitation en-
+ergy of a virtual particle.
+P is a projector to the re-
+stricted Hilbert space, where ˆNf = nf.
+The expecta-
+tion ⟨dµ′
+1(ϵ′
+1)d†
+µ′
+2(ϵ′
+2)⟩ evaluated on the ground state is
+δ(ϵ′
+1 − ϵ′
+2)δµ′
+1µ′
+2. Then we have
+− (4g)2
+(πU)2
+dD
+D ∆(D)
+�
+µ1µ2µ′
+ˆ D−dD
+−D+dD
+dϵ1dϵ2
+�
+∆(ϵ1)∆(ϵ2)
+× d†
+µ1(ϵ1)dµ2(ϵ2)(f †
+µ′fµ1 − xδµ1µ′)(f †
+µ2fµ′ − xδµ2µ′) , (B5)
+where P is omitted as it commutes with f †
+µ2fµ′ and
+f †
+µ′fµ1.
+After a few steps of algebra, the four-fermion
+operator �
+µ′ f †
+µ′fµ1f †
+µ2fµ′ can be simplified to
+f †
+µ2fµ1 +
+�
+µ′
+f †
+µ′fµ′fµ1f †
+µ2 = f †
+µ2fµ1(1−nf)+nfδµ1µ2 , (B6)
+where we have made use of the fact that the Hilbert space
+is restricted to ˆNf = nf. Substituting this into Eq. (B5),
+we obtain the corrections to parameters g and xg as
+dg
+dt
+����
+p
+= 4∆(D(t))
+πU
+((nf − 1) + 2x) g2 ,
+(B7)
+d(xg)
+dt
+����
+p
+= 4∆(D(t))
+πU
+�
+x2 + nf
+�
+g2 .
+(B8)
+Here t is the RG parameter and D(t) = De−t is the
+reduced bandwidth after successive t/dt RG steps.
+We then calculate the contribution from virtual hole
+excitation.
+Following the same process as in the last
+paragraph, we obtain
+− (4g)2
+(πU)2
+dD
+D ∆(−D)
+�
+µ1µ2µ′
+ˆ D−dD
+−D+dD
+dϵ1dϵ2
+�
+∆(ϵ1)∆(ϵ2)
+× dµ1(ϵ1)d†
+µ2(ϵ2)(f †
+µ1fµ′ − xδµ1µ′)P(f †
+µ′fµ2 − xδµ2µ′)
+= (4g)2
+(πU)2 dt∆(−D)
+�
+µ1µ2µ′
+ˆ D−dD
+−D+dD
+dϵ1dϵ2
+�
+∆(ϵ1)∆(ϵ2)
+× d†
+µ2(ϵ2)dµ1(ϵ1)(f †
+µ1fµ′ − xδµ1µ′)(f †
+µ′fµ2 − xδµ2µ′) .
+(B9)
+In the second equation, we have omitted an energy con-
+stant term from the anti-commutator {d†
+µ2(ϵ2), dµ1(ϵ1)}.
+P is the projector to the restricted Hilbert space, where
+ˆNf = nf.
+It is omitted in the second equation be-
+cause it commutes with f †
+µ′fµ2 and f †
+µ1fµ′.
+The four-
+fermion operator �
+µ′ f †
+µ1fµ′f †
+µ′fµ2 can be simplified to
+(N − nf + 1)f †
+µ1fµ2 as the Hilbert space is restricted to
+ˆNf = nf. Then the corrections to g, xg from Eq. (B9)
+can be read out as
+dg
+dt
+����
+h
+= 4∆(−D(t))
+πU
+(N − nf + 1 − 2x) g2 ,
+(B10)
+d(xg)
+dt
+����
+h
+= 4∆(−D(t))
+πU
+�
+−x2�
+g2 .
+(B11)
+
+17
+Adding up the particle and the hole contributions we
+can obtain the RG equations for g and (xg). The Kondo
+energy scale TK can be estimated as the reduced band-
+width D(t) where g diverges. For a constant ∆(ω) = ∆0,
+we obtain
+dg
+dt = 4∆0
+πU Ng2,
+d(xg)
+dt
+= 4∆0
+πU nfg2
+(B12)
+and the solution
+g(t) =
+g(0)
+1 − g(0) 4∆0
+πU N · t ,
+(B13)
+x(t) = x(0)g(0)
+g(t) + nf
+N · g(t) − g(0)
+g(t)
+.
+(B14)
+where g(0) is the initial condition given in Eq. (B3). g(t)
+diverges at tK =
+πU
+4N g(0)∆0 , corresponding the Kondo en-
+ergy scale De−tK = De−
+πU
+4N g(0)∆0 .
+As g(t) diverges as
+t → tK, the second term in x(t) dominates and there
+must be x → nf
+N . In other words, x flows to the occupa-
+tion fraction.
+2.
+Application to the symmetric state at CNP
+We assume a symmetric state of the THF model at
+CNP and examine its Kondo energy scale. Following the
+calculations in Appendix A 3, we obtain the hybridiza-
+tion function contributed by the fully symmetric c-bands
+(Fig. 1(b))
+∆(ω) = Ω0
+4v2⋆
+� ����|ω| − M
+2
+���� θ(|ω| − M)
+�
+γ2 + v′2
+⋆ k2
+F 1
+�
+sin2 θkF 1
+2
+× e−k2
+F 1λ2 +
+����|ω| + M
+2
+����
+�
+γ2 + v′2
+⋆ k2
+F 2
+�
+cos2 θkF 2
+2
+e−k2
+F 2λ2�
+,
+(B15)
+where
+kF 1 = 1
+v⋆
+�
+(|ω| − M/2)2 − (M/2)2,
+(B16)
+kF 2 = 1
+v⋆
+�
+(|ω| + M/2)2 − (M/2)2,
+(B17)
+θk = arccos
+M/2
+�
+M 2/4 + v2⋆k2 .
+(B18)
+We should choose the initial cutoff D
+=
+1
+2U1 be-
+yond which the Schrieffer-Wolff transformation is invalid.
+For these states kF 1,2 ≲
+U1
+2v⋆ and hence v′2
+⋆ k2
+F 1,2 ≲
+119.4meV2, which is significantly smaller than γ2 ≈
+612.6meV2. The damping factors e−λ2kF 1,22 ≳ 0.74 are
+also large. Thus, in the following we approximate ∆(ω)
+(|ω| < U1/2) as
+∆(ω) ≈ Ω0
+4v2⋆
+� ����|ω| − M
+2
+���� θ(|ω| − M)γ2 sin2 θkF 1
+2
++
+����|ω| + M
+2
+���� γ2 cos2 θkF 2
+2
+�
+.
+(B19)
+We first consider the flat-band limit (M = 0), where
+the parameter θk is always π
+2 . Thus, we have
+∆(ω) = b|ω|,
+b = Ω0
+4v2⋆
+γ2 ≈ 0.1290 .
+(B20)
+We also assume that there is no multiplet splitting in the
+symmetric state such that the effective Anderson model
+should be a U(8) theory with nf = 4. Naively applying
+the RG equations derived in the last subsection gives
+d�g
+dt = −�g + 4bD
+πU1
+N �g2,
+(B21)
+where N = 8, D = U1/2, �g = ge−t. Due to the particle-
+hole symmetry at CNP, the initial condition (Eq. (B3))
+is �g(0) = 1. It seems that there would be an unstable
+fixed point �g∗ =
+2π
+4N b, the initial �g below (above) which
+flows to zero (infinity). Using the actual parameters we
+find g∗ ≈ 1.52, hence the system would not be in the
+Kondo phase. This result differs from the standard case
+with a constant ∆(ω), where a positive g always flows
+to infinity. Furthermore, a more careful RG analysis [98]
+shows that the fixed point �g∗ does not really exist. It is
+a false result of the weak coupling expansion, which fails
+for ∆(ω) ∼ |ω|r with r > 1
+2. Thus, a ∆(ω) ∼ |ω| bath
+does not have a strong coupling phase. This conclusion
+is also consistent with numerical studies [95–97].
+We then consider the case with M ̸= 0. We use the
+value M = 3.697meV. The RG process can be divided
+into two stages: (i) When D(t) = 1
+2U1e−t > M, there is
+approximately ∆(D(t)) ≈ bD(t). (ii) When D(t) < M,
+the first line of Eq. (B19) vanishes, and cos2 θkF 2
+2
+in the
+second line equals to
+M
+4ω+2M + 1
+2. Then there is ∆(D(t)) ≈
+∆0(1 + D(t)/M), with ∆0 = Ω0Mγ2
+8v2⋆
+≈ 0.239meV. The
+boundary between the two stages is t1 = ln U1
+2M . The RG
+equation in the first stage reads
+dg
+dt = 2Nb
+π
+g2e−t ⇒ g(t) =
+1
+1 − 2N b
+π (1 − e−t) .
+(B22)
+Due to the particle-hole symmetry, the initial condition
+given by Eq. (B3) is g(0) = 1. We have g1 = g(t1) ≈ 2.34.
+The RG equation in the second stage is given by
+dg
+dt =4N∆0
+πU1
+g2 + 4N∆0
+πU1
+g2 · e−(t−t1)
+⇒ g(t) =
+1
+g−1
+1
+− 4N ∆0
+πU1 (t − t1) − 4N ∆0
+πU1 (1 − e−(t−t1)) .
+(B23)
+
+18
+g(t) diverges at tK − t1 ≈
+πU1
+4g1N ∆0 − y with y = 1, corre-
+sponding to the Kondo energy scale
+kBTK = Mey · e−
+πU1
+4g1N ∆0 ≈ 3.8 × 10−4meV.
+(B24)
+3.
+Application to the effective model for ν > 0
+states
+In the absence of the Hund’s coupling JH, we can re-
+gard (α, s) as a composite index so that ˆHSI (Eq. (20))
+is a U(N) theory with N = 4. Then the flow equations
+in Appendix B 1 apply. For simplicity, we omit the neg-
+ative branch of ∆(ω) (Eq. (19)) at ω < −µc because
+it is far away from the Fermi level for ν > 0 (Fig. 1(d)).
+The positive branch of ∆(ω) can be well approximated by
+∆(ω) = ∆(0)(1+ω/(µc−G)) for |ω| < µc−G (Fig. 1(d)).
+We choose the initial cutoff D to be the minimum value
+of µc − G and ∆E±.
+Substituting this ∆(ω) into the
+general RG equations in Appendix B 1, we obtain
+dg
+dt = 4∆(0)
+πU1 Ng2 +
+4∆(0)D
+πU1(µc − G)(4x + 2nf − 2 − N)g2e−t
+(B25)
+and
+d(xg)
+dt
+= 4∆(0)
+πU1 nfg2 +
+4∆(0)D
+πU1(µc − G)(2x2 + nf)g2e−t (B26)
+The O(e−t) terms will eventually become irrelevant when
+t is sufficiently large.
+After the O(e−t) terms become
+irrelevant, we have
+d(xg)
+dg
+= nf/N, implying x →
+nf
+N
+at the divergence of g. We then approximate the flow
+equation of g by setting x to its fixed point value
+nf
+N ,
+i.e.,
+dg
+dt ≈ 4∆(0)
+πU1 Ng2 +
+4∆(0)D
+πU1(µc − G)(3nf − 6)g2e−t
+(B27)
+The solution of g is
+g(t) ≈
+1
+g−1(0) − 4∆(0)
+πU1 N
+�
+t + ynf (1 − e−t)
+� ,
+(B28)
+where ynf =
+D
+µc−G(3nf − 6). The Kondo energy scale
+is determined t = tK at which g diverges.
+Assuming
+tK ≫ 1, we have
+tK ≈
+πU1
+4Ng(0)∆(0) − ynf
+(B29)
+and hence
+kBTK ≈ D · eynf · e−
+πU1
+4N g(0)∆(0) .
+(B30)
+a.
+The nf = 1, 3 cases
+In the presence of the Hund’s coupling, we have to
+examine the derivations in Appendix B 1 carefully. The
+most important effect of ˆHH is to change the local Hilbert
+space at small energy scales. In general, JH leads to a
+multiplet splitting. When the RG energy scale is smaller
+than the splitting, the higher energy multiplet will be-
+come inaccessible, and the local Hilbert space is effec-
+tively reduced. A minor effect is that the charge gaps
+∆E± will depend on JH and the resulted coupling be-
+tween f-spin and d-spin in the Coqblin–Schrieffer model
+will break the U(N) symmetry.
+In the following, we study how ˆHH changes the RG
+equations. We first consider the nf = 1 case. In the vir-
+tual particle excitation process (Eq. (B5)), the interme-
+diate f-multiplet is given by |F ′⟩ = (f †
+µ2fµ′ − δµ2µ′)|F⟩,
+where F is the initial f-multiplet. (µ should be regarded
+as the composite index (α, s).) As |F ′⟩ has the same par-
+ticle number as |F⟩, it must be one of the four states with
+(n1↑, n1↓; n2↑, n2↓) = (10;00), (01;00), (00;10), (00;01).
+All of the possible intermediate states do not feel the
+Hund’s coupling (JH
+�
+α nα↑nα↓) and hence they have
+the same energy as |F⟩.
+Hence, the excitation energy
+of the intermediate state is purely contributed by d-
+electrons. Then all the following derivations apply. The
+same argument applies to the virtual hole excitation
+(Eq. (B9)). Therefore, the RG equations for nf = 1 will
+not be affected by JH. For the same reason, RG equa-
+tions for nf = 3 will also not be affected by JH, where
+the initial and intermediate states are single-hole states
+that do not feel JH. The TK for nf = 1, 3 is given by
+Eq. (B30).
+b.
+The nf = 2 case
+The Hilbert space with two particles has six states:
+(n1↑, n1↓; n2↑, n2↓) = (10;10), (10;01), (01;10), (01;01),
+(11;00), (00;11). The former four states have the energy
+−2µf + U1, and the latter two states have the energy
+−2µf + U1 + JH. Thus JH leads to a multiplet splitting.
+We divide the RG into two stages. In the first stage D(t)
+is larger than JH, then the splitting JH only plays a
+minor role and can be neglected. Thus the RG equations
+in the first stage are given by Eq. (B27). The first stage
+ends at t1 = ln(D/JH). If g diverges before t reaches
+t1, the Kondo energy scale should be given by Eq. (B30)
+with y2 = 0, i.e.,
+kBT ′
+K = D · e−
+πU1
+4N g(0)∆(0) .
+(B31)
+If g is still finite at t1
+g1 =
+g(0)
+1 − g(0) 4∆(0)
+πU1 N ln D
+JH
+,
+(B32)
+then the RG goes into the second stage.
+The effective cutoff and the initial g of the second stage
+are JH and g1, respectively. We first examine the virtual
+particle excitation process (Eq. (B5)), where the interme-
+diate f-multiplet is given by |F ′⟩ = (f †
+µ2fµ′ − δµ2µ′)|F⟩.
+
+19
+Here F is the initial f-multiplet.
+µ′, µ2 should be re-
+garded as the composite indices (α′, s′), (α2, s2), respec-
+tively. Suppose |F⟩ is one of the four low energy states,
+where each orbital (α = 1, 2) has one electron; then, for
+|F ′⟩ to be a low energy state, the index µ′ must have the
+same orbital index with µ2, i.e., α′ = α2, such that each
+orbital (α = 1, 2) in |F ′⟩ still has one electron.
+With
+this restriction, the four-fermion operator in Eq. (B6)
+becomes
+f †
+α2s2fα1s1 +
+�
+s′
+f †
+α2s′fα2s′fα1s1f †
+α2s2
+(B33)
+�
+s′ f †
+α2s′fα2s′ acting on the bra state (final state) gives
+nf
+α2, which must equal to 1 given that the bra state is one
+of the four low energy states. Thus the four-fermion op-
+erator equals to δα2α1δs2s1. The resulting contributions
+to the RG equation are
+dg
+dt
+����
+p
+= 4∆(D(t))
+πU
+(2x) g2 ,
+(B34)
+d(xg)
+dt
+����
+p
+= 4∆(D(t))
+πU
+�
+x2 + 1
+�
+g2 .
+(B35)
+We second examine the virtual hole excitation process
+(Eq. (B9)), where the intermediate f-multiplet is given
+by |F ′⟩ = (f †
+µ′fµ2 − δµ′µ2)|F⟩. Suppose |F⟩ is one of the
+four low energy states; then, for |F ′⟩ to be in the low
+energy state, the index µ′ must have the same orbital
+index with µ2, i.e., α′ = α2. With this restriction, the
+four-fermion operator in Eq. (B9) can be written as
+�
+s′
+f †
+α1s1fα2s′f †
+α2s′fα2s2 .
+(B36)
+If |F⟩ is one of the four low energy states, it at most
+occupies one electron in the α2 orbital. The α2 orbital
+of fα2s2|F⟩ must be empty, implying �
+s′ fα2s′f †
+α2s′ = 2.
+Thus the four-fermion operator equals 2f †
+α1s1fα2s2. The
+resulting contributions to the RG equation are
+dg
+dt
+����
+h
+= 4∆(D(t))
+πU
+(2 − 2x) g2 ,
+(B37)
+d(xg)
+dt
+����
+h
+= 4∆(D(t))
+πU
+�
+−x2�
+g2 .
+(B38)
+Eqs. (B34), (B35), (B37) and (B38) are identical to equa-
+tions of the U(2) case where N = 2, nf = 1. Following
+the steps of deriving Eq. (B30), we find x still flows to 1
+2,
+and
+kBT ′′
+K ≈ JH · e−
+πU1
+8g1∆(0) .
+(B39)
+The final expression for the Kondo energy scale at nf =
+2 is
+kBTK =
+�
+kBT ′
+K,
+kBTK > JH
+kBT ′′
+K,
+otherwise
+.
+(B40)
+
+20
+[1] Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watan-
+abe, Takashi Taniguchi, Efthimios Kaxiras, and Pablo
+Jarillo-Herrero, “Unconventional superconductivity in
+magic-angle graphene superlattices,” Nature 556, 43–
+50 (2018).
+[2] Yuan Cao, Valla Fatemi, Ahmet Demir, Shiang Fang,
+Spencer L. Tomarken, Jason Y. Luo, Javier D. Sanchez-
+Yamagishi,
+Kenji
+Watanabe,
+Takashi
+Taniguchi,
+Efthimios Kaxiras, Ray C. Ashoori, and Pablo Jarillo-
+Herrero, “Correlated insulator behaviour at half-filling
+in magic-angle graphene superlattices,” Nature 556, 80–
+84 (2018).
+[3] Rafi Bistritzer and Allan H. MacDonald, “Moir´e bands
+in twisted double-layer graphene,” Proceedings of the
+National Academy of Sciences 108, 12233–12237 (2011).
+[4] Xiaobo Lu, Petr Stepanov, Wei Yang, Ming Xie, Mo-
+hammed Ali Aamir, Ipsita Das, Carles Urgell, Kenji
+Watanabe, Takashi Taniguchi, Guangyu Zhang, Adrian
+Bachtold, Allan H. MacDonald,
+and Dmitri K. Efe-
+tov, “Superconductors, orbital magnets and correlated
+states in magic-angle bilayer graphene,” Nature 574,
+653–657 (2019), see Fig. 1(c) and Extended Data Fig.
+5 for gapped insulator at ν = 1, inferred by the low
+temperature resistivity behaviour.
+[5] Matthew Yankowitz, Shaowen Chen, Hryhoriy Polshyn,
+Yuxuan Zhang, K. Watanabe, T. Taniguchi, David
+Graf, Andrea F. Young,
+and Cory R. Dean, “Tuning
+superconductivity in twisted bilayer graphene,” Science
+363, 1059–1064 (2019).
+[6] Yu Saito,
+Jingyuan Ge,
+Kenji Watanabe,
+Takashi
+Taniguchi, and Andrea F. Young, “Independent super-
+conductors and correlated insulators in twisted bilayer
+graphene,” Nature Physics 16, 926–930 (2020), see Fig.
+1 (a) and Extended Data Fig. 4 (a) for the low tem-
+perature resistivity exhibiting insulator behavior near
+ν = 1.
+[7] Petr Stepanov, Ipsita Das, Xiaobo Lu, Ali Fahimniya,
+Kenji Watanabe,
+Takashi Taniguchi,
+Frank H. L.
+Koppens, Johannes Lischner, Leonid Levitov,
+and
+Dmitri K. Efetov, “Untying the insulating and super-
+conducting orders in magic-angle graphene,” Nature
+583, 375–378 (2020), see Fig. 2(a) D3 for insulator at
+ν = 1.
+[8] Xiaoxue Liu, Zhi Wang, K. Watanabe, T. Taniguchi,
+Oskar Vafek,
+and J. I. A. Li, “Tuning electron cor-
+relation in magic-angle twisted bilayer graphene using
+Coulomb screening,” Science 371, 1261–1265 (2021).
+[9] Harpreet Singh Arora, Robert Polski, Yiran Zhang,
+Alex Thomson, Youngjoon Choi, Hyunjin Kim, Zhong
+Lin,
+Ilham Zaky Wilson,
+Xiaodong Xu,
+Jiun-Haw
+Chu, Kenji Watanabe, Takashi Taniguchi, Jason Alicea,
+and Stevan Nadj-Perge, “Superconductivity in metallic
+twisted bilayer graphene stabilized by WSe2,” Nature
+583, 379–384 (2020), number: 7816 Publisher: Nature
+Publishing Group.
+[10] Yuan Cao, Daniel Rodan-Legrain, Jeong Min Park,
+Noah F. Q. Yuan, Kenji Watanabe, Takashi Taniguchi,
+Rafael M. Fernandes, Liang Fu,
+and Pablo Jarillo-
+Herrero, “Nematicity and competing orders in super-
+conducting magic-angle graphene,” Science 372, 264–
+271 (2021), publisher: American Association for the Ad-
+vancement of Science.
+[11] Myungchul Oh,
+Kevin P. Nuckolls,
+Dillon Wong,
+Ryan L. Lee, Xiaomeng Liu, Kenji Watanabe, Takashi
+Taniguchi,
+and Ali Yazdani, “Evidence for unconven-
+tional superconductivity in twisted bilayer graphene,”
+Nature 600, 240–245 (2021), see Fig. 1(c) for the the
+zero-energy peak, where the conduction (valence) bands
+are found to be pinned to the Fermi energy at negative
+(positive) fillings.
+[12] Yonglong Xie, Biao Lian, Berthold J¨ack, Xiaomeng Liu,
+Cheng-Li Chiu, Kenji Watanabe, Takashi Taniguchi,
+B. Andrei Bernevig,
+and Ali Yazdani, “Spectroscopic
+signatures of many-body correlations in magic-angle
+twisted bilayer graphene,” Nature 572, 101–105 (2019).
+[13] Aaron L. Sharpe, Eli J. Fox, Arthur W. Barnard, Joe
+Finney, Kenji Watanabe, Takashi Taniguchi, M. A.
+Kastner, and David Goldhaber-Gordon, “Emergent fer-
+romagnetism near three-quarters filling in twisted bi-
+layer graphene,” Science 365, 605–608 (2019).
+[14] Youngjoon Choi, Jeannette Kemmer, Yang Peng, Alex
+Thomson, Harpreet Arora, Robert Polski, Yiran Zhang,
+Hechen Ren, Jason Alicea, Gil Refael, Felix von Op-
+pen, Kenji Watanabe, Takashi Taniguchi,
+and Stevan
+Nadj-Perge, “Electronic correlations in twisted bilayer
+graphene near the magic angle,” Nature Physics 15,
+1174–1180 (2019).
+[15] Alexander Kerelsky, Leo J. McGilly, Dante M. Kennes,
+Lede
+Xian,
+Matthew
+Yankowitz,
+Shaowen
+Chen,
+K. Watanabe, T. Taniguchi, James Hone, Cory Dean,
+Angel Rubio,
+and Abhay N. Pasupathy, “Maximized
+electron interactions at the magic angle in twisted bi-
+layer graphene,” Nature 572, 95–100 (2019).
+[16] Yuhang Jiang, Xinyuan Lai, Kenji Watanabe, Takashi
+Taniguchi, Kristjan Haule, Jinhai Mao,
+and Eva Y.
+Andrei, “Charge order and broken rotational symmetry
+in magic-angle twisted bilayer graphene,” Nature 573,
+91–95 (2019).
+[17] Dillon Wong, Kevin P. Nuckolls, Myungchul Oh, Biao
+Lian, Yonglong Xie, Sangjun Jeon, Kenji Watanabe,
+Takashi Taniguchi, B. Andrei Bernevig,
+and Ali Yaz-
+dani, “Cascade of electronic transitions in magic-angle
+twisted bilayer graphene,” Nature 582, 198–202 (2020).
+[18] U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao,
+R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. von
+Oppen, Ady Stern, E. Berg, P. Jarillo-Herrero,
+and
+S. Ilani, “Cascade of phase transitions and Dirac re-
+vivals in magic-angle graphene,” Nature 582, 203–208
+(2020).
+[19] Youngjoon Choi, Hyunjin Kim, Cyprian Lewandowski,
+Yang Peng, Alex Thomson, Robert Polski, Yiran Zhang,
+Kenji Watanabe, Takashi Taniguchi, Jason Alicea, and
+Stevan Nadj-Perge, “Interaction-driven band flattening
+and correlated phases in twisted bilayer graphene,” Na-
+ture Physics 17, 1375–1381 (2021), see Fig. 4(a,f-i) for
+the evolutions of the zero-energy peak (around ν = −1)
+and Hubbard bands as the temperature is increased
+from 400mK to 20K.
+[20] M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang,
+J. Zhu, K. Watanabe, T. Taniguchi, L. Balents,
+and
+A. F. Young, “Intrinsic quantized anomalous Hall ef-
+fect in a moir´e heterostructure,” Science 367, 900–903
+
+21
+(2020).
+[21] Kevin P. Nuckolls, Myungchul Oh, Dillon Wong, Biao
+Lian, Kenji Watanabe, Takashi Taniguchi, B. Andrei
+Bernevig, and Ali Yazdani, “Strongly correlated Chern
+insulators in magic-angle twisted bilayer graphene,” Na-
+ture 588, 610–615 (2020), see Fig 1.c for the zero- energy
+peak from ν = ±1 to ±2 at T = 200mK and B⊥ = 1T.
+[22] Youngjoon Choi,
+Hyunjin Kim,
+Yang Peng,
+Alex
+Thomson, Cyprian Lewandowski, Robert Polski, Yi-
+ran Zhang, Harpreet Singh Arora, Kenji Watanabe,
+Takashi Taniguchi, Jason Alicea,
+and Stevan Nadj-
+Perge, “Correlation-driven topological phases in magic-
+angle twisted bilayer graphene,” Nature 589, 536–541
+(2021), see Fig. 1(a) for zero-energy peak around ν = −1
+and Hubbard band at T = 2K.
+[23] Yu Saito, Jingyuan Ge, Louk Rademaker, Kenji Watan-
+abe, Takashi Taniguchi, Dmitry A. Abanin,
+and An-
+drea F. Young, “Hofstadter subband ferromagnetism
+and symmetry-broken Chern insulators in twisted bi-
+layer graphene,” Nature Physics 17, 478–481 (2021).
+[24] Ipsita Das, Xiaobo Lu, Jonah Herzog-Arbeitman, Zhi-
+Da Song, Kenji Watanabe, Takashi Taniguchi, B. An-
+drei Bernevig,
+and Dmitri K. Efetov, “Symmetry-
+broken Chern insulators and Rashba-like Landau-level
+crossings in magic-angle bilayer graphene,” Nature
+Physics 17, 710–714 (2021).
+[25] Shuang
+Wu,
+Zhenyuan
+Zhang,
+K.
+Watanabe,
+T. Taniguchi,
+and Eva Y. Andrei, “Chern insula-
+tors, van Hove singularities and topological flat bands
+in
+magic-angle
+twisted
+bilayer
+graphene,”
+Nature
+Materials 20, 488–494 (2021), fig. 1(b), near ν = ±1 a
+resistivity peak emerges at around T = 11K.
+[26] Jeong Min Park, Yuan Cao, Kenji Watanabe, Takashi
+Taniguchi, and Pablo Jarillo-Herrero, “Flavour Hund’s
+coupling, Chern gaps and charge diffusivity in moir´e
+graphene,” Nature 592, 43–48 (2021), fig. 4(a), near
+ν = ±1, resistivity peaks occur and above around 10K.
+[27] Hryhoriy Polshyn, Matthew Yankowitz, Shaowen Chen,
+Yuxuan Zhang, K. Watanabe, T. Taniguchi, Cory R.
+Dean,
+and
+Andrea
+F.
+Young,
+“Large
+linear-in-
+temperature resistivity in twisted bilayer graphene,”
+Nature Physics , 1–6 (2019), in fig. 1 (a)(b), a peak
+rises as T increases at ν = −1.
+[28] Yuan
+Cao,
+Debanjan
+Chowdhury,
+Daniel
+Rodan-
+Legrain,
+Oriol
+Rubies-Bigorda,
+Kenji
+Watanabe,
+Takashi Taniguchi, T. Senthil,
+and Pablo Jarillo-
+Herrero, “Strange Metal in Magic-Angle Graphene with
+near Planckian Dissipation,” Physical Review Letters
+124, 076801 (2020), publisher: American Physical So-
+ciety.
+[29] Alexandre Jaoui, Ipsita Das, Giorgio Di Battista, Jaime
+D´ıez-M´erida, Xiaobo Lu, Kenji Watanabe, Takashi
+Taniguchi, Hiroaki Ishizuka, Leonid Levitov,
+and
+Dmitri K. Efetov, “Quantum critical behaviour in
+magic-angle twisted bilayer graphene,” Nature Physics
+18, 633–638 (2022), in Fig. 1 (a), near ν = ±1 peaks
+occurs when temperature rises. Notably, the peak near
+ν = 1 is rather sharp in contrast to other experiments
+mentioned, where usually the peak near ν = −1 is rec-
+ognized more easily.
+[30] Asaf Rozen, Jeong Min Park, Uri Zondiner, Yuan
+Cao, Daniel Rodan-Legrain, Takashi Taniguchi, Kenji
+Watanabe, Yuval Oreg, Ady Stern, Erez Berg, Pablo
+Jarillo-Herrero,
+and Shahal Ilani, “Entropic evidence
+for a Pomeranchuk effect in magic-angle graphene,” Na-
+ture 592, 214–219 (2021), see Fig. 2(e) for entropies at
+different fillings and temperatures.
+[31] Yu Saito, Fangyuan Yang, Jingyuan Ge, Xiaoxue Liu,
+Takashi Taniguchi, Kenji Watanabe, J. I. A. Li, Erez
+Berg, and Andrea F. Young, “Isospin Pomeranchuk ef-
+fect in twisted bilayer graphene,” Nature 592, 220–224
+(2021), see Fig. 1(b) and Extended Data Fig. 2 (a-f) for
+resistivity peaks occur at around 10K near ν=-1.
+[32] Zhida Song, Zhijun Wang, Wujun Shi, Gang Li, Chen
+Fang,
+and B. Andrei Bernevig, “All Magic Angles in
+Twisted Bilayer Graphene are Topological,” Physical
+Review Letters 123, 036401 (2019).
+[33] Hoi Chun Po, Liujun Zou, T. Senthil,
+and Ashvin
+Vishwanath, “Faithful tight-binding models and frag-
+ile topology of magic-angle bilayer graphene,” Physical
+Review B 99, 195455 (2019).
+[34] Junyeong Ahn, Sungjoon Park, and Bohm-Jung Yang,
+“Failure of Nielsen-Ninomiya Theorem and Fragile
+Topology in Two-Dimensional Systems with Space-
+Time Inversion Symmetry: Application to Twisted Bi-
+layer Graphene at Magic Angle,” Physical Review X 9,
+021013 (2019).
+[35] Grigory Tarnopolsky, Alex Jura Kruchkov, and Ashvin
+Vishwanath, “Origin of Magic Angles in Twisted Bi-
+layer Graphene,” Physical Review Letters 122, 106405
+(2019).
+[36] Jianpeng Liu, Junwei Liu,
+and Xi Dai, “The pseudo-
+Landau-level representation of twisted bilayer graphene:
+band topology and the implications on the correlated
+insulating phase,” Physical Review B 99, 155415 (2019),
+arXiv: 1810.03103.
+[37] Zhi-Da Song, Biao Lian, Nicolas Regnault, and B. An-
+drei Bernevig, “Twisted bilayer graphene. II. Stable
+symmetry anomaly,” Physical Review B 103, 205412
+(2021), publisher: American Physical Society.
+[38] Jian Kang and Oskar Vafek, “Strong Coupling Phases
+of Partially Filled Twisted Bilayer Graphene Narrow
+Bands,” Phys. Rev. Lett. 122, 246401 (2019), publisher:
+American Physical Society.
+[39] Nick Bultinck, Eslam Khalaf, Shang Liu, Shubhayu
+Chatterjee, Ashvin Vishwanath, and Michael P. Zaletel,
+“Ground State and Hidden Symmetry of Magic-Angle
+Graphene at Even Integer Filling,” Physical Review X
+10, 031034 (2020), publisher: American Physical Soci-
+ety.
+[40] Kangjun Seo, Valeri N. Kotov, and Bruno Uchoa, “Fer-
+romagnetic Mott state in Twisted Graphene Bilayers at
+the Magic Angle,” Phys. Rev. Lett. 122, 246402 (2019),
+publisher: American Physical Society.
+[41] B. Andrei Bernevig, Zhi-Da Song, Nicolas Regnault,
+and Biao Lian, “Twisted bilayer graphene. III. Interact-
+ing Hamiltonian and exact symmetries,” Physical Re-
+view B 103, 205413 (2021), publisher: American Phys-
+ical Society.
+[42] Oskar Vafek and Jian Kang, “Renormalization Group
+Study
+of
+Hidden
+Symmetry
+in
+Twisted
+Bilayer
+Graphene with Coulomb Interactions,” Physical Review
+Letters 125, 257602 (2020), publisher: American Phys-
+ical Society.
+[43] Biao Lian, Zhi-Da Song, Nicolas Regnault, Dmitri K.
+Efetov, Ali Yazdani, and B. Andrei Bernevig, “Twisted
+bilayer graphene. IV. Exact insulator ground states and
+phase diagram,” Physical Review B 103, 205414 (2021),
+
+22
+publisher: American Physical Society.
+[44] Fang Xie, Aditya Cowsik, Zhi-Da Song, Biao Lian,
+B. Andrei Bernevig,
+and Nicolas Regnault, “Twisted
+bilayer graphene. VI. An exact diagonalization study at
+nonzero integer filling,” Physical Review B 103, 205416
+(2021), publisher: American Physical Society.
+[45] Ming Xie and A. H. MacDonald, “Nature of the Cor-
+related Insulator States in Twisted Bilayer Graphene,”
+Phys. Rev. Lett. 124, 097601 (2020), publisher: Amer-
+ican Physical Society.
+[46] Jianpeng Liu and Xi Dai, “Theories for the correlated
+insulating states and quantum anomalous Hall effect
+phenomena in twisted bilayer graphene,” Phys. Rev. B
+103, 035427 (2021), publisher: American Physical So-
+ciety.
+[47] Tommaso Cea and Francisco Guinea, “Band structure
+and insulating states driven by Coulomb interaction
+in twisted bilayer graphene,” Physical Review B 102,
+045107 (2020), publisher: American Physical Society.
+[48] J¨orn W. F. Venderbos and Rafael M. Fernandes, “Corre-
+lations and electronic order in a two-orbital honeycomb
+lattice model for twisted bilayer graphene,” Physical Re-
+view B 98, 245103 (2018), publisher: American Physical
+Society.
+[49] Masayuki Ochi, Mikito Koshino, and Kazuhiko Kuroki,
+“Possible correlated insulating states in magic-angle
+twisted bilayer graphene under strongly competing in-
+teractions,” Phys. Rev. B 98, 081102 (2018), publisher:
+American Physical Society.
+[50] Yi Zhang, Kun Jiang, Ziqiang Wang,
+and Fuchun
+Zhang, “Correlated insulating phases of twisted bilayer
+graphene at commensurate filling fractions: A Hartree-
+Fock study,” Physical Review B 102, 035136 (2020),
+publisher: American Physical Society.
+[51] Shang Liu, Eslam Khalaf, Jong Yeon Lee, and Ashvin
+Vishwanath, “Nematic topological semimetal and insu-
+lator in magic-angle bilayer graphene at charge neutral-
+ity,” Physical Review Research 3, 013033 (2021), pub-
+lisher: American Physical Society.
+[52] Yuan Da Liao, Jian Kang, Clara N. Breiø, Xiao Yan
+Xu, Han-Qing Wu, Brian M. Andersen, Rafael M. Fer-
+nandes,
+and Zi Yang Meng, “Correlation-Induced In-
+sulating Topological Phases at Charge Neutrality in
+Twisted Bilayer Graphene,” Physical Review X 11,
+011014 (2021), publisher: American Physical Society.
+[53] Jian Kang and Oskar Vafek, “Non-Abelian Dirac node
+braiding and near-degeneracy of correlated phases
+at odd integer filling in magic-angle twisted bilayer
+graphene,” Physical Review B 102, 035161 (2020), pub-
+lisher: American Physical Society.
+[54] Tomohiro Soejima, Daniel E. Parker, Nick Bultinck, Jo-
+hannes Hauschild,
+and Michael P. Zaletel, “Efficient
+simulation of moir´e materials using the density matrix
+renormalization group,” Physical Review B 102, 205111
+(2020), publisher: American Physical Society.
+[55] Kasra Hejazi, Xiao Chen,
+and Leon Balents, “Hybrid
+Wannier Chern bands in magic angle twisted bilayer
+graphene and the quantized anomalous Hall effect,”
+Physical Review Research 3, 013242 (2021), publisher:
+American Physical Society.
+[56] Yuan Da Liao, Zi Yang Meng, and Xiao Yan Xu, “Va-
+lence bond orders at charge neutrality in a possible
+two-orbital extended hubbard model for twisted bilayer
+graphene,” Phys. Rev. Lett. 123, 157601 (2019).
+[57] Dante M. Kennes, Johannes Lischner,
+and Christoph
+Karrasch, “Strong correlations and d + id superconduc-
+tivity in twisted bilayer graphene,” Physical Review B
+98, 241407 (2018), publisher: American Physical Soci-
+ety.
+[58] Laura Classen, Carsten Honerkamp,
+and Michael M.
+Scherer, “Competing phases of interacting electrons on
+triangular lattices in moir´e heterostructures,” Physical
+Review B 99, 195120 (2019), publisher: American Phys-
+ical Society.
+[59] B. Andrei Bernevig, Biao Lian, Aditya Cowsik, Fang
+Xie, Nicolas Regnault, and Zhi-Da Song, “Twisted bi-
+layer graphene. V. Exact analytic many-body excita-
+tions in Coulomb Hamiltonians: Charge gap, Goldstone
+modes, and absence of Cooper pairing,” Physical Re-
+view B 103, 205415 (2021), publisher: American Phys-
+ical Society.
+[60] Eslam Khalaf, Nick Bultinck, Ashvin Vishwanath, and
+Michael P Zaletel, “Soft modes in magic angle twisted
+bilayer graphene,” arXiv preprint arXiv:2009.14827
+(2020).
+[61] Zhi-Da Song and B. Andrei Bernevig, “Magic-Angle
+Twisted Bilayer Graphene as a Topological Heavy
+Fermion
+Problem,”
+Physical
+Review
+Letters
+129,
+047601 (2022).
+[62] Hao Shi and Xi Dai, “Heavy-fermion representation for
+twisted bilayer graphene systems,” Physical Review B
+106, 245129 (2022), publisher: American Physical So-
+ciety.
+[63] Ph Nozi`eres and A. Blandin, “Kondo effect in real met-
+als,” Journal de Physique 41, 193–211 (1980), publisher:
+Soci´et´e Fran¸caise de Physique.
+[64] A.M. Tsvelick and P.B. Wiegmann, “Exact results in
+the theory of magnetic alloys,” Advances in Physics 32,
+453–713 (1983), publisher:
+Taylor & Francis
+eprint:
+https://doi.org/10.1080/00018738300101581.
+[65] Piers Coleman, “New approach to the mixed-valence
+problem,” Physical Review B 29, 3035–3044 (1984),
+publisher: American Physical Society.
+[66] N. E. Bickers, “Review of techniques in the large-$N$
+expansion for dilute magnetic alloys,” Reviews of Mod-
+ern Physics 59, 845–939 (1987).
+[67] Ian Affleck and Andreas W. W. Ludwig, “Critical theory
+of overscreened Kondo fixed points,” Nuclear Physics B
+360, 641–696 (1991).
+[68] Ian Affleck and Andreas W. W. Ludwig, “The Kondo
+effect, conformal field theory and fusion rules,” Nuclear
+Physics B 352, 849–862 (1991).
+[69] V. J. Emery and S. Kivelson, “Mapping of the two-
+channel Kondo problem to a resonant-level model,”
+Physical Review B 46, 10812–10817 (1992), publisher:
+American Physical Society.
+[70] A. M. Tsvelik and M. Reizer, “Phenomenological the-
+ory of non-Fermi-liquid heavy-fermion alloys,” Physical
+Review B 48, 9887–9889 (1993), publisher: American
+Physical Society.
+[71] Akira Furusaki and Naoto Nagaosa, “Kondo effect in
+a Tomonaga-Luttinger liquid,” Physical Review Letters
+72, 892–895 (1994), publisher: American Physical Soci-
+ety.
+[72] D. L. Cox and A. Zawadowski, “Exotic Kondo effects
+in metals: Magnetic ions in a crystalline electric field
+and tunnelling centres,” Advances in Physics 47 (1998),
+10.1080/000187398243500.
+
+23
+[73] Olivier Parcollet, Antoine Georges, Gabriel Kotliar,
+and Anirvan Sengupta, “Overscreened multichannel
+$\mathrm{SU}(N)$ Kondo model: Large-$N$ solution
+and conformal field theory,” Physical Review B 58,
+3794–3813 (1998).
+[74] Piers Coleman and Andrew J Schofield, “Quantum crit-
+icality,” Nature 433, 226–229 (2005).
+[75] G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko,
+O. Parcollet, and C. A. Marianetti, “Electronic struc-
+ture calculations with dynamical mean-field theory,”
+Reviews of Modern Physics 78, 865–951 (2006), pub-
+lisher: American Physical Society.
+[76] Philipp Werner, Armin Comanac, Luca de’ Medici,
+Matthias Troyer,
+and Andrew J. Millis, “Continuous-
+Time Solver for Quantum Impurity Models,” Physical
+Review Letters 97, 076405 (2006), publisher: American
+Physical Society.
+[77] Philipp Gegenwart, Qimiao Si,
+and Frank Steglich,
+“Quantum criticality in heavy-fermion metals,” Nature
+Physics 4, 186–197 (2008).
+[78] Qimiao
+Si
+and
+Frank
+Steglich,
+“Heavy
+fermions
+and
+quantum
+phase
+transi-
+tions,”
+Science
+329,
+1161–1166
+(2010),
+https://www.science.org/doi/pdf/10.1126/science.1191195.
+[79] Maxim Dzero, Kai Sun, Victor Galitski, and Piers Cole-
+man, “Topological Kondo Insulators,” Physical Review
+Letters 104, 106408 (2010), publisher: American Phys-
+ical Society.
+[80] Feng Lu, JianZhou Zhao, Hongming Weng, Zhong Fang,
+and Xi Dai, “Correlated Topological Insulators with
+Mixed Valence,” Physical Review Letters 110, 096401
+(2013), publisher: American Physical Society.
+[81] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wil-
+son, “Renormalization-group approach to the Anderson
+model of dilute magnetic alloys. I. Static properties for
+the symmetric case,” Physical Review B 21, 1003–1043
+(1980), publisher: American Physical Society.
+[82] H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wil-
+son, “Renormalization-group approach to the Anderson
+model of dilute magnetic alloys. II. Static properties for
+the asymmetric case,” Physical Review B 21, 1044–1083
+(1980), publisher: American Physical Society.
+[83] Ralf Bulla, Theo A. Costi, and Thomas Pruschke, “Nu-
+merical renormalization group method for quantum im-
+purity systems,” Reviews of Modern Physics 80, 395–
+450 (2008), publisher: American Physical Society.
+[84] Jian Kang and Oskar Vafek, “Symmetry, Maximally Lo-
+calized Wannier States, and a Low-Energy Model for
+Twisted Bilayer Graphene Narrow Bands,” Phys. Rev.
+X 8, 031088 (2018).
+[85] Mikito Koshino, Noah F. Q. Yuan, Takashi Koretsune,
+Masayuki Ochi, Kazuhiko Kuroki, and Liang Fu, “Max-
+imally Localized Wannier Orbitals and the Extended
+Hubbard Model for Twisted Bilayer Graphene,” Physi-
+cal Review X 8, 031087 (2018).
+[86] Noah F. Q. Yuan and Liang Fu, “Model for the metal-
+insulator transition in graphene superlattices and be-
+yond,” Physical Review B 98, 045103 (2018).
+[87] Liujun Zou, Hoi Chun Po, Ashvin Vishwanath,
+and
+T. Senthil, “Band structure of twisted bilayer graphene:
+Emergent symmetries,
+commensurate approximants,
+and Wannier obstructions,” Physical Review B 98,
+085435 (2018), publisher: American Physical Society.
+[88] Jiawei Zang, Jie Wang, Antoine Georges, Jennifer Cano,
+and Andrew J. Millis, “Real space representation of
+topological system: twisted bilayer graphene as an ex-
+ample,” (2022), arXiv:2210.11573 [cond-mat].
+[89] Kazuyuki Uchida, Shinnosuke Furuya, Jun-Ichi Iwata,
+and Atsushi Oshiyama, “Atomic corrugation and elec-
+tron localization due to moir´e patterns in twisted bilayer
+graphenes,” Phys. Rev. B 90, 155451 (2014).
+[90] MM Van Wijk, A Schuring, MI Katsnelson, and A Fa-
+solino, “Relaxation of moir´e patterns for slightly mis-
+aligned identical lattices: graphene on graphite,” 2D
+Materials 2, 034010 (2015).
+[91] Shuyang Dai, Yang Xiang,
+and David J Srolovitz,
+“Twisted bilayer graphene: Moir´e with a twist,” Nano
+letters 16, 5923–5927 (2016).
+[92] Sandeep K Jain, Vladimir Juriˇci´c,
+and Gerard T
+Barkema, “Structure of twisted and buckled bilayer
+graphene,” 2D Materials 4, 015018 (2016).
+[93] Samuel V. Gallego, Emre S. Tasci, Gemma de la Flor,
+J. Manuel Perez-Mato, and Mois I. Aroyo, “Magnetic
+symmetry in the Bilbao Crystallographic Server: a com-
+puter program to provide systematic absences of mag-
+netic neutron diffraction,” Journal of Applied Crystal-
+lography 45, 1236–1247 (2012).
+[94] Jie Wang, Yunqin Zheng, Andrew J. Millis,
+and Jen-
+nifer Cano, “Chiral approximation to twisted bilayer
+graphene: Exact intravalley inversion symmetry, nodal
+structure, and implications for higher magic angles,”
+Phys. Rev. Res. 3, 023155 (2021).
+[95] Kan Chen and C. Jayaprakash, “The Kondo effect in
+pseudo-gap Fermi systems:
+a renormalization group
+study,” Journal of Physics: Condensed Matter 7, L491
+(1995).
+[96] Carlos
+Gonzalez-Buxton
+and
+Kevin
+Ingersent,
+“Renormalization-group
+study
+of
+Anderson
+and
+Kondo impurities in gapless Fermi systems,” Physical
+Review B 57, 14254–14293 (1998).
+[97] Kevin Ingersent and Qimiao Si, “Critical Local-Moment
+Fluctuations, Anomalous Exponents, and ω/T Scaling
+in the Kondo Problem with a Pseudogap,” Physical Re-
+view Letters 89, 076403 (2002).
+[98] Lars Fritz and Matthias Vojta, “Phase transitions in
+the pseudogap Anderson and Kondo models: Critical
+dimensions, renormalization group, and local-moment
+criticality,” Physical Review B 70, 214427 (2004).
+[99] Piers
+Coleman,
+Introduction
+to
+many-body
+physics
+(Cambridge University Press, 2015) see section 17.3 for
+the Kondo temperature in the large-N limit.
+[100] R. Bulla, T. A. Costi,
+and D. Vollhardt, “Finite-
+temperature numerical renormalization group study of
+the Mott transition,” Physical Review B 64, 045103
+(2001), publisher: American Physical Society.
+[101] Ralf Bulla, Hyun-Jung Lee, Ning-Hua Tong,
+and
+Matthias Vojta, “Numerical renormalization group for
+quantum impurities in a bosonic bath,” Physical Review
+B 71, 045122 (2005), publisher: American Physical So-
+ciety.
+[102] Tie-Feng Fang, Ning-Hua Tong, Zhan Cao, Qing-Feng
+Sun, and Hong-Gang Luo, “Spin susceptibility of An-
+derson impurities in arbitrary conduction bands,” Phys.
+Rev. B 92, 155129 (2015), publisher: American Physical
+Society.
+[103] Ulrich Gerland, Jan von Delft, T. A. Costi,
+and Yu-
+val Oreg, “Transmission phase shift of a quantum dot
+with kondo correlations,” Phys. Rev. Lett. 84, 3710–
+
+24
+3713 (2000).
+[104] Yang-Zhi Chou and Sankar Das Sarma, “Kondo lattice
+model in magic-angle twisted bilayer graphene,” arXiv
+preprint arXiv:2211.15682 (2022).
+[105] Haoyu Hu, G. Rai, L. Crippa, T. Wehling, G. Sangio-
+vanni, R. Valen´t, Alexei M. Tsvelik,
+and B. Andrei
+Bernevig, (2023), to appear.
+[106] Haoyu Hu, B. Andrei Bernevig, and Alexei M. Tsvelik,
+(2023), to appear.
+[107] Piers Coleman, (2023), to appear.
+[108] Jiabin Yu,
+Ming Xie,
+B. Andrei Bernevig,
+and
+Sankar Das Sarma, “Magic angle twisted symmetric tri-
+layer graphene as a topological heavy fermion problem,”
+(2023), to appear.
+