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+Symmetric Kondo Lattice States in Doped Strained Twisted Bilayer Graphene
+H. Hu,1 G. Rai,2 L. Crippa,3 J. Herzog-Arbeitman,4 D. C˘alug˘aru,4 T. Wehling,2, 5
+G. Sangiovanni,3 R. Valent´ı,6 A. M. Tsvelik,7 and B. A. Bernevig4, 1, 8, ∗
+1Donostia International Physics Center, P. Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain
+2I. Institute of Theoretical Physics, University of Hamburg, Notkestrasse 9, 22607 Hamburg, Germany
+3Institut f¨ur Theoretische Physik und Astrophysik and W¨urzburg-Dresden Cluster
+of Excellence ct.qmat, Universit¨at W¨urzburg, 97074 W¨urzburg, Germany
+4Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
+5The Hamburg Centre for Ultrafast Imaging, 22761 Hamburg, Germany
+6Institut f¨ur Theoretische Physik, Goethe Universit¨at Frankfurt,
+Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany
+7Division of Condensed Matter Physics and Materials Science,
+Brookhaven National Laboratory, Upton, NY 11973-5000, USA
+8IKERBASQUE, Basque Foundation for Science, Bilbao, Spain
+We use the topological heavy fermion (THF) model [1] and its Kondo Lattice (KL) formulation [2] to study
+the possibility of a symmetric Kondo state in twisted bilayer graphene. Via a large-N approximation, we find
+a symmetric Kondo state in the KL model at fillings ν = 0, ±1, ±2 where a KL model can be constructed [2].
+In the symmetric Kondo state, all symmetries are preserved and the local moments are Kondo screened by the
+conduction electrons. At the mean-field level of the THF model at ν = 0, ±1, ±2, ±3 we also find a similar
+symmetric state that is adiabatically connected to the symmetric Kondo state [3]. We study the stability of the
+symmetric state by comparing its energy with the ordered (symmetry-breaking) states found in Ref. [1] and
+find the ordered states to have lower energy at ν = 0, ±1, ±2. However, moving away from integer fillings
+by doping holes to the light bands, our mean-field calculations find the energy difference between the ordered
+state and the symmetric state to be reduced, which suggests the loss of ordering and a tendency towards Kondo
+screening. We expect that including the Gutzwiller projection in our mean-field state will further reduce the
+energy of the symmetric state. In order to include many-body effects beyond the mean-field approximation, we
+also performed dynamical mean-field theory (DMFT) calculations on the THF model in the non-ordered phase.
+The spin susceptibility follows a Curie behavior at ν = 0, ±1, ±2 down to ∼ 2K where the onset of screening
+of the local moment becomes visible. This hints to very low Kondo temperatures at these fillings, in agreement
+with the outcome of our mean-field calculations. At non-integer filling ν = ±0.5, ±0.8, ±1.2 DMFT shows
+deviations from a 1/T-susceptibility at much higher temperatures, suggesting a more effective screening of local
+moments with doping. Finally, we study the effect of a C3z-rotational-symmetry-breaking strain via mean-field
+approaches and find that a symmetric phase (that only breaks C3z symmetry) can be stabilized at sufficiently
+large strain at ν = 0, ±1, ±2. Our results suggest that a symmetric Kondo phase is strongly suppressed at
+integer fillings, but could be stabilized either at non-integer fillings or by applying strain.
+Introduction— The experiments on magic-angle (θ
+=
+1.05◦) twisted bilayer graphene (MATBLG) [4–6] have es-
+tablished the existence of a variety of interesting phases [7–
+28], including correlated insulating phases [29–39] and super-
+conductivity [40–44]. Their discovery has been followed by
+considerable theoretical efforts [45–69] aimed at understand-
+ing their origin. An extended Hubbard model has been con-
+structed to analyze the interacting physics [60, 70–82], how-
+ever, due to the non-trivial topology of the flat bands [83–
+91], certain symmetries become non-local. Alternatively, an
+approach based on a momentum space model has been con-
+sidered [92–100], in which correlated insulators [101–108],
+superconductivity [109–114], and other correlated quantum
+phases [115–119] have been identified and studied. Besides,
+various numerical calculations [120–127] have also been per-
+formed to investigate the correlated nature of the phenom-
+ena. However, the active phase diagram including the states
+at non-integer fillings is not well understood. The exact map-
+ping between the MATBLG and topological heavy-fermion
+∗ bernevig@princeton.edu
+model constructed in Ref. [1] could be used for develop-
+ments in this direction. This mapping establishes a bridge
+between heavy-fermions [3, 128–131] and moir´e systems [1,
+2, 132]. The presence of localized moments in MATBG is
+supported by recent entropy measurements which have found
+a Pomeranchuk-type transition [19, 133]. A large entropy ob-
+served at high-temperatures, originates from weakly interact-
+ing local moments whose fluctuations are quenched at low
+temperatures [19, 133]. Since a similar behavior is observed
+in heavy fermion systems [3, 128], where the fluctuating lo-
+cal moments are screened by conduction electrons (Kondo
+effect), this observation is suggestive of a Kondo state with
+screened local moments in MATBLG [128, 134].
+In this paper we use the KL model [2], to describe and study
+the symmetric Kondo (SK) state. We focus on integer fillings
+ν = 0, ±1, ±2, where a KL model can be constructed [2] (a
+KL description fails at ν = ±3 as demonstrated in Ref. [2]).
+The SK phase preserves all symmetries; the local moments
+are screened. We discuss the topology and the band struc-
+ture of the SK state and extend the study to the THF model
+where we identify the symmetric state that is adiabatically
+connected to the SK state [3]. In order to address integer
+arXiv:2301.04673v1  [cond-mat.str-el]  11 Jan 2023
+
+2
+and fractional fillings on equal footing, we perform both a
+mean-field and a dynamical mean-field theory (DMFT) calcu-
+lations of the THF defining a “periodic Anderson model” with
+a momentum-dependent hybridization between the correlated
+f- and the dispersive c-electrons in the non-ordered state.
+Our mean-field calculations indicate that the energy of the
+symmetric state is higher than that of the ordered (symmetry-
+breaking) states found in Ref. [1] at integer filling. We thus
+conclude that ordered states are more energetically favored
+at integer fillings. DMFT supports this picture as we obtain
+a Curie behavior of the local spin susceptibility at integer
+fillings, down to very low temperatures ∼ 2K, hinting to a
+very small Kondo scale (lower than ∼ 2K). Together with the
+mean-field results we would then expect an ordered state to be
+favored at low temperatures for these fillings.
+Turning to the effect of doping, instead, from our mean-
+field analysis, we find that the energy difference between the
+symmetric phase and the ordered phase can be sizeably re-
+duced. Doping hence suppresses the ordering and enhances
+the Kondo screening. This conclusion is further supported
+by the DMFT results at non-integer fillings. Here, we find
+clear deviations from the Curie behavior in the entire range
+from 10K down to ∼1K. Even though it is computationally
+too demanding to go further down in temperature, we point
+out that our evidence of a clear-cut difference in the screening
+properties between integer and fractional fillings is reliable.
+DMFT treats indeed local quantum fluctuations exactly [135]
+and hence takes into account the many-body processes that
+can potentially lead to the screening of local moments at any
+filling.
+Since realistic samples have intrinsic strains, we finally
+study the effect of a C3z-breaking strain on the symmetric
+phase. Our mean-field calculations show that the order is sup-
+pressed by the strain effect and a symmetric state can be sta-
+bilized at a sufficiently large strain at ν = 0, ±1, ±2.
+In summary, we conclude that a symmetric Kondo phase
+is absent at integer fillings of MATBLG, but could in princi-
+ple be stabilized either at non-integer fillings or by applying
+strain.
+Topological Heavy Fermion model and the Kondo lattice
+model— The THF model [1] contains two types of electrons:
+topological conduction c-electrons (ck,aηs) and localized f-
+electrons (fR,αηs). The operator ck,aηs annihilates conduc-
+tion c-electron with momentum k, orbital a ∈ {1, 2, 3, 4},
+valley η ∈ {+, −} and spin s ∈ {↑, ↓}. At the ΓM-point for
+each valley and each spin projection, c-electrons in the orbital
+1 and 2 transform according to the Γ3 irreducible represen-
+tation (of magnetic space group P6′2′2) [1]. The remaining
+c-electrons (a = 3, 4) at the same valley with the same spin
+projection transform in the Γ1 ⊕ Γ2 reducible representation
+(of magnetic space group P6′2′2) [1]. We will call them Γ3 c-
+electrons (a = 1, 2) and Γ1⊕Γ2 c-electrons (a = 3, 4) respec-
+tively. fR,αηs is the annihilation operator of the f-electron at
+the moir´e unit cell R with orbital α ∈ {1, 2}, valley η and
+spin s. The Hamiltonian of the THF model [1, 136] is
+ˆHT HF = ˆHc + ˆHfc + ˆHU + ˆHW + ˆHV + ˆHJ
+(1)
+where ˆHc describes the kinetic term of conduction elec-
+trons,
+ˆHfc describes the hybridization between f-c elec-
+trons [1, 136]. The interactions include an on-site Hubbard
+interaction of f-electrons ( ˆHU with U = 57.95meV), a re-
+pulsion between f- and c-electrons ( ˆHW with W = 48meV),
+a Coulomb interaction between c-electrons ( ˆHV with V (q =
+0)/Ω0 = 48.33meV and Ω0 the area of moir´e unit cell), and a
+ferromagnetic exchange coupling between f-and c-electrons
+( ˆHJ with J = 16.38meV) [1, 136].
+Based on the THF model [1], a KL model of MATBLG
+has been constructed via a generalized Schrieffer–Wolff (SW)
+transformation as shown in Ref. [2]. The KL model is de-
+scribed by the following Hamiltonian
+ˆHKondo = ˆHc + ˆHcc + ˆHK + ˆHJ .
+(2)
+where
+ˆHc, ˆHJ come from the original THF model and
+ˆHcc, ˆHK emerge from the SW transformation. ˆHcc is the one-
+body scattering term of Γ3 c-electrons with the form of
+ˆHcc =
+�
+|k|<Λc
+�
+a,a′∈{1,2}
+η,s
+e−|k|2λ2 : c†
+k,aηsck,a′ηs :
+�
+−1
+Dνc,νf
++
+−1
+Dνc,νf
+� �
+γ2/2
+γv′
+⋆(ηkx − iky)
+γv′
+⋆(ηkx + iky)
+γ2/2
+�
+a,a′ .
+(3)
+λ is the damping factor of the f-c hybridization in the THF
+model. γ, v′
+⋆ characterize the zeroth order and linear order
+f-c hybridization of the THF model with v′
+⋆ characterizing
+a k-dependent hybridization matrix [1, 136]. D1,νc,νf and
+D2,νc,νf are defined as
+D1,νc,νf = (U − W)νf − U
+2 + (−V (0)
+Ω0
++ W)νc
+D2,νc,νf = (U − W)νf + U
+2 + (−V (0)
+Ω0
++ W)νc ,
+(4)
+where νf, νc are the filling of f- and c-electrons deter-
+mined from the calculations of the THF model at the zero-
+hybridization limit [2]. We point out that in the single-orbital
+Kondo model, the one-body scattering term merely introduces
+a chemical potential shift [3, 137] of the c-electrons and is
+usually omitted. However, in our model, ˆHcc cannot be ig-
+nored for two reasons. First, ˆHcc is k-dependent and thus
+introduces additional kinetic energy to the conduction elec-
+trons. From Eq. S24, we observe the k-dependency mainly
+comes from the linear k term that is proportional to v′
+⋆ and
+can be traced back to the k-dependency of the hybridization
+matrix in the THF model. Secondly, even if we drop the v′
+⋆
+term in Eq. S24 (v′
+⋆ = 0 corresponding to the chiral limit [1]),
+ˆHcc still produces an energy shift for the Γ3 c-electrons. Thus
+ˆHcc leads to the energy splitting between Γ3 and Γ1 ⊕ Γ2 c-
+electrons and cannot be simply treated as a shift of the chem-
+ical potential.
+ˆHK is the Kondo interaction between f- and Γ3 c-electrons
+whose explicit form is given in Refs. [2, 136]. The Kondo
+interaction consists of two parts: the zeroth order Kondo in-
+teraction proportional to γ2/Dνc,νf and the first order Kondo
+
+3
+interaction proportional to γv′
+⋆/Dνc,νf , where D−1
+νc,νf
+=
+−D−1
+1,νc,νf + D−1
+2,νc,νf . The zeroth order Kondo interaction
+term describes the antiferromagnetic interaction between the
+U(8) moments of the f- and the Γ3 c-electrons and has a U(8)
+symmetry. The linear-order Kondo interaction only has a flat
+U(4) symmetry and is k-dependent [1, 136].
+ˆHJ is the fer-
+romagnetic exchange interaction between Γ1 ⊕ Γ2 c- and f-
+electrons that already exists in the TFH model [1, 136]. We
+also note that, for both the THF model and the KL model,
+ground states at filling ν and −ν are connected by a charge-
+conjugation transformation [1]. This can be broken by other
+one-body terms which we did not consider here. Therefore, in
+what follows, we only focus on ν ≤ 0.
+Mean-field Hamiltonian of the Kondo model— We next per-
+form a mean-field study of the KL model [3]. This MF sup-
+presses the RKKY interaction and essentially restores the hy-
+bridization term ˆHfc of the original periodic Anderson model,
+but in a renormalized form. It becomes exact in the N → ∞
+limit (we have N = 4 here which corresponds to the approxi-
+mate flat U(4) symmetry of the KL Hamiltonian in Eq. 2). At
+the mean-field level, the Kondo interaction ˆHK can be written
+as (see Supplementary Materials (SM))
+ˆHMF
+K
+=
+�
+R,|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NMDνc,νf
+�
+− f †
+R,αηsck,aηs
+�
+γ2V ∗
+1 + γv′
+⋆V ∗
+2
+V ∗
+1 (ηkx − iky)
+V ∗
+1 (ηkx + iky) γ2V ∗
+1 + γv′
+⋆V ∗
+2
+�
+α,a
++ h.c.
+�
++ NM
+�
+γ2|V ∗
+1 |2 + γv′
+⋆(V ∗
+1 V ∗
+2 + V ∗
+2 V ∗
+1 )
+�
++ H.T.
+(5)
+where we have introduced the following mean fields
+V ∗
+1 =
+�
+R,|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NMNM
+⟨Ψ|f †
+R,αηsck,αηs|Ψ⟩
+V ∗
+2 =
+�
+R,|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2
+√NMNM
+(ηkxσx + kyσy)αa
+⟨Ψ|f †
+r,αηsck,αηs|Ψ⟩
+(6)
+with |Ψ⟩ being the mean-field ground state, and H.T. denotes
+the Hartree term (⟨f †f⟩, ⟨c†c⟩) whose explicit formula is in
+the Supplementay Materials (SM) [136]. Several points are
+in order. First, as we have mentioned above, the mean field
+restores the hybridization of the original Anderson model, but
+in a renormalized form. V ∗
+1 , V ∗
+2 describe the renormalized
+hybridization between the f- and Γ3 c-electrons driven by the
+Kondo interactions between two types of electrons(f and Γ3
+c) [3, 138]. Second, it is necessary to keep the Hartree con-
+tributions. In the canonical Kondo model, the Hartree term
+merely produces a chemical potential shift (in the case without
+symmetry breaking) and hence can be omitted. Here, Hartree
+contributions (see SM [136]) are k-dependent because of the
+k-dependency of the Kondo interactions, and thus contribute
+to the dispersion of the conduction c-electrons. Furthermore,
+since only Γ3 c-electrons contribute to the Kondo interaction,
+the Hartree term also produces an energy splitting between the
+Γ3 and the Γ1 ⊕ Γ2 c-electrons.
+As for ˆHJ, we perform a similar mean-field decoupling
+ˆHMF
+J
+=J
+�
+R,|k|<Λc,αηs
+eik·R
+√NM
+�
+V3δ1,η(−1)α+1f †
+R,αηsck,α+2ηs
++ V4δ−1,η(−1)α+1ηf †
+R,αηsck,α+2ηs + h.c.
+�
+− JNM
+�
+|V3|2 + |V4|2
+�
++ H.T.
+(7)
+where we have introduced the following two mean-field aver-
+ages that describe the f-c hybridization:
+V3 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+√NMNM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩
+V4 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+√NMNM
+⟨Ψ|ηf †
+R,αηsck,α+2ηs|Ψ⟩ ,
+(8)
+To impose the filling of the f-electrons to be νf, we intro-
+duce the Lagrange multiplier [134, 136, 138]:
+ˆHλf =
+�
+R,αηs
+λf
+�
+: f †
+R,αηsfR,αηs : −νf
+�
+(9)
+with λf to be determined self-consistently [136]. Finally, we
+introduce a chemical potential µc to the c-electrons
+ˆHµc = −µc
+�
+|k|<Λc,aηs
+: c†
+k,aηsck,aηs : .
+(10)
+In the calculation, we tune µc and λc together to fix both the
+total filling ν = νf + νc and the filling of f-electrons [136].
+The final mean-field Hamiltonian of the KL model now is
+ˆHMF
+Kondo = ˆHc + ˆHcc + ˆHMF
+K
++ ˆHMF
+J
++ ˆHλf + ˆHµc .
+(11)
+We then self-consistently solve the mean-field equations
+(see SM [136]). At ν = νf = 0, −1, −2 (where a KL model
+can be constructed), we identify a SK state that preserves
+all the symmetries and is characterized by V ∗
+1
+̸= 0, V ∗
+2
+̸=
+0, V ∗
+3 = 0, V ∗
+4 = 0 [136]. We comment that the exchange
+interaction ˆHJ [1] between f- and Γ1 ⊕ Γ2 c-electrons is fer-
+romagnetic, and hence disfavors the singlet formation or hy-
+bridization (V3, V4) between f- and Γ1 ⊕ Γ2 c-electrons. We
+find that V3, V4 vanish (their numerical amplitudes are smaller
+than 10−5). In fact, ˆHJ favors the triplet formation or pair-
+ing formation (f †c†), where both lead to a symmetry-breaking
+state at the mean-field level and are beyond our current con-
+sideration of SK state.
+Properties of the symmetric Kondo phase— In Fig. 1, we
+plot the band structure of the SK phase and compare it with
+the non-interacting THF model. We find the hybridization in
+
+4
+(a)
+(b)
+(c)
+(d)
+FIG. 1.
+(a) Band structure of the non-interaction THF model at ν = 0. (b), (c), (d) Band structure of the SK phase at ν = 0, −1, −2
+respectively.
+the SK state defined in Eq. 5 to be enhanced compared to the
+non-interacting limit of THF model, which is clear from the
+increase of the gap of the Γ3 states at the Γ point [1] from
+its non-interacting value 24.75meV at ν = 0, to 168meV,
+190meV, 213meV at ν = 0, −1, −2 respectively. We also
+find that in the SK phase the bandwidths of the flat bands at
+ν = −1, −2 become 16meV, 53meV, which are (much) larger
+than the non-interacting flat-band bandwidth (= 7.4meV) of
+the THF model (Fig. 1). However, at ν = 0, the flat-band
+bandwidth is the same as the non-interacting flat-band band-
+width. This is because, at ν = 0, the one-body scattering
+term and the Hartree contributions from ˆHK, ˆHJ all van-
+ish [136], and the enhanced hybridization pushes the remote
+bands away from the Fermi energy and does not change much
+the band structures of the flat bands. In addition, unlike the
+non-interacting case, here we found the flat bands are mostly
+formed by Γ1⊕Γ2 c-electrons with orbital weights larger than
+70% at ν = 0, −1, −2. This is because the large f-c hy-
+bridization induced by V1, V2 (Eq. 5) pushes the energy of Γ3
+c- and f-electrons away from the Fermi energy and reduces
+their orbital weights [136].
+The flat bands in the SK phase form Γ1 ⊕ Γ2, M1 ⊕ M2
+and K2K3 representations at ΓM, MM, KM respectively, and
+have the same topology as the flat bands in the non-interacting
+THF model [1]. More explicitly, the flat bands for each val-
+ley and each spin projection belong to a fragile topology [1]
+at ν = −1, −2. At ν = 0, due to the additional particle-hole
+symmetry, flat bands have a stable topology [1, 85, 91, 136],
+which is characterized by the odd winding number of the Wil-
+son loop as shown in supplementary material [136]. We men-
+tion that the interplay between Kondo effect and the topologi-
+cal bands has also been studied in various other systems [139–
+144].
+Symmetric phase in the topological heavy-fermion model—
+We next investigate the similar symmetric phase in the THF
+model Eq. 1.
+We first focus on integer fillings ν
+=
+0, −1, −2, −3 and perform the mean-field calculations of
+THF as introduced in Ref. [1, 136]. By enforcing the mean-
+field Hamiltonian to preserve all the symmetries, we are able
+to identify a symmetric state that preserves all the symmetries
+at ν = 0, −1, −2, −3. To observe the stability of the sym-
+metric phase, we compare its energy (Esym) with the energy
+(Eorder) of the ordered (symmetry-breaking) ground states
+derived in Ref. [1]. The ordered ground states in Ref. [1]
+are a Kramers inter-valley-coherent (KIVC) state at ν = 0,
+0.5
+0.0
+0.5
+0
+10
+20
+30
+40
+50
+E/meV
+= -3
+0.5
+0.0
+0.5
+0
+10
+20
+30
+40
+50
+= -2
+0.5
+0.0
+0.5
+0
+10
+20
+30
+40
+50
+= -1
+0.5
+0.0
+0.5
+0
+10
+20
+30
+40
+50
+= 0
+FIG. 2. Doping dependence of the ground state energy difference
+∆E = Esym − Eorder near integer fillings νt = 0, −1, −2, −3.
+a KIVC+valley polarized (VP) state at ν = −1, a KIVC state
+at ν = −2 and a VP state at ν = −3. We point out that at
+ν = −3 other states with lower energy exist [145]. In our
+numerical calculations, we find ∆E = Esym − Eorder =
+47meV, 40meV, 33meV, 23meV at ν = 0, −1, −2, −3 re-
+spectively. In all integer filling cases, the symmetric states
+have higher energy, and the ground states cannot be the sym-
+metric state, which is consistent with the previous calcula-
+tions of Ref. [1, 2, 103]. Note that our mean-field calcula-
+tion does not include a Gutzwiller projection to fix the fill-
+ing of f-electrons at each site, and hence we expect the en-
+ergy of projected symmetric states will be lower. However, as
+we show later, after including the effect of local correlations
+via the DMFT approach, we confirm that the Kondo phase,
+which is adiabatically connected to the symmetric phase in
+the mean-field calculations, is strongly suppressed at integer
+fillings (down to temperatures ∼ 1-2K). This further supports
+the development of ordering at integer fillings.
+Effects of doping— We next investigate the effects of dop-
+ing, first at the level of mean-field theory. We stick to a nar-
+row region ν ∈ [νint−0.5, νint+0.5] near each integer filling
+νint = 0, −1, −2, −3 and compare the energies of the ordered
+states Eord and the symmetric states Esym in the THF model.
+To find the ordered state solutions, we first initialize the cal-
+culations with the mean-field solutions at integer filling νint
+and fill the mean-field bands up to current filling ν. We then
+self-consistently solve the mean-field equations and calculate
+the energy of the resulting states. We obtain the symmetric-
+state solution in a similar manner but take the symmetric so-
+lution at νint as initialization and enforce the symmetry of
+the mean-field Hamiltonian during the calculations. Fig. 2
+displays a plot of the difference of the ground state energies
+
+5
+∆E = Esym −Eorder as a function of doping ∆ν = ν −νint
+near νint = 0, −1, −2, −3. We observe that hole doping at
+ν = 0, −1, −2, −3 and electron doping at ν = 0 decreases
+the ∆E. Doping holes at ν = 0, −1, −2, −3 and doping elec-
+trons at ν = 0 to the ordered states is equivalent to doping the
+light (dispersive) bands which are mostly formed by conduc-
+tion c-electrons. After doping, the conduction electrons will
+stay close to the Fermi energy, and then enhance the tendency
+towards the Kondo effect.
+However, doping electrons at ν = −1, −2 to the ordered
+states is equivalent to doping heavy (flat) bands which mostly
+come from the f-electrons. Because of the flatness of the
+band, we find the nature of the ordered states will change with
+doping (see SM [136]). For example at ν = 2, the KIVC
+order is suppressed by the electron doping (see SM [136]).
+Thus, ∆E will be affected by both, changes of ordering and
+doping. However, a sizeable change of the order parameters
+is not observed for hole doping at ν = 0, −1, −2, −3 and also
+electron doping at ν = 0 (see SM [136]), because we are dop-
+ing conduction c-electrons in both cases. We also point out
+the complexity of ν = −3. First, other states that break trans-
+lational invariant [145] could have lower energy than the VP
+state we currently considered. Second, even for the VP state,
+doping electrons is equivalent to doping both heavy and light
+bands [1], since both light and heavy bands appear in the elec-
+tron doping case [1]. In practice, as we increase ∆ν, we find
+that, at ν = −1, −2, ∆E will first decrease and then increase
+and, at ν = −3, ∆E will always increase.
+In summary, we conclude that hole doping can suppress the
+long-range order and enhance the tendency towards the Kondo
+effect near ν = 0, −1, −2. Electron doping, depending on the
+fillings, could also enhance the tendency toward the Kondo ef-
+fect. However, on the electron doping side, the change of or-
+der moments indicates the importance of the correlation effect
+which could be underestimated in the mean-field approach. In
+the next section, we provide a more comprehensive study of
+the doping effect using the DMFT calculation.
+Dynamical mean-field theory results of the THF model—
+In the following, we present the dynamical mean-field the-
+ory resultss of the THF model (Eq. 1), where we describe the
+local quantum many-body effects of the density-density Hub-
+bard term ˆHU within the f-subspace. The ˆHW and ˆHV in-
+teractions involving density fluctuations on the c orbitals are
+accounted for at the static mean-field level. We neglect or-
+dered phases and perform calculations in the non-ordered one.
+There, we focus in particular on lifetime effects, quasiparticle
+weights and exploit the ability of DMFT to take local vertex
+corrections to the spin-spin correlation function into account.
+First, DMFT finds a qualitative difference between the
+strong quasiparticle renormalization when the f+c manifold
+is occupied with an integer number of electrons and a lighter
+Fermi liquid at fractional fillings: this can be seen from the
+scattering rate Γf = −ImΣf(ω = 0) which is shown as a
+function of the total filling ν at T = 11.6K (light blue empty
+circles) in Fig.3(a). The largest scattering rates are found close
+to ν = 0.0, -1.0 and -2.0, progressively decreasing as one
+moves away from the charge neutrality point. Correspond-
+ingly, the spectral weight at the Fermi level (black and grey
+solid circles) is suppressed at these fillings, with a residual
+nonzero value due to the finite temperature on the one hand
+and the resilient f/c hybridization on the other.
+Fig.3(b) illustrates the temperature-dependent screening of
+the local magnetic moment on the f orbitals at different fill-
+ings. A flat T · χloc
+spin(ω = 0) indicates Curie behavior and a
+well-defined effective local moment, while deviations signal
+the onset of screening and a crossover towards a renormalized
+Pauli-like behavior, in agreement with the general expecta-
+tion of zero-temperature Fermi-liquid in the periodic Ander-
+son model [146]. While at ν = 0.0, -1.0 and -2.0 the 1/T
+local spin susceptibility persists down to 1-2 K, the fractional
+fillings deviate from Curie at much higher temperatures, in
+line with the better Fermi-liquid nature signaled by the single-
+particle quantities in Fig.3(a).
+As in the Hartree-Fock treatment of the THF model, DMFT
+confirms the difference between electron doping and hole dop-
+ing (particle-hole asymmetry) near integer filling ν = −1 and
+-2. Here, DMFT reveals particle-hole asymmetric scattering
+rates (Fig. 3(a)) and also in the difference of effective local
+moments at ν = −0.8 and −1.2(Fig. 3(b)).
+In summary, our DMFT calculations confirm that the
+Kondo phase is strongly suppressed at integer fillings ν =
+0, −1, −2, increasing the propensity towards long-range order
+at these fillings. However, by doping the system, the develop-
+ment of Kondo screening (starting from ∼ 10K) is observed,
+which suggests that doping could enhance the Kondo effect.
+This picture is consistent with our mean-field calculations.
+Effects of strain— Since twisted bilayer graphene samples
+exhibit intrinsic strain [147] and the ordered states are disfa-
+vored by strain, we investigate the effect of strain on the sym-
+metric state of THF model via mean-field approach. We focus
+on ν = 0, −1, −2, −3 and introduce the following Hamilto-
+nian [136] that qualitatively characterizes the effect of strain
+ˆHstrain = α
+�
+R,ηs
+(f †
+R,1ηsfR,2ηs + h.c.)
+where α is proportional to the strain amplitude (we leave the
+construction of a realistic strain Hamiltonian [148–150] for
+a future study). A non-zero α breaks the C3z symmetry but
+preserves all other symmetries [136]. We compare the ground
+state energies of the symmetric states (Estrain
+sym
+) and the or-
+dered states (Estrain
+ord
+) at non-zero strain. To obtain the sym-
+metric state solution, we solve the mean-field equations by
+requiring the mean-field Hamiltonian to satisfy all symme-
+tries except for the C3z. We obtain the solution of the or-
+dered states by initializing the mean-field calculations with
+the ordered ground states at zero strain and then perform self-
+consistent calculations. In Fig. 4, we plot the difference be-
+tween the ground state energies of the symmetric and the or-
+dered states ∆Estrain = Estrain
+sym
+− Estrain
+order as a function of
+the effective strain amplitude α with 0meV < α < 20meV at
+ν = 0, −1, −2, −3. We observe that ∆E at ν = 0, −1 van-
+ishes at sufficiently large strain. A detailed analysis [136] of
+the wavefunction shows that the ordered state cannot be sta-
+bilized and converged to a C3z broken symmetric solution at
+large strain. By further increasing strain, we find a symmet-
+ric state at ν = −2 can also be stabilized at α ∼ 45meV
+
+6
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+A(
+= 0)
+a
+Atot
+Af
+Ac
+f
+0
+2
+4
+6
+8
+10
+T [K]
+0.5
+1.0
+1.5
+2.0
+T
+loc
+spin(
+= 0)
+b
+total
+eff
+= 1.39
+total
+eff
+= 1.34
+total
+eff
+= 1.19
+= 0.00
+=
+0.50
+=
+0.80
+=
+1.00
+=
+1.20
+=
+2.00
+2
+1
+0
+2
+0
+f/c
+f
+c
+0
+20
+40
+f [meV]
+FIG. 3. DMFT solution of the THF model. (a) Doping ν dependent low-energy spectral function at the Fermi level (A(ω = 0)) for the
+full system Atot, the c- (Ac) and the f-electrons (Af) at 11.6 K. Also shown is the scattering rate Γf as extracted from the local f-electron
+self-energy. (b) Effective local moment T · χloc
+spin(ω = 0) as a function of temperature T for different doping levels ν.
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+/meV
+0
+2
+4
+6
+8
+10
+12
+14
+16
+Estrain/meV
+=
+0
+=
+1
+=
+2
+=
+3
+FIG. 4. Energy difference ∆Estrain = Estrain
+sym
+− Estrain
+ord
+between
+the symmetric state that only breaks C3z symmetry (Estrain
+sym
+) and
+the ordered state (Estrain
+ord
+) as a function of α - a parameter charac-
+terizing the strain amplitude. We note that even at zero strain α = 0,
+a symmetric state that only breaks C3z symmetry has lower energy
+than the fully symmetric state. Thus ∆Estrain at α = 0 is smaller
+than the corresponding ∆E in Fig. 2.
+(see SM [136]). We conclude that a symmetric phase can be
+stabilized by sufficiently large strain at ν = 0, −1, −2. As
+for ν = −3, we mention that other ordered states, that break
+translational symmetry and have lower energy than the VP
+state, exist even at zero strain. We leave a systematical analy-
+sis of ν = −3 for future study. Finally, we comment that even
+at zero strain, a symmetric state that breaks C3z symmetry has
+lower energy than the fully symmetric state that preserves all
+the symmetries (including C3z). Therefore, ∆Estrain (energy
+difference between a symmetric state that only breaks C3z and
+an ordered state) at zero strain α = 0 in Fig. 4 is smaller
+than the corresponding ∆E (energy difference between a fully
+symmetric state and an ordered state) in Fig. 2.
+Discussion and summary— Our main result is that an or-
+dered state, instead of a SK state, will be the ground state of
+the system at integer filling ν = 0, −1, −2, −3. Our mean-
+field calculations of THF model indicate ground state energy
+of the symmetric state is higher than the one of the ordered
+states at these fillings. Via DMFT calculations, we find the
+Kondo temperature to be substantially smaller than 2K. Thus,
+we conclude the Kondo effect is suppressed at integer filling
+ν = 0, −1, −2, −3, and the ground state is likely to be an
+ordered state. However, our mean-field calculations suggest
+doping can reduce the energy difference between the symmet-
+ric state and the ordered state enhancing the tendency towards
+the SK state. This has also been confirmed by the DMFT
+calculations which show a strong deviation from the Curie
+Weiss law at non-integer fillings ν = −0.5, −0.8, −1.2 al-
+ready around 10K. Furthermore, we show that a sufficiently
+large C3z breaking strain could also stabilize a symmetric
+state that only breaks the C3z symmetry at ν = 0, −1, −2.
+Therefore, we conclude both doping and strain enhance the
+Kondo effect and could, in principle, stabilize a SK state. Our
+results may explain the recent entropy experiments [18, 19]
+which reveal a high-temperature phase with fluctuating mo-
+ments and a low-temperature Fermi liquid phase with unpolar-
+ized isospins. This could be understood as a sign of screening
+of the local moments and the development of the SK phase at
+low temperatures.
+As far as the SK state is concerned, we have performed a
+systematic study of its band structure and topology. Via the
+mean-field approach, we successfully identified the SK state
+in the KL model, and a symmetric state, that is adiabatically
+connected to the SK state, in the THF model. For the SK
+state in the KL model, we find that the Γ3 states near the ΓM
+point have been pushed away, and the bandwidth of the flat
+bands is enlarged at ν = −1, −2. The hybridization between
+f-electrons and Γ3 c-electons is enhanced by the Kondo in-
+teractions. Consequently, the flat bands are mostly formed by
+Γ1 ⊕ Γ2 c-electrons. The topology of the flat bands remains
+the same as in the non-interacting case. However, for the sym-
+metric state in the THF model, the enhanced f-c hybridization
+does not appear. We mention that the mean-field solution of
+
+7
+the symmetric state in the THF model underestimates the cor-
+relation effect, which could be the origin of the weak f-c hy-
+bridization. We expect introducing a Gutzwiller projector will
+give a more precise description of the symmetric state in the
+THF model.
+Note added— After finishing this work, we learned that re-
+lated, but not identical, results have also recently been ob-
+tained by the S. Das Sarma’s [151], P. Coleman’s [134], and
+Z. Song’s groups [152]. We also mention that results from Z.
+Song’s group are compatible with our DMFT results.
+Acknowledgements— B. A. B.’s work was primarily sup-
+ported by the DOE Grant No.
+DE-SC0016239, the Si-
+mons Investigator Grant No. 404513. H. H. was supported
+by the European Research Council (ERC) under the Euro-
+pean Union’s Horizon 2020 research and innovation program
+(Grant Agreement No. 101020833). J. H. A. was supported
+by a Hertz Fellowship.
+D. C. was supported by the DOE
+Grant No. DE-SC0016239 and the Simons Investigator Grant
+No. 404513. A. M. T. was supported by the Office of Ba-
+sic Energy Sciences, Material Sciences and Engineering Divi-
+sion, U.S. Department of Energy (DOE) under Contract No.
+DE-SC0012704. G. R, L. K, T. W., G. S. and R. V. thank
+the Deutsche Forschungsgemeinschaft (DFG, German Re-
+search Foundation) for funding through QUAST FOR 5249-
+449872909 (Projects P4 and P5). G.R. acknowledges funding
+from the European Commission via the Graphene Flagship
+Core Project 3 (grant agreement ID: 881603). T.W. acknowl-
+edges support by the Cluster of Excellence “CUI: Advanced
+Imaging of Matter” of the Deutsche Forschungsgemeinschaft
+(DFG)–EXC 2056–Project ID No. 390715994.
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+arXiv:2211.01352 (2022).
+[148] Naoto Nakatsuji and Mikito Koshino, “Moir´e disorder ef-
+fect in twisted bilayer graphene,” Phys. Rev. B 105, 245408
+(2022).
+[149] Naoto Nakatsuji and Mikito Koshino, “Moir´e disorder ef-
+fect in twisted bilayer graphene,” Phys. Rev. B 105, 245408
+(2022).
+[150] Oskar Vafek and Jian Kang, “Continuum effective hamil-
+tonian for graphene bilayers for an arbitrary smooth lat-
+tice deformation from microscopic theories,” arXiv preprint
+arXiv:2208.05933 (2022).
+[151] Yang-Zhi Chou and Sankar Das Sarma, “Kondo lattice model
+in magic-angle twisted bilayer graphene,” (2022).
+[152] Geng-Dong Zhou and Zhi-Da Song, to be published.
+
+12
+[153] Nicolaus Parragh, Alessandro Toschi, Karsten Held,
+and
+Giorgio
+Sangiovanni,
+“Conserved
+quantities
+of
+su(2)-
+invariant interactions for correlated fermions and the advan-
+tages for quantum monte carlo simulations,” Phys. Rev. B 86,
+155158 (2012).
+[154] Markus Wallerberger, Andreas Hausoel, Patrik Gunacker,
+Alexander Kowalski, Nicolaus Parragh, Florian Goth, Karsten
+Held,
+and Giorgio Sangiovanni, “w2dynamics: Local one-
+and two-particle quantities from dynamical mean field theory,”
+Computer Physics Communications 235, 388–399 (2019).
+[155] Olivier Parcollet, Michel Ferrero, Thomas Ayral, Hartmut
+Hafermann, Igor Krivenko, Laura Messio, and Priyanka Seth,
+“TRIQS: A Toolbox for Research on Interacting Quantum
+Systems,” Computer Physics Communications 196, 398–415
+(2015), arXiv:1504.01952 [cond-mat, physics:physics].
+[156] Priyanka Seth, Igor Krivenko, Michel Ferrero,
+and Olivier
+Parcollet, “TRIQS/CTHYB: A Continuous-Time Quantum
+Monte Carlo Hybridization Expansion Solver for Quantum
+Impurity Problems,” Computer Physics Communications 200,
+274–284 (2016), arXiv:1507.00175 [cond-mat].
+[157] Markus Aichhorn, Leonid Pourovskii, Priyanka Seth, Veron-
+ica Vildosola, Manuel Zingl, Oleg E. Peil, Xiaoyu Deng,
+Jernej Mravlje, Gernot J. Kraberger, Cyril Martins, Michel
+Ferrero, and Olivier Parcollet, “TRIQS/DFTTools: A TRIQS
+application for ab initio calculations of correlated materials,”
+Computer Physics Communications 204, 200–208 (2016).
+
+13
+Supplementary Materials
+CONTENTS
+References
+7
+S1. Toplogical heavy-fermion model
+14
+S2. Kondo lattice model
+15
+S3. Symmetry
+16
+S4. Mean-field solutions of the Kondo lattice model
+16
+A. Mean-field decoupling of ˆHK
+17
+1. Fock term
+17
+2. Hartree term
+18
+3. Fock and Hartree terms
+19
+B. Mean-field decoupling of ˆHJ
+20
+1. Fock term
+20
+2. Hartree term
+21
+3. Fock and Hartree terms
+21
+C. Filling constraints and mean-field equations
+21
+D. Mean-field equations of the symmetric Kondo state
+22
+E. Properties of the symmetric Kondo state
+24
+S5. Mean-field solutions of the topological heavy-fermion model
+27
+A. Mean-field equations of fully symmetric state
+27
+B. Mean-field equations of the symmetric state in the presence of strain
+29
+C. Effect of doping
+30
+D. Effect of strain
+31
+1. ν = −1
+32
+2. ν = −2
+34
+3. Discussions about ν = −3
+36
+E. Strain
+36
+S6. Dynamical mean field theory: implementation
+39
+
+14
+S1.
+TOPLOGICAL HEAVY-FERMION MODEL
+The topological heavy-fermion (THF) model introduced in Ref. [1] takes the following Hamiltonian
+ˆH = ˆHc + ˆHfc + ˆHU + ˆHJ + ˆHW + ˆHV + ˆHµ
+(S12)
+The single-particle Hamiltonian of conduction c-electrons has the form of
+ˆHc =
+�
+η,s,a,a′,|k|<Λc
+H(c,η)
+a,a′ (k)c†
+kaηscka′ηs
+,
+H(c,η)(k) =
+�
+02×2
+v⋆(ηkxσ0 + ikyσz)
+v⋆(ηkxσ0 − ikyσz)
+Mσx.
+�
+(S13)
+where σ0,x,y,z are identity and Pauli matrices. ckaηs represents the annihilation operator of the a(= 1, 2, 3, 4)-th conduction
+band basis of the valley η(= ±) and spin s(=↑, ↓) at the moir´e momentum k. At ΓM point (k = 0) of the moir´e Brillouin zone
+(MBZ), ck1ηs, ck2ηs form a Γ3 irreducible representation (of P6′2′2 group), ck3ηs, ck4ηs form a Γ1 ⊕ Γ2 reducible (into Γ1 and
+Γ2 - as they are written, the ck3ηs, ck4ηs are just the σx linear combinations of Γ1 ± Γ2 ) representation (of P6′2′2 group). Λc is
+the momentum cutoff for the c-electrons. N is the total number of moir´e unit cells. The parameter values are v⋆ = −4.303eV·˚A,
+M = 3.697meV.
+The hybridization between f and c electrons has the form of
+ˆHfc =
+1
+√NM
+�
+|k|<Λc
+R
+�
+αaηs
+�
+eik·R− |k|2λ2
+2
+H(fc,η)
+αa
+(k)f †
+Rαηsckaηs + h.c.
+�
+,
+(S14)
+where fRαηs represents the annihilation operators of the f electrons with orbital index α(= 1, 2), valley index η(= ±) and spin
+s(=↑, ↓) at the moir´e unit cell R. NM is the number of moir´e unit cells and λ = 0.3376aM is the damping factor, where aM is
+the moir´e lattice constant. The hybridization matrix H(fc,η) has the form of
+H(fc,η)(k) =
+�γσ0 + v′
+⋆(ηkxσx + kyσy), 02×2
+�
+(S15)
+which describe the hybridization between f electrons and Γ3 c electrons (a = 1, 2). The parameter values are γ = −24.75meV,
+v′
+⋆ = 1.622eV · ˚A.
+ˆHU (U = 57.89meV) describes the on-site interactions of f-electrons.
+ˆHU = U
+2
+�
+R
+: nf
+R :: nf
+R :,
+(S16)
+where nf
+R = �
+αηs f †
+RαηsfRαηs is the f-electrons density and the colon symbols represent the normal ordered operator with
+respect to the normal state: : f †
+Rα1η1s1fRα2η2s2 := f †
+Rα1η1s1fRα2η2s2 − 1
+2δα1η1s1;α2η2s2.
+The ferromagnetic exchange interaction between f and c electrons ˆHJ is defined as
+HJ = −
+J
+2NM
+�
+Rs1s2
+�
+αα′ηη′
+�
+|k1|,|k2|<Λc
+ei(k1−k2)·R(ηη′ + (−1)α+α′) : f †
+Rαηs1fRα′η′s2 :: c†
+k2,α′+2,η′s2ck1,α+2,ηs1 :
+(S17)
+where J = 16.38meV and : c†
+k2,α′+2,η′s2ck1,α+2,ηs1 := c†
+k2,α′+2,η′s2ck1,α+2,ηs1 − 1
+2δk1,k2δα,α′δη,η′δs1,s2
+The repulsion between f and c electrons ˆHW has the form of
+ˆHW =
+�
+η,s,η′,s′,a,α
+�
+|k|<Λc,|k+q|<Λc
+Wae−iq·R : f †
+R,aηsfR,aηs :: c†
+k+q,aη′s′ck,aη′s′ :
+(S18)
+where we take W1 = W2 = 44.03meV and W3 = W4 = 50.20meV.
+The Coulomb interaction between c electrons has the form of
+ˆHV =
+1
+2Ω0N
+�
+η1s1a1
+�
+η2s2a2
+�
+|k1|,|k2|<Λc
+�
+q
+|k1+q|,|k2+q|<Λc
+V (q) : c†
+k1a1η1s1ck1+qa1η1s1 :: c†
+k2+qa2η2s2ck2a2η2s2 :
+(S19)
+
+15
+where Ω0 is the area of the moir´e unit cell and V (q = 0)/Ω0 = 48.33meV. We will always treat ˆHV at mean-field level (int
+both the THF model and the Kondo lattice (KL) model) [1]
+ˆHV ≈ V (0)
+Ω0
+νc
+�
+|k|<Λc,a,η,s
+c†
+k,aηsck,aηs − V (0)
+2Ω0
+NMν2
+c + V (0)
+Ω0
+�
+|k|<Λc
+8νc
+(S20)
+where νc is the filling of c electrons νc =
+1
+NM
+�
+|k|<Λc,a,η,s⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩ with |ψ⟩ the ground state.
+Finally, we introduce a chemical potential term
+ˆHµ = −µ
+�
+|k|<Λc,aηs
+c†
+k,aηsck,aηs − µ
+�
+R,αηs
+f †
+R,αηsfR,αηs .
+(S21)
+S2.
+KONDO LATTICE MODEL
+The Kondo lattice model is derived by performing a generalized Schrieffer-Wolff (SW) transformation on the topological
+heavy fermion model (detailed derivation in Ref. [2]). The Hamiltonian has the form of
+ˆHKondo = ˆHc + ˆHV + ˆHW + ˆHJ + ˆHK + ˆHcc − ˆHµc
+(S22)
+where ˆHc (Eq. S13), ˆHV (Eq. S19) and ˆHW (Eq. S18) and ˆHJ (Eq. S17) come from the original TFH model. The Kondo
+interactions and the one-body scattering term are
+ˆHK =
+�
+R,|k|<Λc,|k′|<Λc
+�
+α,α′,a,a′,η,η′,s,s′
+ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2
+NMDνc,νf
+: f †
+R,αηsfR,α′η′s′ :: c†
+k′,a′η′s′ck,aηs :
+�
+γ2δα′,a′δα,a + γv′
+⋆δα,a[η′k′
+xσx − k′
+yσy]α′a′ + γv′
+⋆δα′,a′[ηkxσx + kyσy]αa
+�
+(S23)
+and
+ˆHcc = −
+�
+|k|<Λc,η,s
+�
+a,a′∈{1,2}
+e−|k|2λ2�
+1
+D1,νc,νf
++
+1
+D2,νc,νf
+� �
+γ2/2
+γv′
+⋆(ηkx − iky)
+γv′
+⋆(ηkx + iky)
+γ2/2
+�
+a,a′ : c†
+k,aηsck,a′ηs : (S24)
+where
+D1,νc,νf = (U − W)νf − U
+2 + (−V0
+Ω0
++ W)νc
+,
+D2,νc,νf = (U − W)νf + U
+2 + (−V0
+Ω0
++ W)νc
+Dνc,νf =
+�
+−
+1
+D1,νc,νf
++
+1
+D2,νc,νf
+�−1
+.
+(S25)
+We point out that, at ν = νf = νc = 0, D1,νc,νf = −D2,νc,νf and the on-body term ˆHcc(= 0) vanishes.
+We note that in the Kondo model the filling of f electron at each site is fixed to be νf. Then we can replace �
+αηs :
+f †
+R,αηsfR,αηs : with νf and ˆHW becomes
+ˆHW =
+�
+|k|<Λc,|k′|<Λc,aηs
+�
+Q
+Wνf : c†
+k,aη′s′ck′,aηsδk,k′+Q
+(S26)
+where Q ∈ {mbM1 + mbM2|m, n ∈ Z} and bM1, bM2 are the reciprocal lattice vectors. If we focus on the conduction
+electrons within the first MBZ, we can replace δk,k′+Q by δk,k′ and
+ˆHW =
+�
+|k|<Λc,aηs
+�
+Q
+Wνf : c†
+k,aη′s′ck,aηs
+(S27)
+which is a chemical shift of conduction electrons. We also set W1 = W2 = W3 = W4 = W = 47.12meV in ˆHW to simplify
+the SW transformation. The realistic values of W1,2,3,4 are not identical but the difference is about 15%.
+Finally, we introduce a chemical potential µc to tune the filling of the system
+ˆHµc = −µc
+�
+|k|<Λc,aηs
+: c†
+k,aηsck,aηs :
+(S28)
+
+16
+S3.
+SYMMETRY
+We now provide the symmetry transformation of electron operators.
+For a given symmetry operation g, we let
+Df(g), Dc′(g), Dc′′(g) denote the representation matrix of f-, Γ3 c- and Γ1 ⊕ Γ2 c-electrons:
+gf †
+R,αηsg−1 =
+�
+α′η′s′
+f †
+gR,α′η′s′Df(g)α′η′s′,αηs
+gc†
+k,aηsg−1 =
+�
+a′∈{1,2},η′s′
+c†
+gk,a′η′s′Dc′(g)a′η′s′,aηs,
+a ∈ {1, 2}
+gc†
+k,aηsg−1 =
+�
+a′∈{3,4},η′s′
+c†
+gk,a′η′s′Dc′′(g)a′+2η′s′,a+2ηs,
+a ∈ {3, 4}
+(S29)
+We consider the following symmetry operations as given in Ref. [1].
+T, C3z, C2x, C2zT
+(S30)
+with the following representation matrices
+T :
+Df(T) = σ0τxς0,
+Dc′(T) = σ0τxς0,
+Dc′′(T) = σ0τxς0
+C3z :
+Df(C3z) = ei 2π
+3 σzτzς0,
+Dc′(C3z) = ei 2π
+3 σzτzς0,
+Dc′′(C3z) = σ0τ0ς0
+C2x :
+Df(C2x) = σxτ0ς0,
+Dc′(C2x) = σxτ0ς0,
+Dc′′(C2x) = σxτ0ς0
+C2zT :
+Df(C2xT) = σxτ0ς0,
+Dc′(C2xT) = σxτ0ς0,
+Dc′′(C2zT) = σxτ0ς0
+(S31)
+where σx,y,z,0, τx,y,z,0, ςx,y,z,0 are Pauli or identity matrices of orbital, valley and spin degrees of freedom respectively.
+At M ̸= 0, v′
+⋆ ̸= 0, we also have U(1)c charge symmetry, U(1)v valley symmetry and SU(2)η spin symmetry for each
+valley η. We also mention that at M = 0, we have an enlarged flat U(4) symmetry and at v′
+⋆ = 0 we have an enlarged
+chiral U(4) symmetry [1, 2]. At M = 0, v′
+⋆ = 0, we have a U(4) × U(4) symmetry [1, 2]. Here, we consider the case of
+M ̸= 0, v′
+⋆ ̸= 0, where we only have a U(1)c ×U(1)v ×SU(2)η=+ ×SU(2)η=− symmetry. We comment that M = 3.698meV
+is relatively small and we have an approximate flat U(4) symmetry. Under U(1)c transformation gU(1)c(θc) (characterized
+by a real number θc), U(1)v transformation gU(1)v(θv) (characterized by a real number θv) and SU(2)η spin transformation
+gSU(2)η(θµ
+η ) (characterized by three real numbers θµ
+η , µ ∈ {x, y, z} ), we have
+U(1)c :
+Df(gU(1)c((θc)) = e−iθcσ0τ0ς0,
+Dc′(gU(1)c((θc)) = e−iθcσ0τ0ς0,
+Dc′′(gU(1)c((θc)) = e−iθcσ0τ0ς0
+U(1)v :
+Df(gU(1)v((θv)) = σ0e−iθvτzς0,
+Dc′(gU(1)v((θv)) = σ0e−iθvτzς0,
+Dc′′(gU(1)v((θv)) = σ0e−iθvτzς0
+SU(2)η :
+Df(gSU(2)η(θµ
+η )) = σ0e−i �
+µ θη
+µ
+τ0+ητz
+4
+ςµ,
+Dc′(gSU(2)η(θµ
+η )) = σ0e−i �
+µ θη
+µ
+τ0+ητz
+4
+ςµ,
+Dc′′(gSU(2)η(θµ
+η )) = σ0e−i �
+µ θη
+µ
+τ0+ητz
+4
+ςµ
+(S32)
+S4.
+MEAN-FIELD SOLUTIONS OF THE KONDO LATTICE MODEL
+The Kondo Hamiltonian in Eq. S22 contains two single-particle term ˆHc and ˆHcc and two interaction terms ˆHK + ˆHJ. We
+now discuss the mean-field decoupling of ˆHK, ˆHJ.
+
+17
+A.
+Mean-field decoupling of ˆHK
+We treat the interaction terms via mean-field decoupling
+ˆHK ≈ ˆHMF
+K
+=
+�
+R,|k|<Λc,|k′|<Λc
+�
+α,α′,a,a′,η,η′,s,s′
+ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2
+NMDνc,νf
+�
+γ2δα′,a′δα,a + γv′
+⋆δα,a[η′k′
+xσx − k′
+yσy]α′a′ + γv′
+⋆δα′,a′[ηkxσx + kyσy]αa
+�
+�
+⟨f †
+R,αηsck,aηs⟩⟨c†
+k′,a′η′s′fR,α′η′s′⟩ − ⟨f †
+R,αηsck,aηs⟩c†
+k′,a′η′s′fR,α′η′s′ − f †
+R,αηsck,aηs⟨c†
+k′,a′η′s′fR,α′η′s′⟩
+− ⟨: f †
+R,αηsfR,α′η′s′ :⟩⟨: c†
+k′,a′η′s′ck,aηs :⟩ + ⟨: f †
+R,αηsfR,α′η′s′ :⟩ : c†
+k′,a′η′s′ck,aηs : + : f †
+R,αηsfR,α′η′s′ : ⟨: c†
+k′,a′η′s′ck,aηs :⟩
+�
+(S33)
+where for an operator O, ⟨O⟩ = ⟨Ψ|O|Ψ⟩ with |Ψ⟩ the mean-field ground state.
+1.
+Fock term
+We first consider the Fock term (F.T.), which takes the form of
+F.T. =
+�
+R,|k|<Λc,|k′|<Λc
+�
+α,α′,a,a′,η,η′,s,s′
+ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2
+NMDνc,νf
+�
+γ2δα′,a′δα,a + γv′
+⋆δα,a[η′k′
+xσx − k′
+yσy]α′a′ + γv′
+⋆δα′,a′[ηkxσx + kyσy]αa
+�
+�
+⟨f †
+R,αηsck,aηs⟩⟨c†
+k′,a′η′s′fR,α′η′s′⟩ − ⟨f †
+R,αηsck,aηs⟩c†
+k′,a′η′s′fR,α′η′s′ − f †
+R,αηsck,aηs⟨c†
+k′,a′η′s′fR,α′η′s′⟩
+�
+=
+�
+R
+1
+Dνc,νf
+�
+γ2⟨
+�
+|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+f †
+R,αηsck,aηs⟩⟨
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′⟩
+−
+�
+γ2 �
+|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+f †
+R,αηsck,aηs⟨
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′⟩ + h.c.
+�
++ γv′
+⋆⟨
+�
+|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+f †
+R,αηsck,αηs⟩⟨
+�
+|k′|<Λc
+�
+a′α′η′s′
+e−ik′·R−|k′|2λ2/2[η′k′
+xσx − k′
+yσy]α′a′
+√NM
+c†
+k′,a′η′s′fR,α′η′s′⟩
++ γv′
+⋆⟨
+�
+|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2[ηkxσx + kyσy]αa
+√NM
+f †
+R,αηsck,αηs⟩⟨
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′⟩
+− γv′
+⋆
+� �
+|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2[ηkxσx + kyσy]αa
+√NM
+f †
+R,αηsck,αηs⟨
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′⟩
++
+�
+|k|<Λc
+⟨
+�
+αaηs
+eik·R−|k|2λ2/2[ηkxσx + kyσy]αa
+√NM
+f †
+R,αηsck,αηs⟩
+�
+|k′|<Λc
+�
+α′η′s′
+e−ik′·R−|k′|2λ2/2
+√NM
+c†
+k′,α′η′s′fR,α′η′s′ + h.c.
+��
+(S34)
+
+18
+We introduce the following mean-field expectation values
+V1 =
+�
+R,|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+NM
+√NM
+⟨Ψ|f †
+R,αηsck,αηs|Ψ⟩
+V2 =
+�
+R,|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2
+NM
+√NM
+(ηkxσx + kyσy)αa⟨Ψ|f †
+R,αηsck,aηs|Ψ⟩
+(S35)
+and assume the ground state is translational invariant such that
+�
+|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+⟨Ψ|f †
+R,αηsck,αηs|Ψ⟩ =
+1
+NM
+�
+R,|k|<Λc
+�
+αηs
+eik·R−|k|2λ2/2
+√NM
+⟨Ψ|f †
+R,αηsck,αηs|Ψ⟩ = V1
+�
+|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2
+√NM
+(ηkxσx + kyσy)αa⟨Ψ|f †
+r,αηsck,αηs|Ψ⟩
+= 1
+NM
+�
+R,|k|<Λc
+�
+αaηs
+eik·R−|k|2λ2/2
+√NM
+(ηkxσx + kyσy)αa⟨Ψ|f †
+r,αηsck,αηs|Ψ⟩ = V2
+(S36)
+Then the Fock term (Eq. S34) becomes
+F.T. = −
+γ2
+Dνc,νf
+�
+R,|k|<Λc
+�
+αη,s
+eik·R−|k|2λ2/2
+√NM
+�
+V ∗
+1 f †
+R,αηsck,αηs + h.c.
+�
++ NMγ2|V1|2
+Dνc,νf
+−
+γv′
+⋆
+Dνc,νf
+�
+R,|k|<Λc
+�
+α,a,η,s
+eik·R−|k|2λ2/2
+√NM
+�
+V ∗
+1 (ηkxσx + kyσy)αaf †
+R,αηsck,αηs + h.c.
+�
++ NMγv′
+⋆V ∗
+1 V2
+Dνc,νf
+−
+γv′
+⋆
+Dνc,νf
+�
+R,|k|<Λc
+�
+α,η,s
+eik·R−|k|2λ2/2
+√NM
+�
+V ∗
+2 f †
+R,αηsck,αηs + h.c.
+�
++ NMγv′
+⋆V ∗
+2 V1
+Dνc,νf
+(S37)
+2.
+Hartree term
+For the Hartree term (H.T.), we introduce the following density matrices Of, Oc′,1, Oc′,2, where Of have also been used in
+the mean-field calculations of the THF model as shown in Ref. [1] (however, Oc′,1, Oc′,2, V1, V2 are absent in the THF model)
+Of
+αηs,α′η′s′ =
+1
+NM
+�
+R
+⟨Ψ| : f †
+R,αηsfR,α′η′s′ : |Ψ⟩
+Oc′,1
+aηs,a′η′s′ =
+1
+NM
+�
+|k|<Λc
+e−|k|2λ2⟨Ψ| : c†
+k,aηsck,a′η′s′ : |Ψ⟩,
+a, a′ ∈ {1, 2}
+Oc′,2
+a′η′s′,αηs =
+1
+NM
+�
+|k|<Λc
+�
+a=1,2
+e−|k|2λ2(ηkxσx + kyσy)αa⟨Ψ| : c†
+k,a′η′s′ck,aηs : |Ψ⟩,
+a′, α ∈ {1, 2} .
+(S38)
+We then assume the ground state is translational invariance such that
+⟨Ψ| : f †
+R,αηsfR,α′η′s′ : |Ψ⟩ =
+1
+NM
+�
+R
+⟨Ψ| : f †
+R,αηsfR,α′η′s′ : |Ψ⟩ = Of
+αηs,α′η′s′ .
+(S39)
+
+19
+Using Eq. S38 and Eq. S39, the Hartree term can be written as
+H.T.
+=
+�
+R,
+|k|<Λc,|k′|<Λc
+�
+α,α′,a,a′,
+η,η′,s,s′
+ei(k−k′)R−|k|2λ2/2−|k′|2λ2/2
+NMDνc,νf
+�
+γ2δα′,a′δα,a + γv′
+⋆δα,a[η′k′
+xσx − k′
+yσy]α′a′ + γv′
+⋆δα′,a′[ηkxσx + kyσy]αa
+�
+�
+− ⟨: f †
+R,αηsfR,α′η′s′ :⟩⟨: c†
+k′,a′η′s′ck,aηs :⟩ + ⟨: f †
+R,αηsfR,α′η′s′ :⟩ : c†
+k′,a′η′s′ck,aηs : + : f †
+R,αηsfR,α′η′s′ : ⟨: c†
+k′,a′η′s′ck,aηs :⟩
+�
+=
+�
+α,α′,
+η,η′,s,s′
+NM
+Dνc,νf
+�
+− γ2Of
+αηs,αη′s′Oc′1
+α′η′s′,αηs −
+�
+γv′
+⋆Of
+αηs,α′η′s′Oc′,2
+α′η′s′,αηs + h.c.
+��
++
+�
+|k|<Λc
+�
+α,α′,
+η,η′,s,s′
+�
+Of
+αηs,α′η′s′e−|k|2λ2 : c†
+k,a′η′s′ck,aηs : δα,aδα′,a′ +
+�
+γv′
+⋆Of
+αηs,α′η′s′δα,a[η′kxσx − kyσy]α′a′e−|k|2λ2
+: c†
+k,a′η′sck,aηs : +h.c.
+��
++
+�
+R
+�
+α,α′,
+η,η′,s,s′
+�
+: f †
+R,αηsfR,α′η′s′ : Oc′,1
+α′η′s′,αηs +
+�
+γv′
+⋆ : f †
+R,αηsfR′,α′η′s′ : Oc′,2
+α′η′s′,αηs + h.c.
+��
+(S40)
+3.
+Fock and Hartree terms
+Combining Fock and Hartree (Eq. S37 and Eq. S37) terms, we have
+ˆHK ≈ ˆHMF
+K
+=F.T. + H.T.
+= −
+γ2
+Dνc,νf
+�
+R,|k|<Λc
+�
+αη,s
+eik·R−|k|2λ2/2
+√NM
+��
+V ∗
+1 f †
+R,αηsck,αηs + h.c.
+��
++ NMγ2|V1|2
+Dνc,νf
+−
+γv′
+⋆
+Dνc,νf
+�
+R,|k|<Λc
+�
+α,a,η,s
+eik·R−|k|2λ2/2
+√NM
+��
+V ∗
+1 (ηkxσx + kyσy)αaf †
+R,αηsck,αηs + h.c.
+��
++ NMγv′
+⋆V ∗
+1 V2
+Dνc,νf
+−
+γv′
+⋆
+Dνc,νf
+�
+R,|k|<Λc
+�
+α,η,s
+eik·R−|k|2λ2/2
+√NM
+��
+V ∗
+2 f †
+R,αηsck,αηs + h.c.
+�
+− V ∗
+2 V1
+�
++ NMγv′
+⋆V ∗
+2 V1
+Dνc,νf
++
+�
+α,α′,
+η,η′,s,s′
+NM
+Dνc,νf
+�
+− γ2Of
+αηs,αη′s′Oc′1
+α′η′s′,αηs −
+�
+γv′
+⋆Of
+αηs,α′η′s′Oc′,2
+α′η′s′,αηs + h.c.
+��
++
+�
+|k|<Λc
+�
+α,α′,
+η,η′,s,s′
+�
+Of
+αηs,α′η′s′e−|k|2λ2 : c†
+k,a′η′s′ck,aηs : δα,aδα′,a′ +
+�
+γv′
+⋆Of
+αηs,α′η′s′δα,a[η′kxσx − kyσy]α′a′e−|k|2λ2
+: c†
+k,a′η′sck,aηs : +h.c.
+��
++
+�
+R
+�
+α,α′,
+η,η′,s,s′
+�
+: f †
+R,αηsfR,α′η′s′ : Oc′,1
+α′η′s′,αηs +
+�
+γv′
+⋆ : f †
+R,αηsfR′,α′η′s′ : Oc′,2
+α′η′s′,αηs + h.c.
+��
+(S41)
+V1, V2 describes the Fock contribution that characterize the hybridization between f- and Γ3 c-electrons. Of, Oc′,1, Oc′,2 are
+the mean fields taking the form of ⟨f †f⟩, ⟨c†c⟩ which represent the Fock contribution.
+
+20
+B.
+Mean-field decoupling of ˆHJ
+We now perform a mean-field decoupling of the ferromagnetic exchange coupling term [1]
+ˆHJ ≈ ˆHMF
+J
+= −
+J
+2NM
+�
+R
+�
+αα′ηη′,ss′
+�
+|k|,|k′|<Λc
+ei(k−k′)·R(ηη′ + (−1)α+α′)
+�
+⟨f †
+R,αηsck′,α+2,ηs⟩⟨c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+− ⟨f †
+R,αηsck,α+2,ηs1⟩c†
+k′,α′+2,η′s′fR,α′η′s′ − f †
+R,αηs1ck,α+2,ηs⟨c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+− ⟨: f †
+R,αηsfR,α′η′s′ :⟩⟨: c†
+k′,α′+2,η′s′ck,α+2,ηs :⟩+ : f †
+R,αηsfR,α′η′s′ : ⟨: c†
+k′,α′+2,η′s′ck,α+2,ηs :⟩
++ ⟨: f †
+R,αηsfR,α′η′s′ :⟩ : c†
+k′,α′+2,η′s′ck,α+2,ηs :
+�
+(S42)
+1.
+Fock term
+The Fock term takes the form of
+F.T. = −
+J
+2NM
+�
+R
+�
+αα′ηη′,ss′
+�
+|k|,|k′|<Λc
+ei(k−k′)·R(ηη′ + (−1)α+α′)
+�
+⟨f †
+R,αηsck′,α+2,ηs⟩⟨c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+− ⟨f †
+R,αηsck,α+2,ηs1⟩c†
+k′,α′+2,η′s′fR,α′η′s′ − f †
+R,αηs1ck,α+2,ηs⟨c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+�
+= − J
+�
+R
+�
+ξ=±
+�
+⟨
+�
+|k′|<Λc,αηs
+ei(−k′)·R
+√NM
+δξ,η(−1)α+1f †
+R,αηsck′,α+2,ηs⟩⟨
+�
+|k|<Λc,α′η′s′
+δξ,η′(−1)α′+1 eik·R
+√NM
+c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+−
+�
+|k|<Λc,αηs
+ei(−k′)·R
+√NM
+δξ,η(−1)α+1f †
+R,αηsck′,α+2,ηs⟨
+�
+|k′|<Λc,αηs
+δξ,η′(−1)α′+1 eik·R
+√NM
+�
+α′η′s′
+c†
+k′,α′+2,η′s′fR,α′η′s′⟩
+− ⟨
+�
+k′,αηs
+ei(−k′)·R
+√NM
+δξ,η(−1)α+1f †
+R,αηsck′,α+2,ηs⟩
+�
+|k|<Λc,α′η′s′
+eik·R
+√NM
+δξ,η′(−1)α′+1c†
+k′,α′+2,η′s′fR,α′η′s′
+�
+(S43)
+We then introduce the following mean-fields
+V3 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+NM
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩
+V4 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+NM
+√NM
+⟨Ψ|ηf †
+R,αηsck,α+2ηs|Ψ⟩
+(S44)
+and assume the ground state is translational invariant such that
+�
+|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩ =
+1
+NM
+�
+R
+�
+|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩ = V3
+�
+|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩ =
+1
+NM
+�
+R
+�
+|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩ = V4
+(S45)
+Then the Fock term can be written as
+F.T. = − JNM[|V3|2 + |V4|2]
++ J
+�
+R,|k|<Λc,αηs
+�ei(−k′)·R
+√NM
+�
+δ1,η(−1)α+1f †
+R,αηsck′,α+2,ηsV ∗
+3 + δ−1,η(−1)α+1f †
+R,αηsck′,α+2,ηsV ∗
+4 +
+�
++ h.c.
+�
+(S46)
+
+21
+2.
+Hartree term
+The Hartree term takes the form of
+H.T. = −
+J
+2NM
+�
+R
+�
+αα′ηη′,ss′
+�
+|k|,|k′|<Λc
+ei(k−k′)·R(ηη′ + (−1)α+α′)
+�
+− ⟨: f †
+R,αηsfR,α′η′s′ :⟩⟨: c†
+k′,α′+2,η′s′ck,α+2,ηs :⟩
++ : f †
+R,αηsfR,α′η′s′ : ⟨: c†
+k′,α′+2,η′s′ck,α+2,ηs :⟩ + ⟨: f †
+R,αηsfR,α′η′s′ :⟩ : c†
+k′,α′+2,η′s′ck,α+2,ηs :
+�
+(S47)
+We introduce the following density matric which has also been used in Ref. [1]
+Oc′′
+aηs,a′η′s′ =
+1
+NM
+�
+|k|<Λc
+⟨Ψ| : c†
+k,a+2ηsck,a′+2η′s′ : |Ψ⟩,
+a, a′ ∈ {1, 2} .
+(S48)
+Using Eq. S38 and Eq. S48, the Hartree term becomes
+H.T. = − JNM
+2
+�
+αα′ηη′ss′
+(ηη′ + (−1)α+α′)Of
+αηs,α′η′s′Oc′′
+α′η′s′,αηs
++ J
+2
+�
+Rc,αα′ηη′ss′
+: f †
+R,αηsfR,α′η′s′ : Oc′′
+α′η′s′,αηs + J
+2
+�
+|k|<Λc,αα′ηη′ss′
+Of
+αηs,α′η′s′ : c†
+k′,α′+2,η′s′ck,α+2,ηs :
+(S49)
+3.
+Fock and Hartree terms
+Combing Hartree and Fock terms (Eq. S46 and Eq. S49), we have
+ˆHJ ≈ ˆHMF
+J
+= − JNM
+�
+ξ=±
+|V3|2 + |V4|2
++ J
+�
+R,|k|<Λc,αηs
+�ei(−k′)·R
+√NM
+�
+δ1,η(−1)α+1f †
+R,αηsck′,α+2,ηsV ∗
+3 + δ−1,η(−1)α+1f †
+R,αηsck′,α+2,ηsV ∗
+4
+�
++ h.c.
+�
+− JNM
+2
+�
+αα′ηη′ss′
+(ηη′ + (−1)α+α′)Of
+αηs,α′η′s′Oc′′
+α′η′s′,αηs
++ J
+2
+�
+Rc,αα′ηη′ss′
+: f †
+R,αηsfR,α′η′s′ : Oc′′
+α′η′s′,αηs + J
+2
+�
+|k|<Λc,αα′ηη′ss′
+Of
+αηs,α′η′s′ : c†
+k′,α′+2,η′s′ck,α+2,ηs :
+(S50)
+V3, V4 describes the Fock contribution that characterize the hybridization between f- and Γ1 ⊕ Γ2 c-electrons. Of, Oc′′ are the
+mean fields taking the form of ⟨f †f⟩, ⟨c†c⟩ which represent the Fock contribution and have also been used in Ref. [1].
+C.
+Filling constraints and mean-field equations
+We note that in the Kondo model the filling of f electrons is fixed to be νf at each site. To simplify the calculation, we take
+a common approximation that only requires the average filling of f-electron to be νf [3, 138]. In other words, we only require
+1
+NM
+�
+R,αηs⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩ = νf. We then add the following term to the Hamiltonian
+ˆHλf =
+�
+R,αηs
+λf
+�
+: f †
+R,αηsfR,αηs : −νf
+�
+(S51)
+and determine the Langrangian multiplier λf from the following equation
+1
+NM
+�
+R,αηs
+⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩ = νf
+(S52)
+
+22
+In practice, we perform calculations at fixed total filling ν = νf + νc, where νf and νc are the average fillings of f and c
+electrons respectively. Since νf is also fixed in the Kondo model, we will self-consistently determine the chemical potential µc
+(in Eq. S28) by requiring
+1
+NM
+�
+|k|<Λc,aηs
+⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩ = νc = ν − νf
+(S53)
+Finally, our mean-field Hamiltonian takes the form of
+ˆHMF = ˆHc + ˆHcc + ˆHMF
+K
++ ˆHMF
+J
++ ˆHλf + ˆHµc
+(S54)
+and we determine V1, V2, V3, V4, Of, Oc′,1, Oc′,2, Oc′′, λf, µc from the self-consistent equations (Eq. S35, Eq. S38, Eq. S44,
+Eq. S48, Eq. S52, Eq. S53). During the self-consistent solution, at each step, we will adjust λf, µc according to the current
+filling of f- and c-electrons. We use νi
+f and νi
+c denote the filling of f and c at i-th step. For the i + 1-th step, we will update
+λf, µc as λf → λf + r(νi
+f − νf), µc → µc − r(νi
+c − νc), where r(> 0) will be manually adjusted to improve the convergence
+(in practice, we take r ∼ 0.001).
+D.
+Mean-field equations of the symmetric Kondo state
+We focus on the symmetric Kondo phase without any symmetry breaking. Therefore, we require our density matrix of f-
+and c- electrons (Eq. S38, Eq. S48) to be symmetric. We can then utilize symmetry to simplify the self-consistent equations
+(Eq. S38, Eq. S48). We first consider the U(1)v symmetry. From Eq. S32, a U(1)v symmetric solution satisfies
+Of
+αηs,α′η′s′ = Of
+αηs,α′η′s′e−iθν(η−η′) ⇒ Of
+αηs,α−ηs′ = 0
+Oc′,1
+aηs,a′η′s′ = Oc′,1
+aηs,a′η′s′e−iθν(η−η′) ⇒ Oc′,1
+aηs,a′−η′s′ = 0
+Oc′,2
+aηs,α′η′s′ = Oc′,2
+aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′,2
+aηs,α′−η′s′ = 0
+Oc′′
+aηs,a′η′s′ = Oc′′
+aηs,a′η′s′e−iθν(η−η′) ⇒ Oc′′
+aηs,a′−ηs′ = 0
+(S55)
+and V1, V2, V3, V4 are invariant under U(1)v transformation. From Eq. S55, Of, Oc′,1, Oc′,2, Oc′′ are block diagonalized in
+valley index. We next consider a SU(2)η transformation acting on the valley η. We find
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOf
+αηs,α′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Of
+αηs2,α′ηs′
+2 ⇒ Of
+aηs,a′ηs′ ∝ Is,s′
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′,1
+aηs,a′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′,1
+aηs,a′ηs′ ⇒ Oc′,1
+aηs,a′ηs′ ∝ Is,s′
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′,2
+aηs,a′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′,2
+aηs,a′ηs′ ⇒ Oc′,2
+aηs,a′ηs′ ∝ Is,s′
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′′
+aηs,a′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′′
+aηs,a′ηs′ ⇒ Oc′′
+aηs,a′ηs′ ∝ Is,s′
+(S56)
+where I is an 2 × 2 identical matrix. In addition, V1, V2, V3, V4 are invariant under SU(2)η transformation.
+From Eq. S55 and Eq. S56, the density matrices Of, Oc′,1, Oc′′ are diagonalized in valley and spin incdies. We then introduce
+2 × 2 matrices, of,η, oc′,1,η, oc′,2,η, oc′′,η, such that
+Of
+αηs,α′η′s′ = of,η
+α,α′δη,η′δs,s′,
+Oc′,1
+aηs,a′η′s′ = oc′,1,η
+a,a′ δη,η′δs,s′,
+Oc′,2
+aηs,a′η′s′ = oc′,2,η
+a,a′ δη,η′δs,s′,
+Oc′′
+aηs,a′η′s′ = oc′′,η
+a,a′ δη,η′δs,s′
+(S57)
+We now consider the effect of discrete symmetries in Eq. S30. Using Eq. S31 and Eq. S57, we find
+T :
+(of,η)∗ = of,−η,
+(oc′,1,η)∗ = oc′,1,−η,
+(oc′,2,η)∗ = oc′,2,−η,
+(oc′′,η)∗ = oc′′,−η
+C3z :
+ei 2πη
+3 σzof,ηe−i 2πη
+3 σz = of,η,
+ei 2πη
+3 σzoc′,1,ηe−i 2πη
+3 σz = oc′,1,η,
+ei η2π
+3 oc′,2,ηe−i η2π
+2 σz = oc′,2,η,
+oc′′,η = oc′′,η
+C2x :
+σxof,ησx = of,η,
+σxoc′,1,ησx = oc′,1,η,
+σxoc′,2,ησx = oc′,2,η,
+σxoc′′,ησx = oc′′,η
+C2zT :
+(σxof,ησx)∗ = of,η,
+(σxoc′,1,ησx)∗ = oc′,1,η,
+(σxoc′,2,ησx)∗ = −oc′,2,−η,
+(σxoc′′,ησx)∗ = oc′′,η
+(S58)
+
+23
+From the definition (Eq. S38), Of, Oc′,1, Oc′′ are Hermitian matrices and then of, oc′,1, oc′′ are also Hermitian matrices. Com-
+bining Eq. S58 and the Hermitian properties, we can introduce real numbers χf
+0, χc′,1
+0
+, χc′′
+0 , χc′′
+1
+and then of, oc′, oc′′ take the
+following structure
+of,η = of,−η = χf
+0σ0,
+oc′,1,η = oc′,−η = χc′,1
+0
+σ0,
+oc′,2,η = 0,
+oc′′,η = χc′′
+0 σ0 + χc′′
+1 σx
+(S59)
+where σ0, σx,y,z are identity and Pauli matrices respectively with row and column indices α = 1, 2. Since the filling of f- and
+Γ1 ⊕ Γ2 c- electrons are νf = Tr[Of], νc′′ = Tr[Oc′′] respectively, we find χf
+0 = νf/8, χc′′
+0
+= νc′′/8. Using Eq. S59 and
+Eq. S57, for the symmetric solution, we finally have
+Of
+αηs,α′η′s′ = δα,α′δη,η′δs,s′νf/8
+Oc′,1
+aηs,a′η′s′ = δa,a′δη,η′δs,s′χc′,1
+0
+,
+Oc′,2
+aηs,a′η′s′ = 0,
+a, a′ ∈ {1, 2}
+Oc′′
+aηs,a′η′s′ = δη,η′δs,s′(δa,a′νc′′/8 + δa,3−a′χc′′
+x ),
+a, a′ ∈ {1.2}
+(S60)
+As for the hybridization fields, we find discrete symmetries will not impose constraints on V1, V2. As for V3, V4, we have
+T : V3 = V ∗
+4 ;
+C3z : V3 = ei2π/3V3,
+V4 = ei2π/3V4
+C2x : V3 = V4;
+C2zT : V3 = V ∗
+4
+(S61)
+therefore
+V3 = V4 = 0
+(S62)
+In summary, instead of solving self-consistent equations of Of, Oc′,1, Oc′′, V3, V4 (Eq. S38, Eq. S44, Eq. S48), we can use
+νf =
+1
+NM
+�
+R,αηs
+⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩
+χc′,1
+0
+=
+1
+8NM
+�
+|k|<Λc,a=1,2,ηs
+⟨Ψ|e−|k|2λ2 : c†
+k,aηsck,aηs : |Ψ⟩
+νc′′ =
+1
+NM
+�
+|k|<Λc,a=3,4,ηs
+⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩
+χc′′
+1 =
+1
+8NM
+�
+|k|<Λc,ηs
+⟨Ψ|c†
+k,3ηsck,4ηs + c†
+k,4ηsck,3ηs|Ψ⟩
+V3 = V4 = 0
+(S63)
+and obtain Of, Oc, Oc′′ via Eq. S60. We note that the first equation in Eq. S63 is equivalent to Eq. S52. In summary, combining
+Eq. S35, Eq. S53 and Eq. S63, we have a complete set of mean-field self-consistent equations for the symmetric Kondo state.
+We comment that Eq. S63 are the same mean-field equations as we derived in Sec. S4 A, Sec. S4 B and Sec. S4 C, but with
+additional symmetry requirement, that is the ground states satisfy all symmetries.
+We mention that, at ν = νf = νc = 0, we have Of
+αηs,α′η′s′ = 0, Oc′
+αηs,α′η′s′ = 0 and the Hartree term in Eq. S41 vanishes.
+We now prove the Hartree term in Eq. S50 also vanishes. We note the only non-zero components of Oc′′ are Oc′′
+1ηs,2ηs, Oc′′
+2ηs,1ηs.
+From Eq. S49, the Hartree term takes the form of (with Of = 0)
+−
+J
+2NM
+�
+Rs1s2
+�
+αα′ηη′
+�
+|k1|,|k2|<Λc
+ei(k1−k2)·R(ηη′ + (−1)α+α′)
+�
+: f †
+Rαηs1fRα′η′s2 : Oc′′
+α′η′s2,αηs1
+�
+= −
+J
+2NM
+�
+Rs
+�
+αη
+�
+|k1|,|k2|<Λc
+ei(k1−k2)·R(ηη + (−1)α+3−α)
+�
+: f †
+RαηsfRα′ηs : Oc′′
+3−αηs,αηs
+�
+= −
+J
+2NM
+�
+Rs
+�
+αη
+�
+|k1|,|k2|<Λc
+ei(k1−k2)·R(0)
+�
+: f †
+RαηsfRα′ηs : Oc′′
+3−αηs,αηs
+�
+=0
+(S64)
+and hence vanishes. In summary, at ν = 0, we only need to consider V1, V2, and other mean fields vanish.
+
+24
+E.
+Properties of the symmetric Kondo state
+We solve the self-consistent equations Eq. S35, Eq. S44, Eq. S52, Eq. S53 and Eq. S63 at integer filling ν = 0, −1, −2
+with νf
+= ν, νc = 0.
+We identify the symmetric Kondo (SK) states at ν = 0, −1, −2 which are characterized by
+V1 ̸= 0 (|γ2V1/Dνf ,νc| = 95meV, 111meV, 209meV at ν = 0, −1, −2 respectively), V2 ̸= 0 (|v′
+⋆γV2/Dνf ,νc| =
+80meV, 97meV, 197meV at ν = 0, −1, −2 respectively) and V3 = 0, V4 = 0. Even if we allow non-zero V3, V4 and ini-
+tialize the mean-field calculations with non-zero V3, V4, V3, V4, we still find V3 = V4 = 0 after self-consistent calculations
+(amplitudes smaller than 10−5). This is because ˆHJ describes ferromagnetic interactions and disfavors the development of non-
+zero V3, V4. We also comment that the non-zero V1, V2, introduce an effective f-c hybridization (Eq. S37) and characterize the
+Kondo physics.
+We next discuss the topological feature of the bands. Since the SK states preserve all the symmetries, it is sufficient to only
+consider the bands in valley + and spin ↑. We find at ν = 0, −1, −2, the representations formed by flat bands at Γ, K, M are
+Γ1 ⊕ Γ2, K2K3, M1 ⊕ M2 respectively. We note that the representations formed by flat bands here are equivalent to that of the
+non-interacting THF model. Thus, the flat bands form a fragile topology at ν = −1, −2 and a stable topology at ν = 0 due to
+the additional particle-hole symmetry at ν = 0 [1, 85].
+In addition, we also calculate the Wilson loop of the flat bands (valley + spin ↑). In the calculation of Wilson loop, we
+let k1 ∈ { i
+N }i=1,...,N−1, k2 = { j
+N }j=1,...,N−1 and k1 ∈ { i
+N }i=1,...,N−1, k2 = { j
+N }j=1,...,N−1 and k = k1bM,1 + k2bM,2,
+where bM,1 =
+4π
+3aM (
+√
+3, 0), bM,2 =
+4π
+3aM (
+√
+3
+2 , 3
+2) are two moir´e reciprocal lattice vectors and aM is the moir´e lattice constant.
+We then let |un,k⟩ denote the n-th eigenvectors of the single-particle Hamiltonian H(k) (of valley + spin ↑). We focus on
+the subset of the bands, which we denote with band indices n = 1, .., nband. Here, we take the flat bands as the subset of
+the bands that we are interested in. We then define the matrix Uk as a matrix formed by the eigenvectors of the flat bands
+Uk = [|u1,k⟩, |u2,k⟩, ..., |unband,k⟩]. The Wilson loop [85] along the k2 direction is defined as
+W(k1) = U †
+k1,k2=0
+N−1
+�
+j=1
+�
+Uk1,k2= 2πj
+N U †
+k1,k2= 2π(j+1)
+N
+�
+V (k1=0,k2=2π)Uk1,k2=2π .
+(S65)
+where V G is defined as H(k+G) = V GH(k)V G,†, G = nbM,1 +mbM,2, n, m ∈ Z. We mention that c-electrons are defined
+in the momentum space that can be larger than the first MBZ (depending on the momentum cutoff Λc). Thus we introduce
+V G that maps ck to ck+G to restore the periodic condition H(k + G) = V GH(k)V G,†. The corresponding Wilson loop
+Hamiltonian [85] is
+H(k1) = −i ln(W(k1))
+(S66)
+We plot the Wilson loop spectrum (eigenvalues of H(k1)) in Fig. S5, where we observe the Wilson loop has winding number 1.
+As shown in Ref. [85], in the presence of additional particle-hole symmetry at ν = 0 [1, 85], (−1)n with n the winding number
+of Wilson loop is a stable topological index. We conclude that at ν = 0, the symmetric Kondo state has a stable topology that is
+characterized by the odd winding number of the Wilson loop.
+From Fig. S5, we observe the behaviors of the Wilson loop are similar at different fillings. We check the overlapping of the
+flat-band wavefunctions between different bands. We let {|uν
+i,k⟩}i=1,...,nband denote the wavefunction of flat bands at filling ν.
+We define the overlapping between wavefunctions at fillings ν and ν′ as
+Overlap(ν, ν′) = 1
+N
+�
+k
+�
+i,j∈{1,...,nband}
+⟨uν
+i,k|uν′
+j,k⟩⟨uν′
+j,k|uν
+i,k⟩
+(S67)
+We find Overlap(0, −1) = 99.1%, Overlap(−1, −2) = 91.6%. The large overlapping of wavefunctions between different
+fillings indicates similar behaviors of the Wilson loop at different fillings as we showed in Fig. S5.
+Finally, we analyze the mean-field Hamiltonian of the symmetric Kondo state. The mean-field single-particle Hamiltonian of
+valley η and spin ↑ (spin ↑ and spin ↓ are equivalent) of the Kondo symmetric state can be approximately written as
+˜H(η)(k) =
+� ˜H(f,η)(k)
+˜H(fc,η)(k)
+˜H(fc,η),†(k)
+˜H(c,η)(k)
+�
+(S68)
+˜H(f,η)(k) = EfI2×2
+˜H(c,η)(k) =
+�
+EcI2×2
+v⋆(ηkxσ0 + ikyσz)
+v⋆(ηkxσ0 − ikyσz)
+Ec′′I2×2
+�
+˜H(fc,η)(k) =
+�
+˜γσ0 + ˜v′⋆(ηkxσx + kyσy) 02×2
+�
+(S69)
+
+25
+FIG. S5. Wilson loop spectrum of flat bands of SK states at ν = 0, −1, −2.
+where ˜H(f,η), ˜H(c,η), ˜H(fc,η) denote the single-particle Hamiltonian of the f-block, c-block and fc-block respectively.
+Ef, Ec, Ec′′ denote the energy shifting induced by the Hartree term, one-body scattering term ˆHcc and chemical potential.
+Ef, Ec can be k dependent and we only keep its k-independent part which makes dominant contributions. Ec′′ comes from
+the Hartree contribution of ˆHJ (Eq. S49, which is relatively small and we set Ec′′ = 0. We also set M = 0, since it is small
+compared to the other parameters. ˜γ, ˜v⋆
+′ denote the renormalized f-c hybridization emerged from Kondo interactions (Eq. S37).
+We also drop the damping factor e−|k|2λ2/2 to simplify the analysis. In practice, we find |˜γ| =
+1
+Dνc,νf |γ2V ∗
+1 + γv′
+⋆V ∗
+2 | ≈
+175meV, 209meV, 406meV. In the chiral limit v′
+⋆ = 0, we also have ˜v′⋆ = 0 ( ˜v′⋆ = v′
+⋆ = γv′
+⋆V ∗
+1 /Dνc,νf ). We note that
+|˜v′
+⋆||k| can reach a similar amplitude as the k-independent hybridization |˜γ|. However, we expect in most regions of MBZ, |˜γ|
+makes the dominant contribution. We, therefore, drop the set ˜v′⋆ = 0 or equivalently v′
+⋆ = 0 as an approximation. By setting
+v′
+⋆ = 0, we can further separate ˜H(η)(k) into two blocks. The first block corresponds to the row and column indices 1, 3, 5
+with electron operators,fk,1ηs, ck,1ηs, ck,3ηs. The second block corresponds to the row and column indices 2, 4, 6 with electron
+operators,fk,2ηs, ck,2ηs, ck,4ηs. We focus on the first block whose single-particle Hamiltonian is
+˜h(η)(k) =
+�
+�
+Ef
+˜γ
+0
+˜γ
+Ec
+v⋆(ηkx + iky)
+0
+v⋆(ηkx − iky)
+0
+�
+�
+(S70)
+We next analyze the eigensystems of ˜h(η)(k). We note that ˜γ provides the largest energy scales near ΓM point and will gap
+out f- and Γ3 c-electrons. To observe this, we first consider the first 2 × 2 block of ˜h(η)(k) which describes the single-particle
+Hamiltonian of f- and Γ3 c-electrons
+�
+Ef
+˜γ
+˜γ
+Ec
+�
+(S71)
+The eigenvalues and eigenvectors are
+E1 = Ec + Ef
+2
+−
+�
+˜γ2 + (Ec − Ef)2
+4
+,
+E2 = Ec + Ef
+2
++
+�
+˜γ2 + (Ec − Ef)2
+4
+v1 =
+1
+�
+2Efc(Efc − E3)
+�E3 − Efc ˜γ�T ,
+v2 =
+1
+�
+2Efc(Efc + E3)
+�E3 + Efc ˜γ�T
+(S72)
+where E3 = Ef −Ec
+2
+, Efc =
+�
+˜γ2 + E2
+3. Since ˜γ is larger than Ec, Ef, Γ3 c-electrons and f-electrons are gapped out by the
+hybridization. Consequently, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons. Numerically, we indeed find the orbital
+weights of Γ1 ⊕ Γ2 c-electrons are large (71%, 77%, 89% at ν = 0, −1, −2 respectively).
+We next treat v⋆ perturbatively. We find the dispersion of the flat band becomes
+Eflat
+k
+≈ −Ef|k|2(v⋆)2
+E1E2
+= Ef|k|2(v⋆)2
+˜γ2 − EcEf
+(S73)
+At ν = 0 with particle-hole symmetry, Ef = Ec = 0 and Eflat
+k
+≈ 0. However, at ν = −1, −2, where Ef ̸= 0, Ec ̸= 0, flat
+bands become dispersive. We observe that |Ec| is much smaller than |˜γ| at ν = −1, −2. Ef increases as we change from ν = 0
+to ν = −2, because we are doping more holes to the f-orbitals. At ν = −2, Ef can reach ∼ 0.5|˜γ|, but at ν = −1, Ef ∼ 0.1|˜γ|.
+Approximately, the dispersion of the flat band is Eflat
+k
+≈ (v⋆)2Ef/˜γ2. At ν = −1, we have Ek ≈ 13meV · ˚A2|k|2 and, at
+
+26
+ν = −2, we have Ek ≈ 45meV · ˚A2|k|2. This indicates a larger dispersion at ν = −2, which is consistent with our numerical
+result shown in the main text Fig.1.
+We next analyze the wavefunctions of the flat bands. The corresponding electron operator of the flat band d†
+flat,k is
+d†
+flat,k ≈ 1
+Ak
+�
+c†
+k,3ηs + v⋆(ηkx − iky)
+EcEf − ˜γ2
+�
+− Efc†
+k,1ηs + ˜γf †
+k,1ηs
+��
+(S74)
+where the normalization factor
+Ak =
+�
+1 +
+|v⋆|2|k|2(E2
+f + ˜γ2)
+(EcEf − ˜γ2)2
+(S75)
+We observe that in the large |˜γ| limit, the flat bands are mostly formed by Γ1 ⊕ Γ2 c-electrons (c†
+k,3ηs). We also provide the
+Berry curvature derived from the wavefunction in Eq. S74
+Ω(k) =
+−2(EcEf − ˜γ2)2(E2
+f + ˜γ2)v2
+⋆
+�
+(EcEf − ˜γ2)2 + (E2
+f + ˜γ2)v2⋆|k|2
+�2
+(S76)
+We next calculate the Wilson loop from the wavefunction in Eq. S74. The wavefunctions of d†
+flat,k is
+u(k) = 1
+Ak
+�
+1
+v⋆ηkx−iky
+EcEf −˜γ2 (−Ef)
+v⋆ηkx−iky
+EcEf −˜γ2 ˜γ
+�T
+(S77)
+where the first, second and third rows denote c†
+k,3ηs, f †
+k,1ηs, c†
+k,1ηs respectively. We then parametrize the momentum as
+k = x1aMbM,1 + x2aMbM,2,
+bM,1 =
+4π
+3aM
+(
+√
+3, 0),
+bM,2 =
+4π
+3aM
+(
+√
+3
+2 , 3
+2)
+(S78)
+x1, x2 ∈ [−1
+2, 1
+2] 1
+aM
+(S79)
+and define |u(x1, x2)⟩ as |u(k)⟩ with k = x1aMbM,1 + x2aMbM,2. The Wilson loop can be written as
+W(x1) =
+N−1
+�
+j=0
+⟨u(x1, x2 = x2,i)|u(x1, x2 = x2,j+1⟩⟨u(x1, x2,N)|u(x1, x2,0)⟩,
+x2,i = −
+1
+2aM
++
+1
+aM
+i
+N
+(S80)
+The spectrum of the Wilson loop is
+N(x1) = −i ln(W(x1)) = −i
+�
+1
+2aM
+−
+1
+2aM
+⟨u(x1, x2)|∂x2|u(x1, x2)⟩dx2 − i ln(⟨u(x1, 1/(2aM))|u(x1, −1/(2aM)⟩)
+(S81)
+In the continuous limit with aM → 0, we find
+N(x1) = −i
+� ∞
+−∞
+⟨u(x1, x2)|∂x2|u(x1, x2)⟩dx2 − i ln(⟨u(x1, ∞)|u(x1, −∞⟩)
+(S82)
+Combining Eq. S77 and Eq. S82, we find
+N(x1) =π
+�
+1 +
+v2
+⋆x1
+�
+x2
+1v2⋆ + (˜γ2−EcEf )2
+E2
+f +˜γ2
+�
+(S83)
+Even though Eq. S82 is calculated from the perturbative wavefunction in Eq. S77, it qualitatively captures the behaviors of the
+Wilson loop shown in Fig. S5. We observe that N(−∞) = 0 and N(∞) = 2π, which indicates a 2π winding at ν = 0, −1, −2.
+We also mention that current calculations correspond to one of the two flat bands for each valley and each spin, because we only
+pick one block of the single-particle Hamiltonian as we discussed near Eq. S70. The other flat band can be derived in the same
+manner and has similar behaviors, since it has a similar single-particle Hamiltonian.
+
+27
+S5.
+MEAN-FIELD SOLUTIONS OF THE TOPOLOGICAL HEAVY-FERMION MODEL
+We now discuss the mean-field equations of topological heavy-fermion mode in Eq. S12. We use a similar Hartree-Fock ap-
+proximation as introduced in Ref. [1]. However, we decouple ˆHJ via Eq. S50. The mean-field expectation values we considered
+are
+Of
+αηs,α′η′s′ =
+1
+NM
+�
+R
+⟨Ψ| : f †
+R,αηsfR,α′η′s′ : |Ψ⟩
+Oc′
+aηs,a′η′s′ =
+1
+NM
+�
+|k|<Λc
+⟨Ψ| : c†
+k,aηsck,a′η′s′ : |Ψ⟩,
+a ∈ {1, 2}
+Oc′′
+aηs,a′η′s′ =
+1
+NM
+�
+|k|<Λc
+⟨Ψ| : c†
+k,a+2ηsck,a′+2η′s′ : |Ψ⟩,
+a, a′ ∈ {1, 2}
+Oc′f
+aηs,α′η′s′ =
+1
+√
+NN
+�
+|k|<Λc,R
+e−ik·R⟨Ψ|c†
+k,aηsfR,α′η′s′|Ψ⟩,
+a ∈ {1, 2}
+V3 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ1,η(−1)α+1
+NM
+√NM
+⟨Ψ|f †
+R,αηsck,α+2ηs|Ψ⟩
+V4 =
+�
+R,|k|<Λc
+�
+αη,s
+eik·Rδ−1,η(−1)α+1
+NM
+√NM
+⟨Ψ|ηf †
+R,αηsck,α+2ηs|Ψ⟩
+(S84)
+where Of, Oc′′, V3, V4 have also been used in the Kondo lattice mean-field calculations (Eq. S38, Eq. S48 and Eq. S44).
+In addition, THF model also has a chemical potential term ˆHµ (Eq. S21) and we determine µ by requiring the total filling of
+f- and c-electrons to be ν:
+ν = Tr[Of] + Tr[Oc′] + Tr[Oc′′]
+(S85)
+where we note that the filling of f-, Γ3 c- and Γ1 ⊕ Γ2 c-electrons are
+νf = Tr[Of],
+νc′ = Tr[Oc′],
+νc′′ = Tr[Oc′′] ,
+(S86)
+respectively.
+We discuss the difference and similarities between the mean-field equations of the KL model and that of the THF model.
+For the THF model, we introduce mean fields Of, Oc′, Oc′′, Oc′f, V3, V4 (Eq. S84) (for a generic state without enforcing any
+symmetries). As for KL model, we introduce mean fields V1, V2, Of, Oc′,1, Oc′,2, Oc′′, V3, V4 (Eq. S38, Eq. S35, Eq. S48,
+Eq. S44) (for a generic state without enforcing any symmetry).
+• For both models, Oc′′, Of, V3, V4 are part of mean fields and contribute the mean-field decoupling of ˆHJ (Eq. S50).
+• In the THF model, we introduce a chemical potential term µ that couples to both the f-electron density operators and c-
+electron density operator (Eq. S21. We enforce the total filling of f- and c-electrons to be ν by tuning µ. In the KL model,
+we introduce a Lagrangian multiplier λf (Eq. S51) that couples to f-electron density operators, and a chemical potential
+µc (Eq. S28) that couples to the c-electron density operators. We enforce the fillings of f-electrons and c-electrons to be
+νf and νc respectively by tuning λf and µc in the KL model.
+• We also mention that Oc′ in THF model (Eq. S84) and Oc′,1 in the KL model (Eq. S38) are different, where the latter one
+has included an additional damping factor e−|k|2λ2.
+• In the THF model, we do not need hybridization fields V1, V2 (Eq. S35), since both come from the decoupling of Kondo
+interactions that only appear in the KL model.
+A.
+Mean-field equations of fully symmetric state
+We next discuss the solution of the symmetric state The fully symmetric state is characterized by density matrices
+Of, Oc′, Oc′′, Oc′f and hybridization fields V3, V4 that satisfy all symmetries. The structures of Of, Oc′′ in the fully sym-
+metric state are given in Eq. S60. We also prove that V3 = V4 = 0 in a fully symmetric state (near Eq. S62). We now discuss
+
+28
+the symmetry properties of Oc′, Oc′f. From Eq. S32, a U(1)v symmetric solution satisfies
+Oc′
+aηs,α′η′s′ = Oc′
+aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′
+aηs,α−ηs′ = 0
+Oc′f
+aηs,α′η′s′ = Oc′f
+aηs,α′η′s′e−iθν(η−η′) ⇒ Oc′f
+aηs,α−ηs′ = 0
+(S87)
+Then, Oc′, Oc′f is block diagonalized in valley indices. We next consider a SU(2)η transformation acting on the valley η. It
+indicates
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′
+αηs,α′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′
+αηs2,α′ηs′
+2 ⇒ Oc′
+αηs,α′ηs′
+�
+s,s′
+[ei �
+µ θη
+µσµ]s2,sOc′f
+αηs,α′ηs′[ei �
+µ θη
+µσµ]s′,s′
+2 = Oc′f
+αηs2,α′ηs′
+2 ⇒ Oc′f
+αηs,α′ηs′
+(S88)
+Combining Eq. S87 and Eq. S88, we can introduce 2 × 2 matrices oc′,η, oc′f,η, such that
+Oc′
+αηs,α′η′s′ = oc′
+α,α′δs,s′δη,η′,
+Oc′f
+αηs,α′η′s′ = oc′f
+α,α′δs,s′δη,η′
+(S89)
+We now consider the effect of discrete symmetries in Eq. S30. Using Eq. S31 and Eq. S89, we find
+T :
+(oc′,η)∗ = oc′,−η,
+(oc′f,η)∗ = oc′f,−η
+C3z :
+ei2π/3ησzoc′,ηe−i2πη/3σz = oc′,η,
+ei2π/3ησzoc′f,ηe−i2πη/3σz = oc′f,η
+C2x :
+σxoc′,ησx = oc′,η,
+σxoc′f,ησx = oc′f,η
+C2zT :
+σx(oc′,η)∗σx = oc′,η,
+σx(oc′f,η)∗σx = oc′f,η
+(S90)
+Then we can introduce a single real number χc′
+0 , χc′f
+0
+to characterize the density matrices
+Oc′
+αηs,α′η′s′ = χc′
+0 δα,α′δη,η′,
+Oc′f
+αηs,α′η′s′ = χc′f
+0 δα,α′δη,η′δs,s′ .
+(S91)
+Since the filling of Γ3 c-electrons is νc′ = Tr[Oc′] = 8χc′
+0 , we let χc′
+0 = νc′/8. Therefore, instead of calculating the original
+density matrices in Eq. S84, we can calculate the following quantities
+νf =
+1
+NM
+�
+R,αηs
+⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩
+νc′ =
+1
+NM
+�
+|k|<Λc,a=1,2,ηs
+⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩
+νc′′ =
+1
+NM
+�
+|k|<Λc,a=3,4,ηs
+⟨Ψ| : c†
+k,aηsck,aηs : |Ψ⟩
+χc′f =
+1
+8NM
+√NM
+�
+|k|<Λc,R,αηs
+e−ik·R⟨Ψ|c†
+k,αηsfR,αηs|Ψ⟩
+(S92)
+and construct density matrices via Eq. S60 and Eq. S91. In addition, the filling constraints in Eq. S85 becomes
+ν = νf + νc′ + νc′′
+(S93)
+Combining Eq. S92 and Eq. S93, we have a complete set of the mean-field self-consistent equations of symmetric state.
+Here we discuss the differences and similarities between the symmetric solution in the KL model (Sec. S4 E) and the sym-
+metric solution in the THF model as introduced in this section.
+• Both Kondo symmetric (KS) state in KL model and the symmetric state in the THF model preserve all the symmetries.
+• The KS state is adiabatically connected to the symmetric state in the THF model.
+• The mean-field solutions are exact at N = ∞.
+
+29
+FIG. S6. Wilson loop spectrum of symmetric state in the THF model at ν = 0, −1, −2
+• To obtain a more precise description of the Kondo state, we need to introduce a Gutzwiller projector to our symmetric-
+state wavefunction in the THF model. The Gutzwiller projector will suppress the charge fluctuations of f-electrons and is
+expected to further lower the energy of the symmetric state.
+• We also comment that, in the THF model, the flat bands are mainly formed by f-electrons with f-electron orbital weights
+80%, 85% and 87% at ν = 0, −1, −2 respectively. However, in the KL model, the flat bands are mainly formed by
+Γ1 ⊕Γ2 electrons as discussed in Sec. S4 E. This is because, in the KL model, we observed an enhanced f-c hybridization
+driven by Kondo interactions which is absent in the symmetric state solution of THF model. We expect the enhanced
+hybridization will be recovered after introducing the Gutzwiller projector.
+• We find the spectrum of the Wilson loop in the symmetric state of the THF model has winding number one and three
+crossing points(Fig. S6). However, in the SK state of the KL model, the spectrum of the Wilson loop spectrum has
+winding number one and one crossing point (Fig. S5). However, at ν = 0, for both the THF model and the KL model, the
+symmetric state has a stable topology with an odd Winding number of Wilson loop spectrum [85]. We also mention the
+difference in the Wilson loop spectrum comes from the absence of enhanced f-c hybridization in the THF model.
+B.
+Mean-field equations of the symmetric state in the presence of strain
+We now discuss the mean-field solution of the symmetric state in the presence of strain. We add the following term to the
+Hamiltonian (Eq. S12)
+ˆHstrain = α
+�
+R,ηs
+(f †
+R,1ηsfR,2ηs + h.c.)
+(S94)
+We note that ˆHstrain only breaks C3z symmetry. To show this, we rewrite the ˆHstrain as
+ˆHstrain =
+�
+R,αηs,α′η′s′
+αf †
+R,αηs[hstrain]αηs,α′η′s′fR,α′η′s′
+hstrain = σxτ0ς0
+(S95)
+where
+the
+matrix
+structure
+of
+ˆHstrain
+is
+denoted
+by
+hstrainσxτ0ς0.
+We
+now
+show
+it
+commutes
+with
+Df(T), Df(C2x), Df(C2zT), Df(gSU(2)η(θµ
+η )), Df(gU(1)v((θv)), Df(gU(1)c((θc)) (Eq. S31, Eq. S32) and hence ˆHstrain
+preserves all symmetries except for C3z
+[hstrain, Df(T)] = [σxτ0ς0, σ0τxε0] = 0
+[hstrain, Df(C2x)] = [σxτ0ς0, σxτ0ς0] = 0
+[hstrain, Df(C2zT)] = [σxτ0ς0, σxτ0ς0] = 0
+[hstrain, Df(gU(1)c((θc))] = [σxτ0ς0, e−iθcσ0τ0ς0] = 0
+[hstrain, Df(gU(1)v((θv))] = [σxτ0ς0, σ0e−iθvτzς0] = 0
+[hstrain, Df(gSU(2)η(θµ
+η ))] = [σxτ0ς0, σ0e−i �
+µ θη
+µ
+τ0+ητz
+4
+ςµ] = 0
+
+30
+In the presence of strain, the symmetric state is defined as the state that preserves all the symmetries except for C3z which
+is broken by the strain. In the presence of C3z-breaking strain, Eq. S57 and Eq. S89 still hold, because the system still has
+U(1)c × U(1)v × SU(2)+ × SU(2)− symmetry. As for Eq. S58 and Eq. S90, we only need to consider T, C2x, C2zT and we
+find
+Of
+αηs,α′η′s′ =
+�
+χf
+0σ0 + χf
+1σx
+�
+α,α′δη,η′δs,s′
+Oc′
+αηs,α′η′s′ =
+�
+χc′
+0 σ0 + χc′
+1 σx
+�
+α,α′
+δη,η′δs,s′,
+Oc′′
+αηs,α′η′s′ =
+�
+χc′′
+0 σ0 + χc′′
+1 σx
+�
+α,α′
+δη,η′δs,s′,
+Oc′f
+αηs,α′η′s′ =
+�
+χc′f
+0 σ0 + χc′f
+1 σx
+�
+α,α′δη,η′δs,s′
+(S96)
+where χf
+0, χf
+1, χc′
+0 , χc′
+1 , χc′′
+0 , χc′′
+1 , χc′f
+0 , χc′f
+1
+are real numbers tha characterize the density matrices. Combining Eq. S38, Eq. S48
+and Eq. S96, we find
+χf
+0 =
+1
+8NM
+�
+R,αetas
+⟨Ψ| : f †
+R,αηsfR,αηs : |Ψ⟩,
+χf
+1 =
+1
+8NM
+�
+R,ηs
+⟨Ψ|f †
+R,1ηsfR,2ηs + f †
+R,2ηsfR,1ηs|Ψ⟩
+χc′
+0 =
+1
+8NM
+�
+|k|<Λc,=1,2,ηs
+⟨Ψ| : c†
+k,aηsc†
+k,aηs : |Ψ⟩,
+χc′
+1 =
+1
+8NM
+�
+|k|<Λc,ηs
+⟨Ψ|c†
+k,1ηsck,2ηs + c†
+k,2ηsck,1ηs|Ψ⟩
+χc′′
+0 =
+1
+8NM
+�
+|k|<Λc,=3,4,ηs
+⟨Ψ| : c†
+k,aηsc†
+k,aηs : |Ψ⟩,
+χc′′
+1 =
+1
+8NM
+�
+|k|<Λc,ηs
+⟨Ψ|c†
+k,3ηsck,4ηs + c†
+k,4ηsck,3ηs|Ψ⟩
+χc′f
+0
+=
+1
+8NM
+√NM
+�
+|k|<Λc,R,αηs
+e−ik·R⟨Ψ|c†
+k,αηsfR,αηs|Ψ⟩,
+χc′f
+1
+=
+1
+8NM
+√NM
+�
+|k|<Λc,R,ηs
+e−ik·R⟨Ψ|c†
+k,1ηsfR,2ηs + c†
+k,2ηsfR,1ηs|Ψ⟩
+χc′′f
+0
+=
+1
+8NM
+√NM
+�
+|k|<Λc,R,αηs
+e−ik·R⟨Ψ|c†
+k,α+2ηsfR,αηs|Ψ⟩,
+χc′′f
+1
+=
+1
+8NM
+√NM
+�
+|k|<Λc,R,ηs
+e−ik·R⟨Ψ|c†
+k,3ηsfR,2ηs + c†
+k,4ηsfR,3ηs|Ψ⟩
+(S97)
+where we also have χf
+0 = νf/8, χc′
+0 = νc′/8, χc′′
+0 = νc′′/8. As for V3, V4, we only consider the T, C2x, CzT of Eq. S61, which
+indicates
+V3 = V4 = V ∗
+3 = V ∗
+4
+(S98)
+Combining Eq. S44, Eq. S93, Eq. S97 and Eq. S98 with filling constraints ν = νf + νc′ + νc′′, we have a complete set of the
+mean-field self-consistent equations of symmetric state in the presence of strain. We perform calculations with non-zero strain
+at ν = 0, −1, −2, −3. We initialize the calculations with the fully symmetric solutions derived at zero strain, and the procedure
+converges within 500 iterations. The results are illustrated and discussed in Sec. S5 D.
+C.
+Effect of doping
+We now discuss the effect of doping at zero strain. For hole doping at ν = 0, −1, −2, −3 and electron doping at ν = 0, we
+mainly dope electrons to the light bands that are mostly formed by c-electrons (Fig. S7). Consequently, the energy difference
+between the symmetric state and the ordered state decreases since we have more conduction c-electrons near the Fermi energy,
+and the system favors the symmetric state.
+We now point out the complexity of electron dopings at ν = −1, −2. Doping electrons at ν = −1, −2 is equivalent to dope
+electrons to the heavy bands that are mostly formed by f-electrons (Fig. S7). The heavy (flat) bands become closer to the Fermi
+energy, and hence, the energy cost of putting f-electrons into flat bands will be small. Then we can fill the heavy (flat) bands
+with a small energy cost. By filling the heavy (flat) bands, the type of orders formed by f-electrons can change a lot. To observe
+
+31
+(a)
+(b)
+FIG. S7. Dipsersions of KIVC state at ν = 0, KIVC+VP state at ν = −1, KIVC state at ν = −2 and VP state at ν = −3. The color represents
+the weight of f-(yellow) and c-(blue) electrons.
+FIG. S8. Evolution of order parameter as a function of doping at ν = 0, −1, −2
+the change of the ordered states, we consider the following order parameters
+Ox =
+1
+NM
+�
+R,αηs,α′η′s′
+f †
+R,αηs[ox]αηs,α′η′s′fR,α′η′s′,
+x ∈ {KIV C, Sz, Vz, Vy}
+oKIV C = σyτyς0,
+oSz = σ0τ0ςz,
+oVz = σ0τzς0,
+oVy = σ0τyς0
+(S99)
+We measure the expectation values of Ox=KIV C,Sz,Vz,Vy with respect to the ordered states at each filling. In Fig. S8, we show
+the evolution of ⟨Ox⟩ as a function of doping. We find for hole doping at ν = 0, −1, −2 and electron doping at ν = 0 (where
+carriers go to light bands in both cases), the system stays in the same ordered states (compared to the integer filling). However,
+for electron doping at ν = −1, −2, we can observe the changes of the order parameters. This is because we are mainly dope
+f-electrons for electron doping at ν = −1, −2. We thus conclude that electron doping at ν = −1, −2 will introduce sizeable
+changes of the order parameters. Both the change of order parameters and the doping effect will affect the energy competition
+between the symmetric state and the ordered state.
+Finally, we comment on the ν = −3 case. At ν = −3, the actual ground state might be a CDW state which breaks the
+translational symmetry [145] and is beyond our current consideration. In addition, at ν = −3, even for the valley polarized state
+we currently considered, electron doping is equivalent to doping both heavy and light bands (Fig. S7), which is different from
+ν = −1, −2. We leave the detailed study of ν = −3 for future study.
+D.
+Effect of strain
+We next analyze the effect of the strain. We first note that as we increase α (Eq. S95), the strain will gradually suppress the
+KIVC order (OKIV C, Eq. S99). This can be observed from
+{oKIV C, hstrain} = {σyτyς0, σxτ0ς0} = 0
+(S100)
+Heuristically, the anti-commuting nature indicates the competition between oKIV C and hstrain. Thus, as we increase hstrain,
+oKIV C will be suppressed. We also find the spin-polarization OSz and valley polarization OVz commute with hstrain
+[oSz, hstrain] = [σ0τ0ςz, σxτ0ς0] = 0,
+[oVz, hstrain] = [σ0τzς0, σxτ0ς0] = 0
+(S101)
+
+32
+Heuristically, this indicates the valley and spin polarization do not directly compete with hstrain. However, as we will show in
+this section, a sufficiently large strain could still destroy the valley and spin polarization in the THF model.
+For future convenience, we also introduce the eigenstates of the strain Hamiltonian ˆHstrain (Eq. S95)
+d†
+R,1ηs =
+1
+√
+2(f †
+R,1ηs − f †
+R,2ηs),
+d†
+R,2ηs =
+1
+√
+2(f †
+R,1ηs + f †
+R,2ηs)
+(S102)
+We will call d†
+R,1ηs and d†
+R,2ηs as d1 and d2 electrons (orbitals), respectively, for short. We mention that d1, d2 are f-electrons.
+The strain Hamiltonian can then be written as
+ˆHstrain = α
+�
+R,αηs
+(−d†
+R,1ηsdR,1ηs + d†
+R,2ηsdR,2ηs)
+(S103)
+Thus, for a positive strain amplitude α > 0, the energy of d1 electrons will be lowered and the energy of d2 electrons will be
+raised. We introduce ⟨hstrain⟩ to characterize the population imbalance between d1 and d2 electrons
+⟨hstrain⟩ = ⟨ 1
+NM
+�
+R,αηs,α′η′s′
+f †
+R,αηs[hstrain]αηs,α′η′s′fR,α′η′s′⟩ =
+1
+NM
+�
+R,αηs
+⟨d†
+R,1ηsdR,1ηs − d†
+R,2ηsdR,2ηs⟩
+(S104)
+In Fig. S9, we plot the evolution of various order parameters and also |⟨hstrain⟩| where the expectation value is taken with
+respect to the ordered state solution. In all cases, |⟨hstrain⟩| increases as we increase α, since α linearly coupled to hstrain
+term. The KIVC order will be suppressed and fully destroyed at sufficient strong strain at ν = 0, −1, −2. At ν = 0, after the
+destruction of the KIVC order, self-consistent calculation produces a symmetric ground state that only breaks C3z symmetry,
+even though we initialize the mean-field calculation with an ordered state.
+However, at ν = −1, −2, after the destruction of KIVC order, the spin polarization and valley polarization still exist. By
+further increasing the strain, the ordered states will finally become unstable (Fig. S9), which means the mean-field calculations
+that are initialized with ordered solutions converge to a symmetric state.
+We next analyze the transition from an ordered state to a symmetric state at a large strain at ν = −1, −2.
+1.
+ν = −1
+We first consider the ν = −1 with 4meV ≲ α ≲ 18meV. In this parameter region, the KIVC order is destroyed but valley
+and spin polarization persist (Fig. S9). In Fig. S10 (a) (b), we plot the band structures in this parameter region. We note that
+flat bands that are mostly formed by f-electrons (marked by red circles, Fig. S10) move towards the Fermi level, as we increase
+strain. Near the transition point to the symmetric state, the flat bands are very close to the Fermi level. This signals an instability
+of the ordered states since we can fill the flat band without any energy cost. By diagonalizing the mean-field Hamiltonian, we
+find the flat bands (marked by red circles, Fig. S10) correspond to d1 electrons (Eq. S103). By filling the flat bands, we have
+more populations in d1 orbitals, which increase |⟨hstrain⟩| (Eq. S104) and drive the system to a symmetric state.
+We now estimate the critical value of strain αc at which a transition from an ordered state to a symmetric state happens. At
+αc, the flat bands (marked by red circles, Fig. S10) are very close to the Fermi energy and induce the transition. To estimate αc,
+we calculate the excitation gap of the flat bands (marked by red circles, Fig. S10): ∆Eflat. Then we have
+∆Eflat
+����
+α≈αc
+= 0
+(S105)
+We estimate ∆Eflat using the zero-hybridization limit [2] of the model, where γ = 0, v′
+⋆ = 0 (Eq. S15). In addition, we also
+set J = 0 to simplify the calculation (Eq. S17). The zero-hybridization model with non-zero strains are
+ˆHzero-hyb = ˆHU + ˆHW + ˆHV + ˆHstain + ˆHc + ˆHµ
+(S106)
+where ˆHU, ˆHW , ˆHV , ˆHstrain, ˆHc, ˆHµ are defined in Eq. S16, Eq. S18, Eq. S19, Eq. S95, Eq. S13 and Eq. S12 respectively,
+and ˆHV are treated with mean-field methods. In the zero-hybridization model, the filling of f-electrons νf and c-electrons-νc
+are good quantum numbers. We solve the zero-hybridization model at fixed total filling ν with the assumption that the ground
+state does not break translational symmetry (fillings of f-electrons are uniform) [2]. To estimate the excitation gap of the flat
+bands, we calculate the energy cost of adding one dR,1ηs electron. We mention that, in our mean-field calculations with finite
+f-c hybridization, the relevant flat bands (marked by red circles in Fig. S10) correspond to d1 electrons. We let |Ψzero-hyb⟩ denote
+the ground state of the zero-hybridization model. The state with one-more dR,1ηs is
+|Ψexct
+zero-hyb⟩ = d†
+R,1ηs|Ψzero-hyb⟩ .
+(S107)
+
+33
+We next calculate
+∆Eflat = ⟨Ψexct
+zero-hyb| ˆHzero-hyb|Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHzero-hyb|Ψzero-hyb⟩
+(S108)
+The energy loss from Hubbard interaction term is
+∆EU = ⟨Ψexct
+zero-hyb| ˆHU|Ψexct
+U
+⟩ − ⟨Ψzero-hyb| ˆHU|Ψzero-hyb⟩ = U
+2 (νf + 1)2 − U
+2 ν2
+f = U(νf + 1
+2)
+The energy loss from ˆHW term is
+∆EW =⟨Ψexct
+zero-hyb| ˆHW |Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHW |Ψzero-hyb⟩
+=
+�
+a=1,2,3,4
+Waνc,a(νf + 1) −
+�
+a=1,2,3,4
+Waνc,aνf =
+�
+a=1,2,3,4
+Waνc,a
+(S109)
+where νc,a denotes the filling of c-electrons in orbital a. The energy change from ˆHV is
+∆EV = ⟨Ψexct
+zero-hyb| ˆHV |Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHV |Ψzero-hyb⟩ = 0
+(S110)
+The energy change from ˆHstrain is
+∆Estrain = ⟨Ψexct
+zero-hyb| ˆHstrain|Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHstrain|Ψzero-hyb⟩ = −α
+(S111)
+The energy change from chemical potential ˆHµ is
+∆Eµ = ⟨Ψexct
+zero-hyb| ˆHµ|Ψexct
+zero-hyb⟩ − ⟨Ψzero-hyb| ˆHµ|Ψzero-hyb⟩ − µ
+(S112)
+Then the excitation energy of adding one dR,1ηs electron is
+∆Eflat = ∆EU + ∆EW + ∆strain + ∆Eµ = U
+2 (νf + 1/2) +
+�
+a
+Waνc,a − µ − α
+(S113)
+We further take the following approximation: W1,2,3,4 = W = 47meV (the difference between W1,2,3,4 is about 15%). Then
+∆Eflat ≈ U
+2 (νf + 1/2) + Wνc − µ − α
+(S114)
+At ν = −1 and 0meV ≤ α ≤ 43meV, the ground state of the zero-hybridization model has νf = −1, νc = ν − νf = 0
+(Fig. S11). Then
+∆Eflat = −U
+2 − µ − α
+(S115)
+We next determine chemical potential µ. Chemical potential µ is determined by requiring the c-electrons filling to be νc = 0.
+The single-particle Hamiltonian of c-electron in the zero-hybridization limit takes the form of
+ˆHc,zero-hyb = ˆHc +
+�
+k,aηs
+(Wνf + V (0)
+Ω0
+νc − µ)c†
+k,aηsck,aηs
+(S116)
+where we have set W1,2,3,4 = W. We note that when
+Wνf − V (0)
+Ω0
+νc − µ = 0
+(S117)
+ˆHc,zero-hyb = ˆHc and we have νc = 0. Therefore,
+µ = Wνf + V (0)
+Ω0
+νc = −W
+(S118)
+
+34
+where we take νc = 0, νf = −1 (Fig. S11). Using Eq. S115 and Eq. S118, we find
+∆Eflat = W − U
+2 − α
+(S119)
+Then the flat bands reach Fermi energy when ∆Eflat = 0, which leads to
+∆Eflat = 0 ⇒ αc = W − U
+2 = 18meV
+(S120)
+which is close to the value (also around α = 18meV as shown in Fig. S9 (b)) from self-consistent calculations of the finite-
+hybridization model. Here, the finite-hybridization model refers to the original THF model with finite γ, v′
+⋆. Therefore, we
+conclude the transition from an ordered state to a symmetric state happens at α = αc ≈ 18meV at ν = −1.
+We also discuss the solutions of the zero-hybridization model here. In Fig. S11 (a), we show the ground state properties of the
+zero-hybridization model at various strains and ν = −1, where
+νf
+1 =
+1
+NM
+�
+R,ηs
+: d†
+R,1ηsdR,1ηs :,
+νf
+2 =
+1
+NM
+�
+R,ηs
+: d†
+R,2ηsdR,2ηs :
+(S121)
+denotes the filling of d1 and d2 electrons respectively with νf = νf
+1 + νf
+2 . We find a transition happens at α ≈ 25meV.
+We note that this transition is described by filling one more dR,1ηs electrons at each site. After the transition, there will be
+4 f-electrons filling d1 orbitals, and zero f-electrons filling the d1 orbitals. Thus for d1 orbitals, all the valleys and spins are
+filled, but for d2 orbitals all the valleys and spins are empty. Therefore, there is no room to develop order and the ground state
+is a symmetric state. We note that the transition in the zero-hybridization limit and the transition in the finite-hybridization
+model (at ν = −1, α ≈ 16meV, Fig. S9) share the same origin. They are both driven by filling electrons in d1 orbitals (in the
+finite-hybridization model, we fill the flat bands) and, after the transition, both ground states are symmetric. Thus, the results
+between zero-hybridization and finite-hybridization models are consistent. However, the critical values αc for the two models
+are different, since we have finite f-c hybridization in the finite-hybridization model.
+2.
+ν = −2
+We next discuss the transition from an ordered state to a symmetric state at ν = −2. We focus on the parameter region
+10meV ≲ α ≲ 45meV, where the KIVC order is destroyed but valley and spin polarization exist (Fig. S9). In Fig. S10 (c) (d),
+we plot the band structures in this parameter region. As we increase strain, we note that flat bands (marked with red circles,
+Fig. S10), move towards the Fermi level. Similar to the ν = −1 case, when the flat bands reach the Fermi energy, a transition to
+the symmetric state happens. However, at ν = −2, we need a much larger strain to destroy the ordered state as shown in Fig. S9
+(d). To understand this, we start from the zero-hybridization limit of the model (Eq. S106). As shown in Eq. S114, the excitation
+energy of the relevant flat bands (marked by red circles in Fig. S10)
+∆Eflat = U
+2 (νf + 1/2) + Wνc − µ − α
+(S122)
+By solving the zero-hybridization model, we find the ground states have νf = −1 and νc = −1 in the parameter region we
+focused 4meV ≲ α ≲ 18meV, as shown in Fig. S11. Then
+∆Eflat = −U
+2 − W − µ − α
+(S123)
+We now calculate the chemical potential. µ is determined by requiring the filling of c-electrons to be νc = −1. The single-
+particle Hamiltonian of c-electron in the zero-hybridization limit (Eq. S116) takes the form of
+ˆHc,zero-hyb = ˆHc +
+�
+k,aηs
+(Wνf + V (0)
+Ω0
+νc − µ)c†
+k,aηsck,aηs
+(S124)
+where we have set W1,2,3,4 = W, and take the mean-field treatment of ˆHV (Eq. S20). At M = 0 limit (M = 3.697meV, which
+is relatively small), the dispersion of c-electrons are Ek = ±v⋆|k| − Ec, where we define
+Ec = Wνf + V (0)
+Ω0
+νc − µ
+(S125)
+
+35
+Then all the c-states with energy smaller than 0 will be filled. The corresponding Fermi momentum kF is
+|v⋆kF | = Ec ⇒ kF =
+1
+|v⋆|Ec
+(S126)
+Then the filling of c-electrons is
+νc = −
+8
+AMBZ
+�
+|k|<kF
+dkxdky = − 8πk2
+F
+AMBZ
+= −
+8πE2
+c
+AMBZ|v⋆|2
+(S127)
+where the prefactor 8 comes from the 8-fold degeneracy of the bands (2 spins, 2 valleys, and 2 orbitals) and AMBZ is the area
+of MBZ. We require νc = −1 which indicates
+νc = −
+8πE2
+c
+AMBZ|v⋆|2 = −1 ⇒ Ec =
+�
+AMBZ|v⋆|2
+8π
+(S128)
+Using Eq. S125 and Eq. S128, we find
+µ = −W − V (0)
+Ω0
+−
+�
+AMBZ|v⋆|2
+8π
+(S129)
+where we take νf = −1, νc = −1. Combining Eq. S123 and Eq. S128, we find
+∆Eflat = −U
+2 + V (0)
+Ω0
++
+�
+AMBZ|v⋆|2
+8π
+− α
+(S130)
+And the transition happens at
+∆Eflat = 0 ⇒ αc = −U
+2 + V (0)
+Ω0
++
+�
+AMBZ|v⋆|2
+8π
+= 62meV
+(S131)
+We note that our estimation of transition in Eq. S131 is based on the single-particle picture and has not included the effect
+of hybridization. Thus the estimated value is larger than the transition value α ≈ 45meV from mean-field self-consistent
+calculations with finite f-c hybridization.
+We also discuss the solution obtained from the zero-hybridization model at ν = −2. In Fig. S11 (b), we show the ground state
+properties of the zero-hybridization model at various strains and ν = −2. We observe two transitions, both characterize by the
+increments of νf
+1 . After the first transition α ∼ 15meV, we have three d1 electrons at each site. Then not all valleys and spins
+of d1 orbitals are filled. This gives the possibility of developing spin and valley polarized order after we turn on the ˆHJ and f-c
+hybridization. Indeed, from Fig. S9, we can observe the non-vanishing order at 15meV ≲ α ≲ 40meV. By further increasing
+strain, at α ≈ 50meV, we observe a second transition in the zero-hybridization model. After the second transition, we have
+νf
+1 = 2, νf
+2 = −2 and there is no room for f-electrons to develop order. Correspondingly, for the finite-hybridization model, we
+also observe a transition to the symmetric stat at α ≈ 45meV. Thus we conclude the consistency between zero-hybridization and
+finite-hybridization models. We also point out that, the transition value αc ≈ 62meV suggested from single-particle excitation
+in Eq. S131 is also larger than the transition value αc ≈ 50meV obtained by solving many-body wavefunction in the zero-
+hybridization model. This points out the limitation of the single-particle picture.
+Finally, we comment on the difference between ν = −1 and ν = −2. Eq. S131 and Eq. S120 indicate that a relatively large
+strain is required to destroy the ordered state at ν = −2 compared to ν = −1. This is because, near the transition point with
+α < αc, c-electrons have filling νc = 0 at ν = −1, but have filling νc = −1 at ν = −2 (Fig. S11). In other words, we dope
+more holes to the c-bands at ν = −2 compared to ν = −1. In order to dope more holes to c-bands, the chemical potential (or
+Fermi energy) needs to be increased. The increment of chemical potential also leads to a larger Fermi surface, which has been
+observed in the model with finite hybridization (Fig. S10), where the Fermi surface at ν = −2 is larger than the Fermi surface
+at ν = −1. In the zero-hybridization model, the change of chemical potential can be observed analytically from Eq. S118 and
+Eq. S129. We note that the excitation gap of the flat band (∆Eflat) is measured with respect to the Fermi energy (or chemical
+potential). Then at fixed α, we find the value of ∆Eflat at ν = −2 is much larger than its value at ν = −1 (Eq. S119 and
+Eq. S130) due to the larger chemical potential at ν = −2. Thus, at ν = −2, a much larger strain is needed to make ∆Eflat = 0
+and destroy the ordering.
+
+36
+(a)
+(b)
+(c)
+(d)
+FIG. S9. (a), (b), (c) Evolution of order parameters as a function of strain amplitude α at ν = 0, −1, −2. (d) Evolution of order parameters at
+ν = −2 with an extended parameter region 0meV ≤ α ≤ 55meV
+(a)
+(b)
+(c)
+(d)
+FIG. S10. Band structure at ν = −1 (a), (b) and ν = −2 (c), (d). Red circles mark the relevant flat bands.
+3.
+Discussions about ν = −3
+We now discuss ν = −3. We first comment that, at ν = −3, other low energy states that break translational symmetry exists
+even at zero strain [145]. Furthermore, even in the zero-hybridization limit at zero strain, ν = −3 is close to the transition point
+between νf = −3 and νf = −2 states [1]. These all point out the complexity of ν = −3 even in the absence of strain. Thus, we
+leave a more systematical analysis at ν = −3 for future study.
+E.
+Strain
+We now discuss the derivation of strain term in the Eq. S95. Here we will take a minimal model to capture the effect of strain,
+and leave a detailed analysis (based on Ref. [150]) for future study. We take the single-particle Hamiltonian of twisted bilayer
+(a)
+(b)
+FIG. S11. Evolution of filling as a function of the strain in the zero-hybridization model at ν = −1, −2.
+
+37
+graphene at valley η and with non-zero strain [148]
+Hη(k) =
+�Hη
+1 (k)
+U
+U †
+Hη
+2 (k)
+�
+(S132)
+where Hl(k) is a 2 × 2 matrix that characterizes the Hamiltonian of l-layer graphene. U is the interlayer coupling matrix. In
+real space, U takes the form of
+U =
+3
+�
+j=1
+Ujeiηqj·r,
+Uj =
+�
+w0
+w1e−iη2(j−1)π/3
+w1eiη2(j−1)π/3
+w0
+�
+with w0, w1 are two constant parameters that characterize the interlayer couplings. Hl(k) is given by
+Hη
+l=1,2(k) = −ℏvF
+��
+R(∓θ) + El
+�−1
+(k + e
+ℏAl,η)
+�
+· ση
+(S133)
+where ση = (ησx, σy) , R(θ) is a rotation matrix, θ is the twist angle and [148]
+El =
+�
+ϵ(l)
+xx
+−Ω(l) + ϵ(l)
+xy
+−Ω(l) + ϵ(l)
+xy
+ϵ(l)
+yy
+�
+Al,η = η 3βγ0
+2evF
+�
+ϵ(l)
+−
+ϵ(l)
+xy
+�
+,
+γ0 = 2.7eV,
+β ≈ 3.14
+ϵ(l)
+± = 1
+2(ϵ(l)
+xx ± ϵ(l)
+yy) .
+(S134)
+ϵ(l)
+xx, ϵ(l)
+yy are normal strains along x and y directions respectively, ϵ(l)
+xy is the sheer strain. Ω(l) denotes a rotation from the
+twist angles [148]. Since the flat band is sensitive to ϵ− = (ϵ(1)
+− − ϵ(2)
+− )/2 and ϵxy = (ϵ(1)
+xy − ϵ(2)
+xy )/2 [148], we consider the
+heterostrain with Ω(l) = 0, ϵ(l)
++ = 0, ϵ(l)
+− = (−1)l+1ϵ−, ϵ(l)
+xy = (−1)l+1ϵxy. We further focus on the sheer strain ϵxy and set the
+normal anisotropic strain to be zero ϵ− = 0. For small twist angle, we approximately have
+Hη
+l=1,2(k) ≈ − ℏvF
+��
+R(±θ) − El
+�
+(k + e
+ℏAl,η)
+�
+· ση
+≈ − ℏvF
+�
+R(±θ)k − Elk + e
+ℏAl,η
+�
+· ση
+(S135)
+The effect of strain is then captured by the following single particle Hamiltonian of valley η
+Hη
+strain(k) = Hη
+strain,1 + Hη
+strain,2
+Hη
+strain,1 = −3βγ0η
+2
+ϵxy
+�
+σy
+−σy
+�
+,
+Hη
+strain,2 = ℏvF ϵxy
+�
+ηkyσx + kxσy
+−ηkyσx − kxσy
+�
+(S136)
+We next project the Hη
+strain(k) to the Wannier basis of f-electrons. We take the following wavefunction of f electrons at R
+with orbital α, valley η and spin s [1]
+|WR,α=1,η,s⟩ =
+�
+2πλ2
+0
+Ωtot
+�
+l=±
+�
+k
+�
+Q∈Qlη
+ei π
+4 lη−ik·R− 1
+2 λ2
+0(k−Q)2|k, Q, 1, η, s⟩
+|WR,α=2,η,s⟩ =
+�
+2πλ2
+0
+Ωtot
+�
+l=±
+�
+k
+�
+Q∈Qlη
+e−i π
+4 lη−ik·R− 1
+2 λ2
+0(k−Q)2|k, Q, 2, η, s⟩
+(S137)
+where λ0 = 0.1aM, Ωtot is the total area of the sample and Qlη = {±q1 + n1bM,1 + n2bM,2|n1, n2 ∈ Z}, q1 = kθ(0, −1) [1].
+From Eq. S136 and Eq. S137, we can calculate
+hstrain
+αηs,α2η2s2 =
+1
+NM
+�
+R
+⟨WR,α,η,s|
+�
+k,α′l′,α′
+2l′
+2,η′,s′
+ψ†
+k,l,α′,η′,s′[Hη′
+strain(k)]l′α′,l′
+2α′
+2ψk,l′
+2α′,η′,s′|WR,α2,η2,s2⟩
+(S138)
+
+38
+where ψk,l,α,η,s is the electron operator of the original electron basis with momentum k, layer l, sublattice α, valley η and spin
+s. Here, we take the average over all the moir´e unit cells (sum over R), and thus we only keep the momentum-independent (in
+f-electron basis) contributions. We leave the momentum-dependent term for future studies. In the f-basis, the strain can be
+characterized by
+ˆHstrain =
+�
+R,αηs,α2η2s2
+hstrain
+αηs,α2η2s2f †
+R,αηsfR,α2η2s2
+(S139)
+We next evaluate hstrain from Eq. S138. We separate hstrain into two parts (Eq. S136)
+hstrain,1
+αηs,α2η2s2 =
+1
+NM
+�
+R
+⟨WR,α,η,s|
+�
+k,α′l′,α′
+2l′
+2,η′,s′
+ψ†
+k,l,α′,η′,s′[Hη′
+strain,1(k)]l′α′,l′
+2α′
+2ψk,l′
+2α′,η′,s′|WR,α2,η2,s2⟩
+hstrain,2
+αηs,α2η2s2 =
+1
+NM
+�
+R
+⟨WR,α,η,s|
+�
+k,α′l′,α′
+2l′
+2,η′,s′
+ψ†
+k,l,α′,η′,s′[Hη′
+strain,2(k)]l′α′,l′
+2α′
+2ψk,l′
+2α′,η′,s′|WR,α2,η2,s2⟩
+(S140)
+For hstrain,1, using Eq. S136, Eq. S137 and Eq. S140, we find
+hstrain,1
+αηs,α2η2s2
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0
+Ωtot
+−3βγ0
+2
+ϵxy
+�
+l=±
+�
+Q∈Qlη
+�
+k
+i(−1)αlηe−λ2
+0(k−Q)2e−i πηl
+2 (−1)α+1
+=δη,η2δs,s2δα,3−α2
+3πλ2
+0βγ0
+Ωtot
+ϵxy
+�
+k
+� �
+Q∈Q+
+e−λ2
+0(k−Q)2 +
+�
+Q∈Q−
+e−λ2
+0(k−Q)2�
+(S141)
+In a more compact form, we find
+hstrain,1 = ασxτ0ς0
+α = 3πλ2
+0βγ0
+Ωtot
+ϵxy
+�
+k
+� �
+Q∈Q+
+e−λ2
+0(k−Q)2 +
+�
+Q∈Q−
+e−λ2
+0(k−Q)2�
+≈
+�
+1.3 × 104ϵxy
+�
+meV .
+(S142)
+We next calculate hstrain,2, using Eq. S136, Eq. S137 and Eq. S140, we find
+hstrain,2
+αηs,α2η2s2
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0ℏϵxy
+Ωtot
+�
+l=±
+�
+Q∈Qlη
+�
+k
+l
+�
+η(ky − Qy) + i(−1)α(kx − Qx)
+�
+ei π
+2 lη(−1)αe−λ2
+0(k−Q)2
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0ℏϵxy
+Ωtot
+�
+k
+� �
+Q∈Qη
+�
+η(ky − Qy) + i(−1)α(kx − Qx)
+�
+iη(−1)αe−λ2
+0(k−Q)2
++
+�
+Q∈Q−η
+�
+η(ky − Qy) + i(−1)α(kx − Qx)
+�
+iη(−1)αe−λ2
+0(k−Q)2�
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0ℏϵxy
+Ωtot
+�
+k
+�
+Q∈Q0
+�
+η(ky − Qy − ηq1,y) + i(−1)α(kx − Qx − ηq1,x)
+�
+iη(−1)αe−λ2
+0(k−Q−ηq1)2
++
+�
+η(ky − Qy + ηq1,y) + i(−1)α(kx − Qx + ηq1,x)
+�
+iη(−1)αe−λ2
+0(k−Q+ηq1)2�
+(S143)
+where we let Q0 = {n1bM,1 + n2bM,2|n1, n2 ∈ Z}. Since q1 = kθ(0, 1), q1,x = 0, q1,y = kθ, we find
+hstrain,2
+αηs,α2η2s2
+=δη,η2δs,s2δα,3−α2
+2πλ2
+0ℏϵxy
+Ωtot
+�
+k
+�
+Q∈Q0
+�
+i(−1)α(ky − Qy)f+(k − Q) − iq1,y(−1)αf−(k − Q)
+− (kx − Qx)ηf+(k − Q)
+(S144)
+
+39
+where we define
+f±(k − Q) = ±e−λ2
+0(k−Q+q1)2 + e−λ2
+0(k−Q−q1)2
+(S145)
+Since q1 = kθ(0, 1), f+(k − Q) is an even function of both (k − Q)x and (k − Q)y and f−(k − Q) is an even function of
+(k − Q)x and an odd function of (k − Q)y. Therefore,
+�
+k
+�
+Q∈Q0
+(ky − Qy)f+(k − Q) = 0,
+�
+k
+�
+Q∈Q0
+f−(k − Q) = 0,
+�
+k
+�
+Q∈Q0
+(kx − Qx)f+(k − Q) = 0
+(S146)
+Thus,
+hstrain,2
+αηs,α2η2s2 = 0
+(S147)
+However, we mention that, even though hstrain,2 = 0, Hη
+strain,2 in Eq. S136 can contribute a k-denpendent term in the THF
+model which has been omitted here.
+Combining Eq. S142 and Eq. S147, we have
+ˆHstrain =
+�
+R,αηs,α2η2s2
+ααηs,α2η2s2f †
+R,αηsfR,α2η2s2 =
+α
+�
+R,αηs,α2η2s2
+[hstrain]αηs,α2η2s2f †
+R,αηsfR,α2η2s2
+(S148)
+which is equivalent to the Eq. S95. α is connected to ϵxy via
+α = 3πλ2
+0βγ0
+Ωtot
+ϵxy
+�
+k
+� �
+Q∈Q+
+e−λ2
+0(k−Q)2 +
+�
+Q∈Q−
+e−λ2
+0(k−Q)2�
+≈
+�
+1.3 × 104ϵxy
+�
+meV .
+(S149)
+For a typical strain at the order of ϵxy ∼ 10−3 [148], α ∼ 13meV. We mention that the strain could also induce addition terms
+to the conduction c-electron blocks (c†c) and f-c hybridization blocks (c†f, f †c), which are expected to be small. Here, we
+only consider a minimal model, that is zeroth order in k, to capture the essential effect of the strain. We leave a comprehensive
+construction of strain term (based on Ref. [150]) for future study.
+S6.
+DYNAMICAL MEAN FIELD THEORY: IMPLEMENTATION
+We study dynamical effects of the f-f density-density term using DMFT. The c-f and c-c density terms are included in this
+model on the Hartree level. We neglect the f-c exchange interaction ˆHJ. We treat a Hamiltonian of the form ˆHc + ˆHfc + ˆHU +
+ˆHMF
+W
++ ˆHMF
+V
+where the first two terms compose the non-interacting THF model and the remaining terms are given by
+ˆHU =U
+2
+�
+(α,η,σ)̸=(α′,η′,σ′)
+f †
+α,η,σfα,η,σf †
+α′,η′,σ′fα′,η′,σ′
+− 3.5U
+�
+(α,η,σ)
+f †
+α,η,σfα,η,σ
+(S150)
+ˆHMF
+W
+=Wνf
+�
+(a,η,σ)
+c†
+a,η,σca,η,σ + Wνc
+�
+(α,η,σ)
+f †
+α,η,σfα,η,σ,
+(S151)
+ˆHMF
+V
+=V νc
+�
+(a,η,σ)
+c†
+a,η,σca,η,σ.
+(S152)
+Here, the last two terms are mean-field decoupled and therefore composed of single particle terms, and we have neglected all
+constant terms as an overall shift does not affect the observables we are interested in. The two particle term ˆHU is restricted to the
+f- subspace. We proceed with an LDA+DMFT-style calculation with a correlated subspace made up of f- electrons embedded
+in a larger space of f- and c- electrons. By allowing an arbitrary shift of the chemical potential µ and fixing the total filling
+ν = νf + νc, we can consider an effective interacting Hamiltonian restricted to the the f- subspace:
+ˆHeff
+I
+= ˆHff + ˆHDC
+(S153)
+ˆHff =U
+2
+�
+(α,η,σ)̸=(α′,η′,σ′)
+f †
+α,η,σfα,η,σf †
+α′,η′,σ′fα′,η′,σ′
+(S154)
+ˆHDC = (−3.5U + (ν − 2νf)W − V (ν − νf))
+�
+��
+�
+µDC(νf )
+�
+(α,η,σ)
+f †
+α,η,σfα,η,σ.
+(S155)
+
+40
+The f-electron occupation νf is self-consistently calculated along with the the f- subspace self-energy Σf in at each step of
+the DMFT loop for a set of different total fillings (ν) and temperatures (T). Written in this way, the single particle terms can
+be treated as an iteration-dependent double counting potential, µ(i)
+DC = µDC
+�
+ν(i)
+f
+�
+, where the superscript labels the iteration
+number.
+In the ith iteration of the self-consistency loop, we perform the following sequence of steps:
+1. Embed the old impurity self-energy Σ(i−1)
+f
+into the lattice Green’s function associated to the Hamiltonian ˆHc + ˆHfc +
+ˆH(i−1)
+DC
+.
+2. Fix the total filling to ν.
+3. Calculate the new orbital-resolved fillings ν(i)
+f
+and ν(i)
+c .
+4. Determine the new double counting term µ(i)
+DC.
+5. Use a CT-QMC solver to obtain the new self energy Σ(i)
+f .
+We performed our calculations in parallel using the w2dynamics suite [153, 154]1 as well as the TRIQS family of packages
+[155–157].
+In Fig. 3(a) of the main text, the zero frequency spectral weight and the scattering rate are computed by extrapolating the green
+function and the self-energy respectively on the Matsubara axis. We approximate these by using the value at the first Matsubara
+frequency so that
+Γ = −ImΣ(ω = 0) ≈ −ImΣ(ω0),
+(S156)
+A(ω = 0) = − 1
+π
+��
+k
+TrG(k, ω = 0)
+�
+≈ − 1
+π
+��
+k
+TrG(k, ω0)
+�
+.
+(S157)
+1 https://github.com/w2dynamics/w2dynamics
+