Interplay of Charge Density Wave and Magnetism on the Kagomé Lattice
Yu-Han Lin Jin-Wei Dong Ruiqing Fu Xian-Xin Wu Ziqiang Wang Sen Zhou


	

CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China


Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions, High Magnetic Field Laboratory, Chinese Academy of Sciences, Hefei 230031, China


CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China


CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China


Corresponding author: wangzi@bc.edu
Department of Physics, Boston College, Chestnut Hill, MA 02467, USA
	

Corresponding author: zhousen@itp.ac.cn
CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100049, China

§ ABSTRACT 
Motivated by the recent discovery of charge density wave (CDW) order in the magnetic kagomé metal FeGe, we study the single-orbital t-U-V_1-V_2 model on the kagomé lattice, where U, V_1, and V_2 are the onsite, nearest neighbor, and next-nearest-neighbor Coulomb repulsions, respectively.
When the Fermi level lies in the flat band, the instability toward ferromagnetic (FM) order gives rise to a FM half-metal at sufficiently large onsite U. 
Intriguingly, at band filling n=17/24, the Fermi level crosses the van Hove singularity of the spin-minority bands of the half-metal.
We show that, due to the unique geometry and sublattice interference on the kagomé lattice at van Hove singularity, the intersite Coulomb interactions V_1 and V_2 drive a real and an imaginary bond-ordered 2a_0 × 2a_0 CDW instability, respectively.
The FM loop current CDW with complex bond orders is a spin-polarized Chern insulator exhibiting the quantum anomalous Hall effect.
The bond fluctuations are found to be substantially enhanced compared to the corresponding nonmagnetic kagomé metals at van Hove filling, providing a concrete model realization of the bond-ordered CDWs, including the FM loop current CDW, over the onsite charge density ordered states.
When the spins are partially polarized, we find that the formation of bond-ordered CDWs enhances substantially the ordered magnetic moments.
These findings provide physical insights for the emergence of loop-current and bond-ordered CDW and their interplay with magnetism on the kagomé lattice, with possible connections to the magnetic kagomé metal FeGe.

Interplay of Charge Density Wave and Magnetism on the Kagomé Lattice
    Sen Zhou
    
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Introduction. - Kagomé metals  (A = K, Rb, Cs) <cit.> have attracted enormous attention recently as a fertile playground for exploring novel states of matter <cit.> due to the interplay between lattice geometry, band topology, and electron correlation inherent on the kagomé lattice.
A rich array of correlated and topological electronic states has been observed <cit.>, including a novel charge density wave (CDW) metal with 2a_0 × 2a_0 in-plane periodicity below T_cdw, rotation symmetry breaking, charge-stripe order, electronic nematicity, and a superconductor exhibiting pair density wave order.
Central to the physics of  is the evidence for spontaneous time-reversal symmetry (TRS) breaking in the exotic CDW state <cit.>, despite the absence of local moment <cit.> and itinerant magnetism <cit.>. 
A promising candidate for the TRS breaking is the bond CDW with persistent loop current (LC) order <cit.>, a long-sought-after quantum order also relevant for the quantum anomalous Hall insulator <cit.> and the pseudogap phase of cuprate superconductors <cit.>. 
The physical origin, model realization, and broader implications of the proposed LC order have been a subject under intensive research <cit.>.

More recently, a CDW order with identical 2a_0× 2a_0 in-plane periodicity was discovered in the magnetic kagomé metal FeGe below T_cdw∼110 K <cit.>, deep inside the A-type antiferromagnetic (AFM) phase (T_N∼410 K).
Here the magnetic moments order ferromagnetically (FM) within each kagomé layer with the out-of-plane moments anti-aligned between adjacent layers <cit.>. 
Moreover, the ordered magnetic moments increases substantially at the onset of the CDW transition <cit.>, indicating strong coupling between CDW and magnetism. 
Though recent theoretical calculations <cit.> and experimental investigations <cit.> suggest a CDW mechanism in FeGe different than , the presence of van Hove (VH) singularities in the vicinity of the Fermi level revealed in angle-resolved photoemission spectroscopy (ARPES) <cit.>, together with the observed anomalous Hall effect <cit.> and topological edge modes <cit.>, hint at possible connections between the CDWs in magnetic FeGe and nonmagnetic   kagomé metals.
Here, we study the interplay between FM order and CDW on the kagomé lattice and provide a mechanism for the emergence of a topological magnetic CDW state may relevant for FeGe. 

To this end, we consider the minimal single-orbital model on the kagomé lattice depicted in Fig. <ref>a.
The Fermi surface (FS) at van Hove (vH) filling n=5/12 is the hexagon in Fig. <ref>b that connects the three independent vH points labeled by _1,2,3 at the Brillouin zone boundary.
These vH points with divergent density of states are perfectly nested by the 2a_0 × 2a_0 wave vectors _α =1 2_α, where α=1,2,3 and _α is the reciprocal wave vector.
Since the Bloch state at vH point _α is exclusively localized on the αth sublattice due to the sublattice interference <cit.>, as shown in the sublattice-resolved FS in Fig. <ref>b, the leading 2a_0 × 2a_0 instability is believed to be associated with a bond order that connects different sublattices.
Therefore, while onsite Coulomb repulsion U acting on the same sublattice is obstructed, inter-site Coulomb repulsions are important for the formation of 2a_0 × 2a_0 CDWs. 
Weak-coupling and mean-field theories find that inter-site Coulomb repulsions V_1 on nearest neighbor (nn) bonds and V_2 on next-nn (nnn) bonds indeed drives real and complex bond-ordered CDW with LC order <cit.>.
This picture captures the most essential physics of CDW in nonmagnetic .



We propose that for the magnetic FeGe, the Fermi level in the high-temperature symmetric phase lies instead inside the flat band of the Fe d electrons (Fig. <ref>c), which is conjectured based on the observations of the ARPES experiments <cit.>. 
The onsite U, assisted by the divergent uniform spin susceptibility tied to the flat band, drives a FM ordered state with spin-split bands shown in Fig. <ref>d. 
Intriguingly, at band filling n=17/24 and for a sufficiently large U, the strong FM ordered phase becomes a half-metal (Fig. <ref>e) where the spin-majority (↑) bands is pushed completely below the Fermi level that passes through the vH singularity of the spin-minority (↓) bands, i.e. n_↑=1 and n_↓=5/12. 

The spin-polarized bond fluctuations turn out to be substantially enhanced compared to the spin-degnerate nonmagnetic kagomé metals at vH filling <cit.>. 
The leading instability of the FM half-metal is toward nn real and nnn imaginary bond order rather than the onsite charge density ordered (CDO) states. 
When all particle-hole channels in the Wick contraction are treated on equal footing, we show that the intersite Coulomb interactions (V_1,V_2) drive the FM ordered state into 2a_0 × 2a_0 real and complex CDW phases with spin-polarized LC order, providing a concrete model realization of bond ordered CDW and LC order <cit.>.
We obtain and discuss the properties of the stable phases and the phase diagram. 
In particular, we show that at a smaller U where spins are partially polarized, the emergence of bond-ordered CDW in the FM ordered kagomé plane would induce a significant enhancement of the ordered magnetic moments.

Model and mean-field theory. - We consider the single-orbital t-U-V_1-V_2 model on the kagomé lattice

    H =  -t ∑_⟨ i,j⟩, σ  c^†_iσ c_jσ
    +U ∑_i n̂_i↑n̂_i↓
    +V_1 ∑_⟨ i,j⟩n̂_i n̂_j
    +V_2 ∑_⟨⟨ i,j ⟩⟩n̂_i n̂_j,

where c^†_iσ creates a spin-σ electron on lattice site i, the electron density operators n̂_iσ = c^†_iσ c_iσ and n̂_i = ∑_σn̂_iσ.
Hereinafter, we set nn hopping t≡ 1 as the energy unit.
Decoupling the onsite Coulomb repulsion U in terms of local magnetic moment m̂^η_i = ∑_σσ' c^†_iστ^η_σσ' c_iσ' (η=x,y,z), and the inter-site Coulomb repulsions V_1 and V_2 in terms of bonds ^μ_ij = ∑_σσ' c^†_iστ^μ_σσ' c_jσ' (μ=0,x,y,z), with τ^η/μ the corresponding Pauli matrix, one obtains the mean-field Hamiltonian

    H_MF=   -t∑_⟨ i,j⟩^0_ij -U 4∑_i,η[ 2m^η_i m̂^η_i - (m^η_i )^2 ] 
       - V_1 2∑_⟨ i,j⟩, μ[ (^μ_ij)^* ^μ_ij +h.c. - |^μ_ij|^2 ] 
       - V_2 2∑_ i,j, μ[ (^μ_ij)^* ^μ_ij +h.c. - |^μ_ij|^2 ].

The order parameters, magnetic moment m^η_i =⟨m̂^η_i ⟩ and bonds ^μ_ij =⟨^μ_ij⟩, are to be determined self-consistently by minimizing the state energy.



We note that in the mean-field theory used here <cit.>, the direct Hartree terms of the interactions depending on the electron density n_i =⟨n̂_i ⟩ are neglected to avoid double-counting, since their contributions are already considered in the LDA <cit.>.
These terms are expected to be insignificant in these kaogmé metals, since no large charge disproportionation has been observed.
Moreover, the symmetry preserving correlation induced bond orders, i.e., the real uniform components of the bond order parameters ^μ_ij, are subtracted in the interactions, such that the vH singularities and the band structure are maintained without normalizing the bare band parameters <cit.>.
This is consistent with the fact that vH singularity at the  points has been established by ARPES in both  and FeGe.
Implementing this subtraction scheme, the correlation effects in the self-consistent mean-field theory will only generate spontaneous symmetry-breaking states.

We stay at band filling n=17/24, at which the Fermi level of free electrons passes through the flat band, as shown in the tight-binding band dispersion displayed in Fig. <ref>c.
The presence of flat band at Fermi level gives rise to divergent bare susceptibilities at wave vector =0, and consequently leads to colinear FM at any infinitesimal U.
Though the spins in the studied model have global SU(2) symmetry, we restrict, for simplicity, the spins to order along c axis, in connection to FeGe.
Consequently, bond orders _x,ij and _y,ij vanish on all bonds.
It is thus more convenient to define the spin-dependent bonds, ^σ_ij = 1 2(^0_ij +σ^z_ij), which we use hereinafter.
Furthermore, we consider the quantum states described by the Hamiltonian in Eq. (<ref>) to be periodic with an enlarged 2a_0 × 2a_0 unit cell, which allows investigations on the interplay between =0 magnetism and 2a_0 × 2a_0 CDW.
In addition, we focus on states invariant under C_6 rotation and satisfying the current continuity condition <cit.>, which implies that there are, independently, two lattice sites, three nn bonds, and three nnn bonds within each 2a_0 × 2a_0 enlarged unit cell.
In order to obtain all the possible states, we use different initial conditions for solving the self-consistent equations numerically.
When multiple converged states exist for a given set of Coulomb interactions, we compare the state energies to determine the true mean-field ground state.


FM LC CDWs - 
The fully polarized FM phase at sufficiently large U is a half-metal, with the spin-majority (↑) bands pushed completely below the Fermi level.
Remarkably, the Fermi level exactly passes through the vH points of the spin-minority (↓) bands, as shown by the FM band dispersion at U=6 displayed in Fig. <ref>e.
Therefore, the fully polarized t-U-V_1-V_2 model at a fixed large U and filling n=17/24 provides effectively a spinless counterpart of the spinfull t-V_1-V_2 model at vH filling studied in Ref. <cit.>.
Fig. <ref>a presents the mean-field phase diagram at fixed U=6 in the plane spanned by V_1 and V_2.
Indeed, nn V_1 and nnn V_2 drives, respectively, real and complex bond-ordered CDW, coexisting with the fully polarized FM induced by the large onsite U.

The ground state at V=(1.5, 1) is schematized in Fig. <ref>b.
The electron densities on the two inequivalent sites, A and B, are 1.404 and 1.430, and both of them are fully spin polarized.
Consequently, the spin-majority (↑) electrons are completely localized on lattice sites, with ^↑_ij=0 on all bonds.
The nn bonds for spin-minority (↓) electrons are real and  ^↓ = (0.264, 0.268, 0.172), with the stronger bonds forms the inverse Star-of-David (ISD) pattern displayed in Fig. <ref>b.
This coexistent state of fully polarized FM and real bond-ordered CDW is thus referred to as FM+ISD from now on.
Fig. <ref>c plots the corresponding band dispersion along the high-symmetry path in the reduced Brillouin zone (BZ), where the vH points of spin-down bands near the Fermi level are fully gapped by the 2a_0× 2a_0 CDW.

The coexistent state of fully polarized FM and complex bond-ordered CDW obtained at V=(1.2, 2.8) is illustrated in Fig. <ref>d, with n_A = 1.420, n_B = 1.413, and ^↓ = (0.197 + 0.028i, 0.194 - 0.029i, 0.241 + 0.028i).
We call this state as FM+LC hereinafter, since the complex bonds of spin-↓ electrons generate spin-polarized LCs circulating on the kagomé lattice.
Consequently, the spin-↓ bands are topologically nontrivial.
The band dispersion is presented in Fig. <ref>e, with the Chern numbers of the isolated spin-↓ bands indicated by numbers.
Note that while the Chern number of individual band may change its value as V_1,2 change in the FM+LC regime in the phase diagram we explored, the total Chern number of all occupied bands 𝒩 remains unchanged and 𝒩=-1.
As a result, FM+LC is a spin-polarized Chern insulator exhibiting quantum anomalous Hall effect <cit.>.

CDW enhanced magnetism - Next, we set onsite U=3.6 where the magnetic ground state is a partially polarized FM with ordered moment m = 0.328 μ_B, which we label as a pFM to distinguish from FM with full spin polarization.
The band dispersion is shown in Fig. <ref>d.
The vH points of spin-minority electrons are driven closer to, but still quite away from the Fermi level, due to the insufficiency of FM splitting.
Fig. <ref>a displays the ground state phase diagram at fixed U=3.6 in the V_1-V_2 plane.
Clearly, the uniform pFM is the ground state in the regime where inter-site interactions are weak, while FM+ISD and FM+LC set in, respectively, when nn V_1 or nnn V_2 is sufficiently strong.
Intriguingly, at the onset of 2a_0 × 2a_0 bond-ordered CDW, the FM moments are enhanced from a partially polarized value in pFM to fully polarized one in FM+ISD or FM+LC.
This drives the vH points of spin-minority bands to the Fermi level and thus gains energy by gapping out the vH points via the formation of 2a_0 × 2a_0 CDW.

The enhancement of ordered magnetic moments by the emergence of CDW is illustrated more evidently in the temperature evolution of FM+ISD at V=(1, 1) in Fig. <ref>b and FM+LC at V=(0.8, 2) in Figs. <ref>c and <ref>d.
At high temperatures, the system is a paramagnetic (PM) metal at both sets of interactions, with m=0 on all sites and χ^↑_ij =χ^↓_ij on all bonds.
As temperature reduces, pFM with spins partially polarized sets in via a continuous phase transition.
The spin-up and spin-down hoppings start to deviate from each other.
This phase transition breaks the time-reversal symmetry but preserves the translation symmetry.
The latter is broken at a lower temperature, where a first-order phase transition takes place to the coexisting state of FM and 2a_0 × 2a_0 bond-ordered CDW, i.e., FM+ISD at V=(1, 1) and FM+LC at V=(0.8, 2).
This transition is accompanied by a substantial enhancement in the ordered magnetic moments <cit.> and the energy separation between spin-majority and spin-minority bands.
The latter may induce a sudden spectral weight transfer detectable in optical spectroscopy <cit.>. 



Rivalry with onsite charge density order. - So far, the mean-field theory used in this work has subtracted the contributions from onsite Hartree terms and the symmetry preserving bond corrections.
Though these subtractions are reasonable and physical, as we discussed above, it is desirable to investigate theoretically if the exotic LC order can develop spontaneously in the microscopic model within a complete, self-contained mean-field theory.

The perfectly nested FS and its sublattice features shown in Fig. <ref>b inform us that the vH filled kagomé lattice has instabilities at 2a_0 × 2a_0 wavevectors = associated with bond orders connecting two different sublattices.
Similarly, it has instabilities at wavevector =0 that involve onsite density orders acting on the same lattice.
Therefore, the Hartree terms depending on the electron density would drive 1a_0 × 1a_0 CDO.
The =0 CDO maintains the translation symmetry but breaks rotation symmetry, with charge density modulating on the three sublattices in each unit cell.
It is expected to prevail over 2a_0× 2a_0 bond-ordered CDWs in the regime with strong inter-site Coulomb repulsions to gain electrostatic potential energy.
On the other hand, the nnn V_2 introduces dynamically a nonzero nnn hopping integral and thus alleviates or even removes the vH singularity at the  points, further discouraging the emergence of LC.
Indeed, LC CDW fails to stabilize as the ground state in the entire phase diagram in the vH filled t-V_1-V_2 model studied in Ref. <cit.>, as shown in the End Matter.
This raises the concern that if LC order can emerge spontaneously in a microscopic model on the kagomé lattice without subtracting anything for even physical reasons.



The pursuing of spontaneous LC order on the kagomé lattice has led to the recent studies on the spinless model where the spin degree of freedom is removed <cit.>.
As compared to its spinful conterpart, the charge fluctuations on the bonds are effectively enhanced by a factor of 2 with respect to those on lattice sites in the spinless model, and thus raises the hope of LC CDW being the ground state.
Indeed, random phase approximation <cit.> and functional renormalization group <cit.> studies on the spinless t-V_1-V_2 model reveal real and imaginary bond-ordered CDW states at small to intermediate Coulomb repulsions, and they give their way to =0 CDO when V_1,2 are strong.
This is further confirmed by the complete mean-field calculations reported in the End Matter.

Intriguingly, at large enough U, the t-U-V_1-V_2 model provides a physical realization of spinless t-V_1-V_2 model where the LC order can survive in certain parameter regime.
First, we include the Hartree terms and the ground state phase diagram is presented in Fig. <ref>a at n=17/24 and U=6t.
Clearly, as compared to the diagram in Fig. <ref>a, the regime with strong inter-site interactions is now taken over by onsite CDO and its coexistence with FM.
The phase diagram obtained by complete mean-field calculations is displayed in Fig. <ref>b, where some of the area for FM+LC is further replaced by the FM metal.
Nevertheless, LC order survives in a small parameter regime in the t-U-V_1-V_2 model even when magnetism, onsite CDO, and bond orders are treated on equal footing, in sharp contrast to the t-V_1-V_2 model at vH filling where LC order is absent.
The regime for LC order can be enlarged via introducing a finite nnn hopping t' in the bare band parameter, which compensates to some extent the dynamical correction from nnn V_2, as shown in Fig. <ref>c for t'=0.015.
Interestingly, FM+CDO and CDO differ to each other dramatically, as the two equivalent sites in the unit cell have smaller electron density in FM+CDO but larger electron density in CDO, as illustrated in Fig. <ref>d.

Summary and discussions. - The single-orbital t-U-V_1-V_2 model on the kagomé lattice is studied within mean-field theories at n=17/24, at which the noninteracting Fermi level passes the flat band.
At sufficiently large onsite Coulomb repulsion, says U=6, spins are fully polarized. 
The strong FM splitting pushes the spin-majority bands completely below the Fermi level, and the spin-minority bands realize effectively a spinless model at vH filling.
Consequently, ISD and LC with real and complex 2a_0 × 2a_0 bond orders are induced by inter-site Coulomb repulsions, and give rise to fully spin-polarized FM+ISD and FM+LC.
Both FM+ISD and FM+LC are fully gapped insulators.
FM+ISD is topologically trivial, whereas FM+LC is a spin-polarized Chern insulator exhibiting quantum anomalous Hall effect.
Interestingly, FM+LC with 2a_0 × 2a_0 LC orders persists as a ground state presented in the V_1-V_2 phase diagram even when the model is treated by the full mean-field theory, in analogue to the situation in the spinless t-V_1-V_2 model at vH filling.
In addition, when the spins are partially polarized at a smaller U, the formation of bond-ordered CDW triggered by inter-site Coulomb repulsions induces a significant enhancement in the ordered magnetic moments, which is illustrated clearly in the temperature evolution of states.

The physical origin of CDW with 2a_0 × 2a_0 in-plane periodicity in FeGe is still elusive. 
Recent theoretical calculations and experimental measurements <cit.> suggest that CDW in FeGe arises from out-of-plane dimerization of Ge atoms that saves magnetic exchange energy, implying the importance of spin-phonon coupling, instead of electron-phonon coupling in <cit.>.
On the other hand, neither electron-phonon nor spin-phonon coupling alone is capable of producing topological states with anomalous Hall effect.
The presence of vH singularities in the vicinity of Fermi level, together with the observed anomalous Hall effects in both FeGe and , hint at possible connections between the CDWs in magnetic FeGe and nonmagnetic   kagomé metals.
Therefore, though this work of single d-orbital model with electron correlations is an oversimplified model and has limited connections to the real material, it may provide some physical insights on the emergence of CDW in FeGe and its interplay with magnetism.

This work considers a single kagomé layer, where magnetic moments are FM ordered and the orbital magnetic moments generated from circulating LCs are staggered with zero net moments.
Neutron scattering revealed that magnetic moments are anti-aligned between adjacent layers <cit.>.
It is desirable to investigate the inter-layer correlation of the LC order <cit.> in the presence of A-type AFM.

Acknowledgments. -This work is supported by the National Key Research and Development Program of China (Grant No. 2022YFA1403800 and 2023YFA1407300), the National Natural Science Foundation of China (Grant Nos. 12374153, 12047503, and 11974362), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000).
Z.W. is supported by the U.S. Department of Energy, Basic Energy Sciences (Grant No. DE-FG02-99ER45747) and the Research Corporation for Science Advancement (Cottrell SEED Award No. 27856).
Numerical calculations in this work were performed on the HPC Cluster of ITP-CAS.






§ END MATTER



 §.§ A. Mean-field theories for spinful and spinless models

The spinful t-U-V_1-V_2 model on the kagomé lattice reads 

    H = -t ∑_⟨ i,j⟩, σ( c^†_iσ c_jσ +h.c. )  +U ∑_i n̂_i↑n̂_i↓ +V_1 ∑_⟨ i,j⟩n̂_i n̂_j +V_2 ∑_⟨⟨ i,j ⟩⟩n̂_i n̂_j,

where c^†_iσ creates a spin-σ electron on lattice site i, the electron density operators n̂_iσ = c^†_iσ c_iσ and n̂_i = ∑_σn̂_iσ.
Decoupling the interactions in all particle-hole channels on equal footing, one obtains the mean-field Hamiltonian

    H_MF=   -t∑_⟨ i,j⟩( ^0_ij +h.c.) + U 4∑_i [ 2n_i n̂_i - n^2_i ] + V_1 ∑_⟨ i,j⟩[ n_j n̂_i + n_i n̂_j -n_i n_j ] + V_2 ∑_ i,j[ n_j n̂_i + n_i n̂_j -n_i n_j ] 
       -U 4∑_i,η[ 2m^η_i m̂^η_i - (m^η_i )^2 ] - V_1 2∑_⟨ i,j⟩, μ[ (^μ_ij)^* ^μ_ij +h.c. - |^μ_ij|^2 ] - V_2 2∑_ i,j, μ[ (^μ_ij)^* ^μ_ij +h.c. - |^μ_ij|^2 ] .

Here, local magnetic moment m̂^η_i = ∑_σσ' c^†_iστ^η_σσ' c_iσ' (η=x,y,z), nn and nnn bonds ^μ_ij = ∑_σσ' c^†_iστ^μ_σσ' c_jσ' (μ=0,x,y,z), with τ^η/μ the corresponding Pauli matrix.
The order parameters, electron density n_i =⟨n̂_i ⟩, magnetic moment m^η_i =⟨m̂^η_i ⟩, and bonds ^μ_ij =⟨^μ_ij⟩, are to be determined self-consistently by minimizing the mean-field state energy.
Subtracting the direct Hartree terms depending on the electron density n_i from the mean-field Hamiltonian in Eq. (<ref>), one reach the Hamiltonian given in Eq. (2) in the main text.
Subtracting further the symmetry-preserving bond corrections in the interactions  maintains the bare band structure with van Hove singularities.

In the spinless model where the onsite U is absent due to Pauli exclusion, we consider the t-V_1-V_2 model given by

    H = -t ∑_⟨ i,j⟩( c^†_i c_j +h.c. ) +V_1 ∑_⟨ i,j⟩n̂_i n̂_j +V_2 ∑_⟨⟨ i,j ⟩⟩n̂_i n̂_j,

where c^†_i creates a spinless electron on lattice site i, the electron density operators n̂_i = c^†_i c_i. 
Decoupling the interactions in all particle-hole channels, i.e., density n̂_i and bonds _ij = c^†_i c_j, on equal footing, the obtained mean-field Hamiltonian reads

    H_MF=   -t∑_⟨ i,j⟩( _ij +h.c.) + V_1 ∑_⟨ i,j⟩[ n_j n̂_i + n_i n̂_j -n_i n_j ] + V_2 ∑_ i,j[ n_j n̂_i + n_i n̂_j -n_i n_j ] 
       - V_1 ∑_⟨ i,j⟩[ ^*_ij_ij +h.c. - |_ij|^2 ] - V_2 ∑_ i,j[ ^*_ij_ij +h.c. - |_ij|^2 ] .

Please note that, while the prefactors of the Hartree terms from inter-site Coulomb interactions are the same in Eq. (<ref>) and Eq. (<ref>), there is a factor of 2 difference between the prefactors of bond terms.
Therefore, the bond fluctuations is substantially enhanced in the spinless model, as compared to the spinful model.
This is consistent with the analysis in random phase approximation <cit.>.



To demonstrate the importance of this difference, we consider spinful and spinless t-V_1-V_2 model at van Hove filling n=5/12.
The mean-field ground state phase diagrams for the spinful model are presented in Figs. <ref>a-<ref>c, and those for spinless model in Figs. <ref>d-<ref>f.
In Figs. <ref>a and <ref>d, both the Hartree terms and symmetry-preserving bond corrections are subtracted in the calculations.
Since onsite charge order is neglected and the van Hove singularity is maintained, the system converges to bond-ordered CDW in the entire phase diagram, with V_1 and V_2 favors, respectively, real and complex CDW with loop current (LC) order.
It is easy to show that these two diagrams become identical after multiplying the axes by a factor of 2.
Figs. <ref>b and <ref>e subtract the real uniform components of bonds from the interactions, but take into account the direct Hartree terms.
In the spinful model, the bond-ordered CDW is defeated by the onsite charge density ordered (CDO) states almost in the entire phase diagram, leaving a small regime for the real CDW with inverse Star-of-David (ISD) pattern and LC order fails to develop spontaneously anywhere.
In the spinless model, however, bond-ordered CDWs, ISD and LC, survive in the rivalry with CDO in the regime where inter-site Coulomb repulsions are weak, since the bond fluctuations are substantially enhanced here as compared in the spinful model.
LC CDW even present itself as a ground state in the full mean-field calculations where all particle-hole channels are treated on equal footing, as shown in Fig. <ref>f.



 §.§ B. Magnetic transition on the kagomé lattice at band filling n = 17/24

Here, we investigate the magnetic ground state of the t-U model on the kagomé lattice at band filling n = 17/24.
The presence of flat band at Fermi level gives rise to divergent spin susceptibility at wave vector =0.
In addition, onsite Coulomb repulsion U disfavors charge fluctuations.
As a result, the ground states are expected to be 1a_0 × 1a_0 charge uniform magnetic ordered states.
Indeed, self-consistent mean-field calculations manage to obtain three kind of states, fully polarized ferromagnetic (FM), partially polarized FM (pFM), and coplanar 120^∘ antiferromagnetic (AFM) states.
The state energy per site of them with respect to the paramagnetic phase is displayed in Fig. <ref>a as a function of U, and the corresponding ordered moments are shown in Fig. <ref>b.
Clearly, upon increasing U, the ground state changes from pFM to FM via a first-order transition at U≃ 3.603t.




