Theory of High-Temperature Fractional Chern Phases in Monolayer van der Waals Kagome Materials A$_2$T$_3$X$_4$

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Introduction

Fractional quantum Hall phases are among the most remarkable forms of quantum matter, hosting quasiparticles that carry fractional charge and obey anyonic statistics [1, 2, 3, 4]. These exotic excitations lie at the heart of topological quantum computation [5, 6], connecting fundamental many-body physics with the prospect of protected quantum information. Recent experimental breakthroughs in twisted bilayer $\mathrm{MoTe}_{2}$ and rhombohedral multilayer graphene have realized fractional Chern insulators [7, 8, 9, 10, 11, 12, 13, 14, 15], fulfilling a long-standing theoretical vision of fractional quantum Hall physics without an external magnetic field [16, 17, 18]. Yet the prospect of room-temperature fractionalization envisioned in early proposals [19] remains distant, as the long superlattice length scales constrain their characteristic interaction energies. Larger many-body gaps could bring fractionalization within reach of a broader range of experimental probes like photoemission spectroscopy [20] and potentially improve the robustness of prospective quantum technologies. Beyond these advantages, alternative materials can exhibit distinct phase diagrams, offering opportunities to investigate how fractionalized phases compete with other electronic order.

In this work, we identify monolayers of the recently synthesized van der Waals kagome family $A_{2}T_{3}X_{4}$ [21] as a platform for diverse fractionalized phases, higher Chern bands, and symmetry breaking Figure 1. Weak interlayer van der Waals bonding permits mechanical exfoliation into monolayers, enabling electrostatic control of the carrier density [21]. Band structure calculations reveal an isolated kagome band manifold near the Fermi level, with spin–orbit coupling endowing its nearly flat upper band with nontrivial topology Figure 2. The atomic lattice spacing also supports substantially larger Coulomb energy scales than those typical of moiré superlattices, opening a route to fractionalization at elevated temperatures. Together, these features and the tunability of spin-orbit coupling, Coulomb interactions, and band dispersion of the $A_{2}T_{3}X_{4}$ family provide a broad, experimentally accessible parameter space for exploring fractional Chern insulators. To explore these possibilities systematically, we study monolayers of six experimentally synthesized materials [21]: $\mathrm{Cs}_{2}\mathrm{Pd}_{3}\mathrm{S}_{4}$, $\mathrm{Rb}_{2}\mathrm{Pd}_{3}\mathrm{S}_{4}$, $\mathrm{Cs}_{2}\mathrm{Ni}_{3}\mathrm{S}_{4}$, $\mathrm{Cs}_{2}\mathrm{Ni}_{3}\mathrm{Se}_{4}$, $\mathrm{Rb}_{2}\mathrm{Pd}_{3}\mathrm{Se}_{4}$, and $\mathrm{Cs}_{2}\mathrm{Pt}_{3}\mathrm{S}_{4}$. We perform exact diagonalization of tight-binding models fitted to density functional theory calculations, using the Ohno interaction to interpolate between onsite Hubbard repulsion and the long-range Coulomb interaction [22]. The Ohno interaction provides a minimal phenomenological description of interactions in the low-energy $d_{xy}$ bands, capturing the finite onsite repulsion associated with the orbital extent while retaining nonlocal Coulomb tail [23, 24, 25].

Our key findings are as follows. We find robust ferromagnetism over broad filling ranges, supporting a reremarkably rich variety of correlated phases encompassing high Chern bands, Jain-sequence fractional Chern insulators, anomalous composite Fermi liquids, and charge-density waves [Fig. 1(d)-(i)]. Notably, the fractional Chern insulator in Cs$_2$Pd$_3$S$_4$ at $\nu_{\mathrm{tot}} = 4/3$ retains a finite-size many-body gap of approximately 26 meV with band mixing between the two upper kagome bands in the largest accessible system size [Fig. 3(a)], exceeding typical theoretical gaps in moiré systems [26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. We also consider bilayers made of kagome materials with negligible interlayer tunneling across a hexagonal boron nitride (hBN) spacer. Antiparallel layer magnetizations select opposite Chern-band chiralities [Fig. 4(a)], providing a natural route to fractional quantum spin Hall insulators, whose stabilization in real materials remains a major challenge [38, 39, 40]. These results establish van der Waals kagome materials as a versatile platform for fractional quantum matter, combining enhanced energy scales with tunable competition among correlated phases.

Results

Band Structures

We begin by examining the band structures obtained from density functional theory (DFT) and their tight-binding representations

where $c_{\mathbf{k}\alpha\sigma}^{\dagger}$ ($c_{\mathbf{k}\alpha\sigma}$) creates (annihilates) a $d_{xy}$-orbital electron with momentum $\mathbf{k}$ and spin $\sigma$ on sublattice $\alpha$. $h_{\alpha\beta\sigma}(\mathbf{k})$ incorporates symmetry-allowed hopping up to ninth-nearest neighbors, including intrinsic spin–orbit coupling. Diagonalization yields three spin-degenerate kagome bands. The tight-binding and spin-orbit coupling parameters are fitted to the DFT calculations [Fig. 2] and listed in Tables S1 and S2, while the band characteristics are in Table S3 [41].

Let us comment on the distinct roles of A, T, and X sites in shaping the kagome bands. The alkali-metal A sites lie outside the kagome plane and contribute little orbital weight to the low-energy $d_{xy}$ bands, consistent with the nearly identical band structures of Cs$_2$Pd$_3$S$_4$ and Rb$_2$Pd$_3$S$_4$. The kagome-forming T atoms directly control hopping and spin-orbit coupling. For example, in the Cs-based sulfides, replacing Ni with Pd increases the full kagome bandwidth from 0.4 eV to 1.1 eV [Fig. 2(a), (c)], consistent with the greater spatial extent of Pd 4$d$ orbitals relative to Ni 3$d$ orbitals. Pt 5$d$ orbitals produce a larger spin-orbit coupling gap of 0.1 eV [Fig. 2(f)], approximately twice that of the Ni and Pd compounds, while the Pt compound’s full kagome bandwidth lies between those of its Ni and Pd counterparts. The chalcogen X sites mediate hopping between T sites, influencing the destructive interference underlying flat bands [42, 43]. Replacing S with Se broadens the topmost flat bands in both the Ni and Pd series, consistent with the more extended Se 4p orbitals.

Cs$_2$Pd$_3$S$_4$ and Rb$_2$Pd$_3$S$_4$ offer particularly favorable conditions for fractionalizations. Both combine the narrowest upper flat bands with overall kagome bandwidths among the largest in the family, closely approaching the ideal kagome limit [42, 44, 45]. This proximity is accompanied by small values of the trace-condition violation [41, 46]

where $g(\mathbf{k})$ and $\Omega(\mathbf{k})$ are the quantum metric and Berry curvature of the flat bands. Smaller $\mathcal{T}$ indicates a closer approach to the ideal geometry, a feature favorable to fractional Chern insulators [47, 48, 49, 50]. Further, large dispersions of the lower bands can suppress the effect of virtual interband excitations, helping stabilize fractionalization against band mixing.

Ferromagnetism

Having characterized the single-particle bands, we now investigate electron–electron correlations. We adopt the Ohno interaction, motivated by first-principles studies of transition-metal compounds showing that this form captures the spatial dependence of $d$-orbital Coulomb matrix elements under local background screening [51]

where $r = \lvert\mathbf{r} \rvert$, $U_{0}$ is the onsite Hubbard repulsion, and $U_{C} = e^{2}/4\pi\epsilon$ is the Coulomb prefactor, with $\epsilon$ the effective background permittivity. This form recovers $V(0) = U_{0}$ and approaches the Coulomb tail $V(r) \to U_{C}/r$ for $r \gg U_{C}/U_{0}$. We take $\epsilon= 5\epsilon_{0}$ and a lattice constant $a = 6.3$ Å [21], corresponding to a characteristic Coulomb energy $e^{2}/4\pi\epsilon a \simeq0.457$ eV. We set $U_{0} = 3$ eV as a representative value referring to reported screened interactions in transition-metal $d$ orbitals [52, 53, 54, 55, 56].

We use exact diagonalization to investigate spontaneous symmetry breaking in the partially filled flat bands, focusing on ferromagnetism [Fig. 1(d)-(i)]. We project onto the two spin components of the uppermost kagome band, treating the four lower spin-resolved bands as fully occupied and inert. Conservation of the total $S_{z}$ allows us to diagonalize the Hamiltonian independently in each $S_{z}$ sector and compare the resulting many-body spectra at fixed total filling $\nu_{\mathrm{tot}} = \nu_{\uparrow} + \nu_{\downarrow}$, with $0 < \nu_{\mathrm{tot}} < 2$. In the undoped compounds, the Fermi level lies above the kagome bands and therefore we focus on $1 < \nu_{\mathrm{tot}} < 2$, corresponding to less than one hole per unit cell relative to the fully occupied flat bands.

The stability of ferromagnetism correlates with the bandwidth of the flat bands [41]. In Cs₂Pd₃S₄, Rb₂Pd₃S₄, and Cs₂Ni₃S₄, which have the narrowest upper bands, the ground state exhibits maximal spin polarization and a substantial spin gap of approximately 30 meV or larger over broad filling ranges (red regions in Figure 1(d)-(Figure 1i)), except near full filling, $\nu_{\mathrm{tot}} \to2$ [41]. The energy splittings between different spin sectors decrease toward $\nu_{\mathrm{tot}} = 15/8$ and collapse, indicating the emergence of paramagnetism near full filling [41]. By contrast, Cs₂Ni₃Se₄, Rb₂Pd₃Se₄, and Cs₂Pt₃S₄, with broader upper bands, exhibit substantially narrower ferromagnetic regimes. Extended filling intervals host partially spin-polarized states with $\lvert S_{z} \rvert< (N_{\uparrow} - N_{\downarrow})/2$ (where $N_{\sigma}$ is the total spin-$\sigma$ electron number) and small energy splittings between different spin sectors [41] (dashed regions in Figure 1(d)-(Figure 1i)). At higher fillings, this gives way to a paramagnetic regime (blue regions in Figure 1(d)-(Figure 1i)). In passing, we note that the qualitative trends in symmetry breaking remain largely unchanged when the Ohno interaction is replaced by Hubbard repulsion [41].

In these compounds, ferromagnetism is accompanied by Chern bands. Despite their weaker ferromagnetism, Rb$_2$Pd$_3$Se$_4$ and Cs$_2$Pt$_3$S$_4$ host bands with higher Chern numbers, $|C| = 2$ and $|C| = 3$, whereas the remaining four compounds have $|C| = 1$ [41].

Fractionalizations

We now investigate intrinsic topological order and competing phases in regimes of robust ferromagnetism, focusing on total fillings $\nu_{\mathrm{tot}} = 5/4$, $4/3$, $7/5$, $10/7$, $3/2$, $11/7$, $8/5$, $5/3$, and $7/4$. We perform exact diagonalization with the minority-spin upper flat band partially filled and all other bands fully occupied. To capture the leading effects of band mixing, we retain configurations with at most one interband electron–hole excitation [57, 58]. We distinguish fractional Chern insulators, charge-density waves, and anomalous composite Fermi liquids using ground-state degeneracy patterns, spectral flow under flux insertion, many-body Chern numbers, structure factors, momentum-space occupations, and the finite-size scaling of low-energy splittings [18, 59, 60, 61, 62]. Further details are provided in Methods and [41].

Figure 3 summarizes the filling-dependent phases identified by our exact diagonalization calculations. Consistent with their favorable band geometry, Cs$_2$Pd$_3$S$_4$ and Rb$_2$Pd$_3$S$_4$ host the richest sequences of fractional Chern phases (green solid bars in Figure 1(d)–(Figure 1i)). In particular, within the one-excitation truncation and at the largest accessible system size, $N = 24$, Cs$_2$Pd$_3$S$_4$ at $\nu_{\mathrm{tot}} = 4/3$ exhibits a threefold ground-state manifold separated from excited states by a many-body gap of approximately 26 meV [Fig. 3(a)]. Its spectral flow under flux insertion [Fig. 3(b)] and many-body Chern number $C_{\mathrm{MB}} = 1/3$ support its identification as a Laughlin-type fractional Chern insulator [41]. At the Jain sequence fillings $\nu_{\mathrm{tot}} = 10/7$, $11/7$, $8/5$, and $5/3$, analogous degeneracy patterns, spectral flow, and many-body Chern numbers indicate additional fractional Chern phases [41]. Notably, the state at $\nu_{\mathrm{tot}} = 5/3$ also exhibits a substantial finite-size gap of approximately 19 meV. Similar cascades of fractional Chern phases occur in Rb$_2$Pd$_3$S$_4$ [Fig. 1(e)]. Table S5 summarizes the finite-size many-body gaps of the identified fractional Chern states.

$\nu_{\mathrm{tot}}$	4/3	7/5	10/7	11/7	8/5	5/3	12/7
system size	$N = 24$	$N = 25$	$N = 21$	$N = 21$	$N = 25$	$N = 27$	$N = 28$
Cs₂Pd₃S₄	26	-	12	15	3	19	-
Rb₂Pd₃S₄	12	-	4	18	9	19	4
Cs₂Ni₃S₄	-	-	-	-	-	2	-

At half filling $\nu_{\mathrm{tot}} = 3/2$ of Cs$_2$Pd$_3$S$_4$, we find evidence for a gapless anomalous composite Fermi liquid (magenta solid bar in Figure 1(d)). The low-energy many-body spectra [Fig. 3(c)] closely resemble those of the half-filled lowest Landau level [Fig. 3(d)], including the momentum sectors of the lowest-lying states and twofold degeneracies. Furthermore, the occupation number averaged over the ground states [Fig. 3(e)]

remains nearly uniform, satisfying $|n(k) - 1/2| < 0.13$ at the largest accessible system size, $N = 24$, with no discernible electronic Fermi surface. This bound is comparable to the corresponding value of 0.17 reported for

the anomalous composite Fermi liquid in twisted bilayer MoTe$_2$ [61]. Further evidence comes from the projected static structure factor [Fig. 3(f)]

where $\bar{\rho}_{\mathbf{q}}$ is the density operator projected onto the upper flat band. We observe pronounced peaks near $|\mathbf{q}| = 2k_{\mathrm{F}}$ and nearly vanishing weight for $|\mathbf{q}| > 2k_{\mathrm{F}}$ [41]. These density correlations, together with the absence of a discernible electronic Fermi surface, support an emergent composite-fermion Fermi surface [20, 29, 61, 63, 64]. We also find signatures consistent with an anomalous composite Fermi liquid in Cs₂Pd₃S₄ at quarter filling of the flat band ($\nu_{\mathrm{tot}} = 5/4$) [29, 41, 61, 63]. At several filling fractions, Cs₂Pd₃S₄, Rb₂Pd₃S₄, and Cs₂Ni₃S₄ exhibit translational symmetry breaking and form charge-density waves, characterized by multifold ground states and Bragg peaks in $S(\boldsymbol{q})$ (brown solid bars in Figure 1(d)-(Figure 1f)). Detailed characterizations are in [41].

Bilayers. The fractional Chern phases in these kagome monolayers naturally motivate bilayer heterostructures, which can be readily assembled using established van der Waals techniques [21]. Antiparallel layer magnetizations produce opposite Chern-band chiralities, providing a route to fractional quantum spin Hall insulators. We consider two Cs₂Pd₃S₄ monolayers at $\nu_{\mathrm{tot}} = 4/3$ per layer, separated by an hBN spacer that suppresses interlayer tunneling [Fig. 4(a)]. We assume that ferromagnetism remains robust in each layer, and use the Ohno and Coulomb potentials for intralayer and interlayer interactions.

We use exact diagonalization to map the phase diagram as a function of $d/a$ [Fig. 4(c)], where $d$ is the interlayer separation and $a = 6.3$ Å is the intralayer lattice constant. At small $d/a$, the ground state is fully layer polarized and effectively realizes the monolayer fractional Chern insulator at $\nu_{\mathrm{tot}} = 5/3$. As $d/a$ increases, an intermediate layer-balanced metallic phase emerges at $d/a \simeq0.22$ and gives way to a fractional quantum spin Hall insulator at $d/a \simeq0.52$. This insulator effectively realizes two decoupled fractional Chern insulators with opposite chiralities [Fig. 4(b)]. The measured interlayer spacing of approximately 7 Å [21] corresponds to $d/a \simeq1.1$, well within the fractional quantum spin Hall regime of our model [Fig. 4(b),(c)], suggesting an experimentally accessible route to realizing this phase.

Discussions

We identify the van der Waals kagome family A₂T₃X₄ as a platform for diverse correlated topological phases at enhanced energy scales. Within the ferromagnetic regimes, we find spin-polarized bands with Chern numbers up to $\lvert C\rvert= 3$, alongside Jain-sequence fractional Chern insulators, anomalous composite Fermi liquids, and competing charge order. The Pd-based sulfides are particularly favorable: Cs₂Pd₃S₄ exhibits a finite-size many-body gap of approximately 26 meV in the largest accessible system with band mixing that retains at most one interband electron–hole excitation. Bilayers with antiparallel layer magnetizations further support fractional quantum spin Hall insulators at realistic interlayer separations. Chemical diversity, gate-tunable carrier density, and van der Waals layer assembly together make these kagome materials versatile platforms for realizing and controlling fractionalized quantum matter.

These results motivate several directions for future investigation. The most immediate experimental priority is systematic transport characterization, building on preliminary observations of insulating behavior at partial filling of the flat band [21]. Gate-dependent measurements could test for fractional Hall responses and map the competition among the predicted phases. Secondly, calculations on larger systems using methods such as density matrix renormalization group could clarify the phases at additional fillings, where our finite-size spectra and ground-state diagnostics do not yet establish a definite symmetry-breaking pattern or topological order. Finally, flat bands may also promote other collective instabilities, including superconductivity driven by interaction channels beyond those retained in our minimal model. Together, these efforts could reveal how fractionalization, symmetry breaking, and superconductivity compete and intertwine within a single material family.

Acknowledgements

S. O., D. S., and G. Y. C. acknowledge the financial support from Samsung Science and Technology Foundation under Project Number SSTF-BA2401-03, the NRF of Korea (Grants No. RS-2026-25479545, RS-2024-00410027, RS-2026-25467036, RS-2024-00444725, RS-2023-00256050, RS-2025-25453111, RS-2025-08542968) funded by the Korean Government (MSIT), the Air Force Office of Scientific Research under Award No. FA23862514026, and Institute for Basic Science under project code IBS-R014-D1. S. O. acknowledges the support from the Agency for Defense Development through a grant funded by the Defense Acquisition Program Administration (DAPA) (UI257011TE). J. H., J. C., and M. K. are supported by the National Research Foundation (NRF) of Korea through grant nos. RS-2025-23525391 and RS-2026-25496507. M. K. was supported by the Creative-Pioneering Researchers Program through Seoul National University. M. K. acknowledges POSCO Science Fellowship from Cheongam Foundation. This work is in part supported by the Korea Max Planck Quantum Materials Centre (KOMQUEST), the Top-Tier Premier International Research Consortium through Grant No. RS-2026-25585727, and the core center program by the Ministry of Education through Grant No. 2021R1A6C101B418.

Methods

DFT Calculation.

The electronic structures of $\mathrm{A}_2\mathrm{T}_3\mathrm{X}_4$ were calculated independently using Vienna Ab initio Simulation Package (VASP) [65, 66] and Quantum ESPRESSO package (QE) [67, 68], yielding consistent results. To ensure experimental relevance of our calculations, the DFT parameters were refined through comparison with the experimentally measured band structures of $\mathrm{A}_2\mathrm{T}_3\mathrm{X}_4$ [21]. In particular, we found that the Heyd-Scuseria-Ernzerhof (HSE) [69] hybrid exchange-correlation functional, which incorporates a fraction ($a_{\mathrm{HF}}$) of Hartree-Fock exchange, was necessary to accurately reproduce the experimental dispersion of $\mathrm{A}_2\mathrm{T}_3\mathrm{X}_4$. We used $a_{\mathrm{HF}} = 5\%$, $5\%$, $15\%$, $15\%$, and $15\%$ for $\mathrm{Cs}_2\mathrm{Ni}_3\mathrm{S}_4$, $\mathrm{Cs}_2\mathrm{Ni}_3\mathrm{Se}_4$, $(\mathrm{Rb},\mathrm{Cs})_2\mathrm{Pd}_3\mathrm{S}_4$, $\mathrm{Rb}_2\mathrm{Pd}_3\mathrm{Se}_4$, and $\mathrm{Cs}_2\mathrm{Pt}_3\mathrm{S}_4$, respectively. Monolayer calculations were performed after relaxing the lattice structure with the Perdew-Burke-Ernzerhof (PBE) [70] functional using a $\Gamma$-centered $12 \times12 \times1$ mesh, and a vacuum spacing of up to $20$ Å was introduced along the out-of-plane direction. For the electronic structure calculations, VASP used projector augmented-wave datasets [71] with a plane-wave energy cutoff of $450$ eV for $\Gamma$-centered $5 \times5 \times1$ $k$-point mesh, whereas QE used optimized norm-conserving Vanderbilt (ONCV) pseudopotentials [72] with a wave cutoff of $100$ Ry for $\Gamma$-centered $6 \times6 \times1$ $k$-point mesh. Spin-orbit coupling effects were included for all calculations. Following the self-consistent HSE calculations, maximally localized Wannier functions were constructed using the Wannier90 package [73]. The resulting Wannier Hamiltonians of monolayer $\mathrm{A}_2\mathrm{T}_3\mathrm{X}_4$, constructed from transition metal (T) $d$ orbitals and chalcogen (X) $p$ orbitals, were used to interpolate the HSE band structures along high-symmetry paths.

Tight-Binding Parameter Optimization.

For $\mathrm{Cs}_2\mathrm{Pd}_3\mathrm{S}_4$ and $\mathrm{Rb}_2\mathrm{Pd}_3\mathrm{S}_4$, we have adopted the parameters used in [21]. For the remaining materials in the $\mathrm{A}_2\mathrm{T}_3\mathrm{X}_4$ family, we performed a neural-network-assisted optimization of the tight-binding Hamiltonian to obtain an unbiased fit to the DFT band structures. Three independently trainable neural-network modules were assigned to distinct subsets of the tight-binding parameters. The first module (N1) generated the on-site energy and nearest-neighbor hopping integrals; the second module (N2) generated the remaining hopping integrals; and the third module (N3) generated the spin-orbit-coupling parameters. The network parameters were optimized by minimizing the weighted mean squared error between the tight-binding and DFT band structures. Within each optimization cycle, N1 and N2 were optimized sequentially using the DFT bands without spin-orbit coupling, followed by optimization of N3 using the DFT bands with spin-orbit coupling. For each material, we compared two initialization protocols, random initialization and initialization from tight-binding parameters extracted from the Wannier Hamiltonians, yielding consistent results. The optimized tight-binding parameter sets closely reproduce the DFT bands within the target energy window, as shown in Figure 2.

Exact Diagonalization.

We diagonalize the many-body Hamiltonian $H = H_0 + H_{\mathrm{int}}$, where

Here, $\boldsymbol{Q}$ are reciprocal lattice vectors, $\rho_{\boldsymbol{q}}$ denotes the electron density operator, and

is the Fourier transform of the Ohno interaction, Eq. (2), with $p = \lvert\boldsymbol{p}\rvert$. The diagonalization procedure is as follows: after fixing a momentum grid of finite system size $N$, we choose the total electron number in the six kagome bands between $5N$ and $6N$. Initially, we fully fill the lower four bands with $4N$ electrons and restrict excitation to upper bands, to keep the Hilbert space dimension manageable. We perform exact diagonalization within each $S_z$- and momentum-resolved subspace. Once the maximal $S_z$ sector is chosen as a ground state, the three majority-spin bands become all fully filled and this only contributes as a constant Hartree shift. In this regime, the Hilbert space is effectively spinless and this allows us to consider further interband mixing, thereby investigating the spin-polarized sector thoroughly. To select sufficiently large momentum grids while incorporating this band-mixing effect, we truncate the Hilbert space by allowing at most one interband electron–hole excitation from the spin-minority middle band to the topmost flat band. For all the calculations, sparse diagonalization was performed using Krylov methods. Further details of the Hamiltonian and band mixing process are discussed in Supplementary Information.

Bilayer System.

For bilayer systems, we consider the following interaction,

to distinguish intralayer and interlayer interactions. The exact diagonalization follows the same strategy as in the spinful monolayer case, as we suppress the spin degrees of freedom and introduce layer pseudospin indices $\ell,\ell'$, making the computation essentially equivalent. We neglect band mixing in this calculation owing to computational constraints on the accessible system size.

Supplementary Information for “Theory of High-Temperature Fractional Chern Phases in Monolayer van der Waals Kagome Materials A$_2$T$_3$X$_4$”

Soomin Oh; Jiyong Han; Jinhyeok Choi; Donghae Seo; Mingu Kang; Gil Young Cho

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Supplementary Information for ``Theory of High-Temperature Fractional Chern Phases in Monolayer van der Waals Kagome Materials A$_2$T$_3$X$_4$''

Tight-Binding Model

In this supplementary section, we explicitly write the tight-binding Hamiltonian on a kagome lattice, including hoppings up to ninth-nearest neighbors. The unit cell is spanned by two lattice vectors $\mathbf{a}_1 = a(1,0)$, $\mathbf{a}_2 = a(1/2,\sqrt{3}/2)$, where we set $a=1$. We write the single-electron Hamiltonian in a 6-component spinor basis

where $A$, $B$, $C$ denote the three sublattices of a kagome lattice. The Hamiltonian in momentum space is

The diagonal components are

and the off-diagonal components are

Here, $k_i=\mathbf{k}\cdot\mathbf{a}_i$, and $t_n$, $\lambda_n$ represent the $n$th-nearest-neighbor hopping and spin-orbit coupling parameters, respectively. Note that there are two symmetry-inequivalent contributions of third-, sixth- and eighth-nearest-neighbor hoppings, which are labeled $t_3$, $t_6$, $t_8$ and $t'_3$, $t'_6$, $t'_8$. This is because there are two distinct real-space paths through different intermediate atoms. The real space configurations of sublattices, hoppings, and spin-orbit couplings are illustrated in Figure S1. The parameters of the kagome materials A$_2$T$_3$X$_4$ are listed in Table S1 and Table S2.

A$_2$T$_3$X$_4$	$t_1$	$t_2$	$t_3$	$t'_3$	$t_4$	$t_5$	$t_6$	$t'_6$	$t_7$	$t_8$	$t'_8$	$t_9$
Cs$_2$Pd$_3$S$_4$	0.1134	0.0164	0.0255	0.0255	0.0067	-0.0064	-0.0024	0.0009	0.008	0	0	0
Rb$_2$Pd$_3$S$_4$	0.1146	0.0154	0.0195	0.0176	0.0105	-0.0095	0.0014	-0.0037	0.0063	0	0	0
Cs$_2$Ni$_3$S$_4$	0.0323	0.0013	0.0078	0.0047	0.0088	-0.0149	-0.0039	0.0033	0.0034	-0.0024	0.0018	0
Cs$_2$Ni$_3$Se$_4$	0.0351	-0.0146	0.0193	0.0039	0.0129	-0.0154	-0.0060	0.0048	0.0034	-0.0039	0.0046	-0.0004
Rb$_2$Pd$_3$Se$_4$	0.1103	-0.0255	0.0484	0.0019	0.0118	-0.009	0.0055	-0.0096	0.0049	0	0	0
Cs$_2$Pt$_3$S$_4$	0.0757	0.0403	-0.0025	0.0389	-0.0067	0.0002	-0.0061	0.0042	0.0052	-0.0026	-0.002	0.0002

$A_{2}T_{3}X_{4}$	Cs$_{2}$Pd$_{3}$S$_{4}$	Rb$_{2}$Pd$_{3}$S$_{4}$	Cs$_{2}$Ni$_{3}$S$_{4}$	Cs$_{2}$Ni$_{3}$Se$_{4}$	Rb$_{2}$Pd$_{3}$Se$_{4}$	Cs$_{2}$Pt$_{3}$S$_{4}$
Full Bandwidth	1.086	1.059	0.392	0.519	1.158	0.810
SOC gap at Dirac point	0.058	0.057	0.049	0.054	0.144	0.105
Flat Bandwidth	0.151	0.149	0.154	0.264	0.480	0.171

We next diagonalize the Hamiltonian as

where $\alpha,\beta=A,B,C$ denote the sublattices. We define the operators in the band basis $c^\dagger_{nk\sigma}=\sum_{\alpha}u_{n\alpha\sigma}(\mathbf{k})c^\dagger_{k\alpha\sigma}$. With this, $H_0$ becomes

The three kagome bands are spin-degenerate, i.e. $\epsilon_{n\sigma}(\mathbf{k})\equiv\epsilon_n(\mathbf{k})$, before introducing ferromagnetism.

Quantum Band Geometry

In this supplementary section, we compare the quantum geometry of the flat kagome bands across the six compounds to characterize their relative suitability for fractional Chern insulators. We evaluate the quantum metric, $\operatorname{Tr}g(\mathbf{k})$,

TABLE S2: Spin-Orbit Coupling Parameters. The parameters are in eV units.

TABLE S3: Characteristic Energy Scales. Bandwidths and spin-orbit coupling (SOC)-induced gaps of monolayer kagome $A_{2}T_{3}X_{4}$ compounds, obtained from the tight-binding band structures. All values are given in eV.

the Berry curvature, $\Omega(\boldsymbol{k})$, and their difference, $\operatorname{Tr} g(\boldsymbol{k}) - \lvert\Omega(\boldsymbol{k})\rvert$, which are displayed as momentum-space maps in Figure S2. These quantities are evaluated using the gauge-invariant projector expressions [1]

where $\hat{P}(\boldsymbol{k}) = \lvert u_{n\boldsymbol{k}}\rangle\langle u_{n\boldsymbol{k}}\rvert$ projects onto the normalized cell-periodic Bloch eigenstate of the isolated band $n$. Here, $\operatorname{tr}$ denotes the trace over the orbital and spin degrees of freedom of the Bloch Hamiltonian, whereas $\operatorname{Tr} g = g_{xx} + g_{yy}$ denotes the trace over Cartesian momentum indices. All three quantities plotted [Figure S2] are multiplied by $A_{\mathrm{BZ}}/2\pi$, where $A_{\mathrm{BZ}}$ is the area of the first Brillouin zone.

To model ferromagnetic spin polarization, we introduce a sufficiently large Zeeman splitting to fully separate the spin-up and spin-down bands in energy. We then evaluate the quantum geometry of the flat band with a positive Chern number. The results are presented in Figure S2. Cs$_{2}$Pd$_{3}$S$_{4}$ and Rb$_{2}$Pd$_{3}$S$_{4}$ offer small trace-condition violations of about 0.6–0.8, where the rest of the materials range from 1.1 to 9.3.

Many-Body Hamiltonian

In this supplementary section, we specify the interacting Hamiltonians used in the monolayer and bilayer calculations. Rewriting the full monolayer many-body Hamiltonian including the interacting term, we have

where $A$ is the sample area. $\hat{V}^{nn'mm'\sigma\sigma'}_{\boldsymbol{k},\boldsymbol{k}',\boldsymbol{q}}$ is the two-body interaction matrix

where $\boldsymbol{Q}$ are the reciprocal lattice vectors, and the band form factor $\Lambda^{n'n\sigma}_{\boldsymbol{k},\boldsymbol{q}+\boldsymbol{Q}}$ is

Here, $\tau_{\alpha}=A,B,C$ represents the sublattice coordinates, and the interaction $V(\boldsymbol{q}+\boldsymbol{Q})$ is introduced in the main text,

where $p = |\mathbf{p}|$, $U_0$ is the onsite Hubbard repulsion, and $U_C = e^2/4\pi\epsilon$ is the Coulomb prefactor. We use $U_0 = 3$ eV and $\epsilon= 5\epsilon_0$ as in the main text. A uniform neutralizing background removes the spatially uniform component of the interaction, and we set $V(\mathbf{p}=0)=0$. The sum over reciprocal vectors $\mathbf{Q}$ is an infinite sum, but in practice, we truncate it at an appropriate cutoff as the interaction (S13) exponentially decays.

To extend the model to a bilayer, we include the interlayer Coulomb interaction in addition to the intralayer Ohno interaction. We neglect interlayer tunneling, so the particle number in each layer is separately conserved. Further assuming that the ferromagnetism is robust, we drop the spin index $\sigma$ and the Hamiltonian is of the form

where $\ell,\ell'=1,2$ are layer pseudospin indices, and we restrict ourselves to homogeneous bilayers, $\epsilon_n^{(1)}(\mathbf{k})=\epsilon_n^{(2)}(\mathbf{k})$. The interaction matrix is now

where we introduce different types of potentials for intralayer and interlayer interactions,

with the interlayer distance $d$. The form factor $\Lambda$ essentially has the same expression as (S12), replacing the spin index $\sigma$ with the layer index $\ell$. The uniform neutralizing background leaves a finite layer-antisymmetric contribution by the capacitive correction,

Numerical Details

In this supplementary section, we discuss the details of the Hamiltonian and band mixing process that we apply in our exact diagonalization calculations.

Spinful Exact Diagonalization

We describe the effective Hamiltonian used to investigate the possible ferromagnetism. We start from the many-body Hamiltonian (S10). This Hamiltonian contains a total of 6 bands, 3 for each spin species, which is impractical to handle in exact diagonalization. Therefore, as we perform $S_z$-resolved calculations, we retain only the topmost flat band of each spin species as an active degree of freedom. Labeling the three kagome bands by $n=1,2,3$ in descending order of energy [Fig. S3(a)], we assume that bands $n=2,3$ are fully occupied and inert for both spins. Then the spin number $S_z=(N_\uparrow-N_\downarrow)/2$ is determined by the difference of electrons filling band 1 for each spin component.

The effective Hamiltonian is obtained by projecting onto this restricted Hilbert space,

where $\hat{n}_{\alpha\mathbf{k}\sigma}=c_{\alpha\mathbf{k}\sigma}^\dagger c_{\alpha\mathbf{k}\sigma}$. Here, $H_{\mathrm{eff}}$ is understood as an operator acting within the projected subspace. This projection retains the static Hartree and Fock contributions from the filled bands while excluding excitations that create holes in them. Up to a constant, the resulting Hamiltonian reads

where the filled-band self-energy is

The Hartree contribution includes both spin species in the filled bands, whereas the Fock contribution involves only filled-band electrons with the same spin as the electron in band $n=1$.

For the bilayer systems, the strategy we use is essentially the same, as we drop the spin index $\sigma$ and replace it with the layer pseudospin index $\ell$. The interaction matrix $\hat{V}^{1111}_{k,k',q}{}^{\sigma\sigma'}$ in (S18) is replaced with $\hat{V}^{\ell\ell'}_{k,k',q}{}^{1111}$, introduced in (S15). Here, the potential $V^{\ell\ell'}(\mathbf{p})$ differs from the monolayer spinful case.

Band Mixing in Ferromagnetic Regime

We next incorporate interband mixing within the maximally spin-polarized regime, provided that the spin gap is sufficiently large to restrict the low-energy analysis to the maximally spin-polarized sector. For this, we first show that the fully occupied majority-spin bands contribute only an additive constant within the spin-$S_z$ sector. We then retain the two upper minority-spin bands and restrict their configurations to at most one interband electron–hole excitation.

Taking $\downarrow$ to be the majority spin, we introduce a projection operator for the ferromagnetic sector

where the three majority-spin bands are fully filled. Note that the many-body Hamiltonian (S10) can be decomposed into intraspin and interspin components,

Here, the intraspin component, $H_{\sigma=\uparrow,\downarrow}$, is defined as

and $H_{\mathrm{mix}}$ denotes the interspin component which is a sum of $(\sigma,\sigma')=(\uparrow,\downarrow)$ and $(\downarrow,\uparrow)$ terms in $H_{\mathrm{int}}$. As we project onto the subspace of the maximally spin-polarized sector, the effective Hamiltonian becomes

Here, $E_{\downarrow}$ is a constant energy since all spin-$\downarrow$ electrons occupy their orbitals. We also show that the interspin interaction $H_{\mathrm{mix}}$ only gives a constant shift as well. The last term is expressed as

where

is the Hartree self-energy. Calculating this term explicitly,

We have used the completeness of the Bloch eigenvectors, $\sum_{m=1}^{3}u_{m\alpha\downarrow}(p)u^*_{m\beta\downarrow}(p)=\delta_{\alpha\beta}$. Due to the threefold rotational symmetry of the three sublattices $\alpha,\beta=A,B,C$, and provided that $Q$ is summed properly, the term $\sum_{Q\beta}V(Q)e^{iQ\cdot(\tau_\alpha-\tau_\beta)}$ is independent of $\alpha$. As a result, together with $N/A$, this term can be treated as a constant $C_H$ while summing over $\alpha$,

using orthonormality again. Therefore,

which is a constant shift as long as the particle number $N_{\uparrow}$ is fixed. Consequently, the effective Hamiltonian (S20) becomes

The remaining problem is therefore equivalent to a spinless three-band system, and we suppress the minority-spin index $\uparrow$ below.

To incorporate band mixing within a tractable Hilbert space, we keep band 3 fully occupied and allow at most one electron to be promoted from band 2 to band 1 [Fig. S3(b)]. We define

where $\hat{n}_{nk}=c^\dagger_{nk}c_{nk}$ and $\Pi_{N_{h,2}=r}$ projects onto states with exactly $r$ holes in band 2. At fixed minority-spin particle number, the two retained sectors correspond to zero and one interband excitation, respectively. The full projector and the corresponding truncated Hamiltonian are

Here, $H_{\mathrm{trunc}}$ acts within the projected Hilbert space. After suppressing the frozen degrees of freedom and omitting additive constants, it can be written as

where

Here, the fully occupied band 3 contributes the static Hartree–Fock self-energy

Unlike the Hartree contribution from the complete majority-spin sector, this correction need not be proportional to the identity or independent of momentum. The final projection retains all matrix elements within and between the zero- and one-hole sectors of band 2, while excluding states with two or more holes in band 2 or any holes in band 3.

In the calculations described above, the valence bands are fully occupied and inert, and their static Hartree–Fock contribution is retained. A common alternative is to formulate exact diagonalization by projecting onto a target band without an explicit self-energy contribution from the inert bands [2, 3, 4, 5, 6]. These two treatments are analogous, respectively, to the average (AVE) and charge-neutrality (CN) interaction schemes discussed in Refs. [7, 8], which differ in the reference used to define the interaction and the associated one-body terms. Unless otherwise specified, all results presented in this paper retain the inert-band self-energy, analogous to the AVE convention.

Spontaneous Symmetry Breaking

In this supplementary section, we present the spin-resolved many-body spectra used to determine the ferromagnetic regimes displayed in Figure 1 of the main text. Figure S4 shows the lowest energy in each $S_z$ sector at fixed total filling, measured relative to the overall minimum at that filling. For the momentum clusters, $N = 14$ was used for $\nu_{\mathrm{tot}} = \frac{10}{7}, \frac{11}{7}$, $N = 15$ was used for $\nu_{\mathrm{tot}} = \frac{4}{3}, \frac{7}{5}, \frac{8}{5}$, $N = 16$ was used for $\nu_{\mathrm{tot}} = \frac{9}{8}, \frac{5}{4}, \frac{7}{4}$, and $N = 32$ and $N = 64$ were used for $\nu_{\mathrm{tot}} = \frac{15}{8}, \frac{31}{16}$, respectively. $\nu_{\mathrm{tot}} = \frac{31}{16}$ is the largest accessible filling which can be separated into at least three $S_z$ sectors. We note that the spectra at $\nu_{\mathrm{tot}} = \frac{15}{8}, \frac{31}{16}$ either favor paramagnetic sector or have 0.1 meV or less order of separations (the spectra overlap and only the ferromagnetic sector is shown in the figure) for all cases except Cs$_2$Ni$_3$S$_4$ at $\nu_{\mathrm{tot}} = \frac{15}{8}$, and therefore identify them as paramagnetic. Cs$_2$Ni$_3$S$_4$ has a ferromagnetic ground state with a spin gap of about 1 meV at $\nu_{\mathrm{tot}} = \frac{15}{8}$.

Comparing the flat bandwidth presented in Table S3 with the stability of ferromagnetism over different electron fillings, we observe that the robustness of ferromagnetism over a broad range of fillings is correlated with the flatness of the topmost kagome band. The ordering of the flat bandwidths is:

$A_{2}T_{3}X_{4}$	Cs$_{2}$Pd$_{3}$S$_{4}$	Rb$_{2}$Pd$_{3}$S$_{4}$	Cs$_{2}$Ni$_{3}$S$_{4}$	Cs$_{2}$Ni$_{3}$Se$_{4}$	Rb$_{2}$Pd$_{3}$Se$_{4}$	Cs$_{2}$Pt$_{3}$S$_{4}$
Full Bandwidth	1.086	1.059	0.392	0.519	1.158	0.810
SOC gap at Dirac point	0.058	0.057	0.049	0.054	0.144	0.105
Flat Bandwidth	0.151	0.149	0.154	0.264	0.480	0.171

Notably, the ferromagnetic filling range tends to decrease as the flat bandwidth increases. We organize the relation in Table S4:

A$_2$T$_3$X$_4$	Cs$_2$Pd$_3$S$_4$	Rb$_2$Pd$_3$S$_4$	Cs$_2$Ni$_3$S$_4$	Cs$_2$Pt$_3$S$_4$	Cs$_2$Ni$_3$Se$_4$	Rb$_2$Pd$_3$Se$_4$
Flat Bandwidth	0.151	0.149	0.154	0.171	0.264	0.480
Ferromagnetism	$1 < \nu_{\mathrm{tot}} < 15/8$	$1 < \nu_{\mathrm{tot}} < 15/8$	$1 < \nu_{\mathrm{tot}} < 15/8$	$1 < \nu_{\mathrm{tot}} < 11/7$	$1 < \nu_{\mathrm{tot}} < 3/2$	$1 < \nu_{\mathrm{tot}} < 7/5$

We further find that our phase diagram remains robust under varying the interaction type. For onsite Hubbard interaction, the many-body spectra [Fig. S5] reveal a similar trend, and consequently the symmetry breaking remains largely unchanged.

Momentum Grids

For our exact diagonalization calculations, we used finite-size periodic lattices of various system sizes $N$. To diagonalize on an $N$-unit cell lattice with periodic boundary conditions, we define two boundary vectors $\mathbf{L}_1,\mathbf{L}_2$ such that there exists $u_{ij} \in\mathbb{Z}$ with $\mathbf{L}_i = \sum_{j=1,2} u_{ij}\mathbf{a}_j$. Then the number of unit cells corresponds to

and the two vectors $\mathbf{g}_1,\mathbf{g}_2$ that satisfy $\mathbf{L}_i \cdot\mathbf{g}_j = 2\pi\delta_{ij}$ span the discrete Brillouin zone. We further introduce $N_1,N_2$ so that our $u$ satisfies $u_{ij} = N_i\bar{u}_{ij}$ where $N_i,\bar{u}_{ij} \in\mathbb{Z}$ and $\det\bar{u} = 1$. Then $N = N_1N_2$, and we have a parametrization $K = k_1 + N_1k_2$ in our calculation. The grids that we used are plotted in Figure S6.

Additional Results at Fractional Fillings

In this supplementary section, we present additional numerical studies at fractional fillings for different kagome monolayers. All energy spectra presented and discussed in this section adopt band mixing with at most one interband electron–hole excitation, as mentioned in the main text.

Many-Body Chern Number

We calculate the many-body Berry curvature on the torus of boundary twists $\boldsymbol{\Phi} = (\Phi_1,\Phi_2)$, with $\Phi_\mu\in[0,2\pi)$ using the gauge-invariant discretization [9]. For an $N_{\mathrm{GS}}$-state ground-state multiplet, we define the link variables $U_\mu(\boldsymbol{\Phi}) = \det W_\mu(\boldsymbol{\Phi})/|\det W_\mu(\boldsymbol{\Phi})|$, where $W_\mu$ is the overlap matrix between neighboring multiplets, including the evolution of the single-particle basis. The discretized multiplet Berry curvature is

where $\delta_\mu= \Delta\Phi_\mu\hat{e}_\mu$. Its integral divided by the ground-state degeneracy $N_{\mathrm{GS}}$ yields the many-body Chern number,

where $N_{\Phi}$ is the number of grid meshes. Our many-body Chern number calculation was conducted on $N_{\Phi}=12 \times12$ or $20 \times20$ grid depending on the system size. In the following subsection, we plot the normalized average Berry curvature $2\pi F(\Phi_1,\Phi_2)/N_{\mathrm{GS}}$, where its average over the boundary-twist torus equals $C_{\mathrm{MB}}$.

Fractional Chern Insulators

We report additional details of fractional Chern insulators that we discovered in our exact diagonalization study. We first summarize the many-body spectra of Cs₂Pd₃S₄ at Jain sequence fillings $\nu_{\mathrm{tot}} = 4/3$, $7/5$, $10/7$, $11/7$, $8/5$, $5/3$ at the largest accessible system size [Fig. S7]. The fractional Chern insulator at $\nu_{\mathrm{tot}} = 4/3$ [Fig. S7(a)] is also provided in the main text [Fig. 3(a)]. The result for $\nu_{\mathrm{tot}} = 7/5$ [Fig. S7(b)] is shown as an example for an undetermined phase. The fivefold states are not ground states, and hence the corresponding fractional Chern insulator phase is absent from the phase diagram [Fig. 1(d)]. The calculations of spectral flow under flux insertion and many-body Chern number were done for the five identified Jain sequence fractional Chern insulators [Fig. S8], where we used a smaller system size if possible to reduce computational cost. In the spectral flow plot, at most three eigenstates per momentum sector are presented. The spectral flow for $\nu_{\mathrm{tot}} = 4/3$ [Fig. S8(a)] is also provided in the main text [Fig. 3(b)].

We next turn to Rb₂Pd₃S₄, where a similar set of fractional Chern insulators occurs [Fig. S9]. In addition to the five fillings that were also identified in Cs₂Pd₃S₄, we also report a fractional Chern insulator at $\nu_{\mathrm{tot}} = 12/7$ [Fig. S9(e)]. The spectral flow under flux insertion and many-body Chern numbers are further confirmed [Fig. S10]. Finally, we identify an additional fractional Chern insulator phase in Cs₂Ni₃S₄. The quantum geometry of this material is less favorable than that of the preceding two materials (see Figure S2), and we identify a fractional Chern insulator only at $\nu_{\mathrm{tot}} = 5/3$. The spectral flow exhibits sharp variations near $\Phi_1 = 0$ and $\pi$, and a twist grid of $20 \times20$ points also yielded insufficient overlaps near those regions for a reliable many-body Chern number calculation. We therefore adaptively refine the grid in these regions and confirm convergence to $C_{\mathrm{MB}} = 2/3$. The many-body gaps of fractional Chern insulators are reported in Table S5.

Anomalous Composite Fermi Liquids

We next provide results for anomalous composite Fermi liquids in Cs₂Pd₃S₄. In addition to the many-body spectra, ground state occupation number, and structure factor displayed in the main text [Fig. 3(c-f)], we present the same set of results at $N = 20$ [Fig. S12(a-d)], and also $N = 24$ [Fig. S12(e-h)] with a more symmetric grid [Fig. S6(g)]. The calculations in the main text use an $N = 6 \times4$ grid [Fig. S6(h)] to explore the twofold degeneracies of low-lying states. Whereas the structure factor in main text [Fig. 3(f)] largely exhibits a Fermi-liquid-like behavior, there are also peaks at large $|q|$ points. In our additional calculations, we observe that the peaks in the structure factor diminish with

increasing system size [Figure S12,Figure S12)]. We further note that the fluctuations in the occupation number decrease with increasing system size, from $\lvert n(k)-1/2\rvert< 0.22$ at $N = 20$ [Figure S12] to $\lvert n(k)-1/2\rvert< 0.14$ at $N = 24$ [Figure S12 and Figure 3(e)].

We also report a signature consistent with a quarter-filled anomalous composite Fermi liquid at $\nu_{\mathrm{tot}} = 5/4$, $N = 28$ [Figure S12-Figure S12)]. The low-lying momentum sectors [Figure S12] match those of quarter-filled lowest Landau level [Figure S12], and the momentum occupation number [Figure S12] has no sharp discontinuity, although there exists some modulation ($\lvert n(k) - 1/4\rvert< 0.21$). The structure factor [Figure S12] shows peaks near $\lvert q\rvert= 2k_F$ and nearly vanishes for $\lvert q\rvert> 2k_F$.

Charge Density Waves

We summarize the evidence for charge density waves found at several fillings for different materials. The many-body spectra and ground state structure factor are presented [Figure S13]. We first observe charge density waves at $\nu_{\mathrm{tot}} = 7/4$ for three different compounds, Cs₂Pd₃S₄, Rb₂Pd₃S₄, and Cs₂Ni₃S₄ [Figure S13-Figure S13)]. The fourfold ground-state momentum sectors [Figure S13] are related to each other with $M$-point vectors, consistent with the Bragg peaks in $S(q)$ at $M$-points [Figure S13-Figure S13)]. For Cs₂Ni₃S₄, we find two additional charge density waves at $\nu_{\mathrm{tot}} = 3/2$ and $7/5$ [Figure S13,Figure S13)], where the ground-state momenta are again consistent with Bragg peaks [Figure S13,Figure S13,Figure S13,Figure S13)].

Higher Chern Bands

In this supplementary section, we discuss the higher Chern numbers of the flat bands in Cs₂Pt₃S₄ and Rb₂Pd₃Se₄. We adopt the hopping and spin-orbit coupling conventions introduced in Sec. I and assume a sufficiently large Stoner splitting, $H = H_0 + \Delta\sigma_z$ with $\Delta> 0$, to separate the two spin sectors. Throughout this section, we focus on the upper spin sector and list the three Chern numbers in descending energy order, $C = (C_{\mathrm{top}}, C_{\mathrm{mid}}, C_{\mathrm{bot}})$.

First, we consider the ideal nearest-neighbor kagome model as a reference, choosing $t_1 = 1$ and setting all further-neighbor hopping and spin-orbit coupling amplitudes to zero [Figure S14]. As we turn on a positive nearest-neighbor spin-orbit coupling, $\lambda_1 \to0^+$, the quadratic band touching at $\Gamma$ and the Dirac crossings at $K$ and $K'$ are gapped, producing $C = (+1, 0, -1)$ [Figure S14].

Compared to this ideal model, the Chern numbers for the three bands of Cs$_2$Pt$_3$S$_4$ and Rb$_2$Pd$_3$Se$_4$ with the same sign conventions are $C = (-3, 5, -2)$, and $C = (-2, 3, -1)$, respectively, which are quite different. Using our tight-binding model, we examine the origin of this peculiarity, with emphasis on the Cs$_2$Pt$_3$S$_4$ case.

We introduce a kagome model by retaining the fitted hopping parameters of Cs$_2$Pt$_3$S$_4$ (see Table S1) while initially setting all spin-orbit coupling amplitudes to zero (Figure S14(c),(d)). We first notice that the lowest band is directly separated from the other two. This is because the symmetry-enforced doublet at $K$ and $K'$ now belongs to the upper two bands. This reordering can be seen from the spin-orbit-coupling-free eigenvalues at $K$,

where $E_s$ and $E_d$ denote the singlet and doublet energies, respectively, and $d_K$ is their common diagonal contribution. Whereas $f_K = t_1 > 0$ in the nearest-neighbor model places the doublet below the singlet, the fitted Cs$_2$Pt$_3$S$_4$ parameters give $f_K = -0.0079$ eV, reversing their order, mainly due to large $t_2$. Since each spin sector retains an effective spinless time-reversal symmetry in the absence of spin-orbit coupling, the isolated lowest band has $C_{\mathrm{bot}} = 0$. The upper two bands, meanwhile, exhibit additional crossings beyond those of the ideal kagome model. Besides the quadratic touching at $\Gamma$ and the Dirac touchings at $K$ and $K'$, there are six symmetry-related Dirac points along the $\Gamma$–$M$ directions (marked with orange dots in Figure S14,(Figure S14)). As a weak positive nearest-neighbor spin-orbit coupling $\lambda_1 \to0^{+}$ gaps all these crossings, the local Berry fluxes of the top band, in units of $2\pi$, approach $+1$ near $\Gamma$, $+1/2$ near each of $K$ and $K'$, and $-1/2$ near each of the six additional Dirac points. Consequently,

The contribution from $\Gamma$ thus retains the sign found in the ideal model, while the additional Dirac points reverse the sign of the total upper-band Chern number. The middle band receives the opposite contribution, giving $C = (-1, 1, 0)$. No further gap closing occurs as $\lambda_1$ increases to the fitted value 0.0046 eV with all other spin-orbit coupling terms held at zero (Figure S14).

We now fix $\lambda_1 = 0.0046$ eV and increase the next-nearest-neighbor spin-orbit coupling, $\lambda_2$, from zero to its fitted value 0.0335 eV, keeping the remaining spin-orbit coupling amplitudes zero. The first transition occurs at $\lambda_{2,\mathrm{c1}} = \lambda_1/2 = 0.0023$ eV, where the upper two bands touch simultaneously at $K$ and $K'$ (Figure S14). The reversal of the Dirac mass transfers one unit of Chern number from the top band to the middle band at each valley, changing the sequence from $(-1, 1, 0)$ to $(-3, 3, 0)$. Upon further increasing $\lambda_2$, the descending branch of the split doublet reaches the singlet when

This second transition involves the middle and lowest bands at both valleys and changes their Chern numbers by $+2$ and $-2$, respectively (Figure S14). The resulting evolution is

The $\Gamma$-point gap remains open along this path, and the final sequence persists up to $\lambda_2 = 0.0335$ eV. Restoring the remaining fitted spin-orbit coupling terms continuously from zero also leaves the direct gaps open, connecting this simplified interpolation to the full $\mathrm{Cs}_2\mathrm{Pt}_3\mathrm{S}_4$ model (Figure S14).

A related mechanism yields the $\lvert C\rvert= 2$ upper band in $\mathrm{Rb}_2\mathrm{Pd}_3\mathrm{Se}_4$. Its spin-orbit-coupling-free hopping Hamiltonian also contains six additional Dirac crossings between the upper two bands along the symmetry-related $\Gamma$–$M$ directions. Positive nearest-neighbor spin-orbit coupling gaps these crossings, each contributing $-1/2$ to the upper-band Chern number in the weak-spin-orbit-coupling limit. Together with the $+1$ contribution from the quadratic touching at $\Gamma$, this gives $C_{\mathrm{top}} = 1 - 6/2 = -2$. In this material, the $K$ and $K'$ doublets remain in the lower two bands, and the complete Chern sequence is $(-2, 3, -1)$ in the same upper-spin convention. This sequence is obtained with nearest-neighbor spin-orbit coupling alone and remains unchanged upon continuously restoring the other fitted spin-orbit coupling amplitudes.

References