Coulomb Interaction-Stabilized Isolated Narrow Bands with Chern Numbers $\mathcal{C} > 1$ in Twisted Rhombohedral Trilayer-Bilayer Graphene

Cyprian Lewandowski; Vo Tien Phong

REVIEW 2 major objections 5 minor 2 cited by

Coulomb Interaction-Stabilized Isolated Narrow Bands with Chern Numbers $\mathcal{C} > 1$ in Twisted Rhombohedral Trilayer-Bilayer Graphene

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Pith's one-line read In twisted rhombohedral trilayer-bilayer graphene, Coulomb interactions — not the bare band structure — isolate narrow valence bands with Chern numbers 2 and 3, making the stack a concrete candidate for fractionalizing a higher-Chern band.

desk verdict A careful HF study of a concrete TRMG3+2 platform for interaction-stabilized higher-Chern bands; the main caveat is the restricted ground-state search, which the authors honestly flag. read the letter →

arxiv 2505.07981 v2 pith:B7EZLRWV submitted 2025-05-12 cond-mat.mes-hall cond-mat.str-el

Vo Tien Phong , Cyprian Lewandowski This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.str-el

keywords twistedrhombohedraltrilayer-bilayergraphenefractionalCherninsulatorhighernumberquantumanomalousHalleffectHartree-Fockmeanfieldgeometrytraceconditionviolationmoiréflatbands

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper predicts that twisted rhombohedral trilayer-bilayer graphene (TRMG3+2), a five-layer stack in which a rhombohedral trilayer is rotated against a Bernal bilayer at $\theta \sim 1.2^\circ$, is a field-tunable anomalous Chern insulator whose top valence bands carry Chern numbers $C=2$ and $C=3$. Its central claim is that Coulomb repulsion does the essential work: at the Hartree-Fock mean-field level, interactions spectrally isolate these narrow bands from neighboring bands and improve their quantum geometry, lowering the trace-condition violation $\lambda$ that measures how suitable a band is for hosting fractional Chern insulating states. At optimal parameters the bands have bandwidths of a few meV, gaps of roughly 8–11 meV, and $\lambda$ values down to 0.55, comparable to the values computed for the two material families in which fractional quantum anomalous Hall states have already been seen. If the prediction holds, partially doping these bands would produce fractional states with quasiparticle statistics beyond the Landau-level paradigm, in a lattice system requiring no magnetic field.

What carries the argument

The load-bearing object is the valley-projected continuum Hamiltonian of TRMG3+2, built from the two substack Hamiltonians with all known rhombohedral-graphite hoppings, a moiré tunneling matrix with distinct $AA$ and $AB$ amplitudes, and an interlayer displacement field $\Delta$, then solved self-consistently at the Hartree-Fock level with a layer-dependent, dual-gate-screened Coulomb potential. The diagnostic that carries the argument is the trace-condition violation $\lambda = (2\pi)^{-1}\int d^2k\,[\operatorname{Tr} g_{\mu\nu}(k) - |B_z(k)|]$, which quantifies how far a band's quantum metric rises above the Berry-curvature bound and thus how favorable the band is for fractionalization; the paper tracks $\lambda$, bandwidth $W$, and gap $\delta$ through $(\theta,\Delta)$ phase space, combining them into a figure of merit $\delta/W\lambda$. The physical mechanism is that the rhombohedral trilayer's $\pm|k|^3$ dispersion can be kinetically quenched by a negative displacement field, while the moiré hybridization — not the displacement field alone — creates the global gaps; Coulomb exchange then polarizes one isospin flavor and opens the mean-field gap that isolates the Chern band.

What would settle it

Build a TRMG3+2 device with twist angle near $1.2^\circ$ and apply a negative displacement field inside the predicted window (about $-12$ to $-21$ meV): the calculation predicts a quantized anomalous Hall plateau of $2e^2/h$ (antiparallel or parallel stack) or $3e^2/h$ (parallel stack) at one electron per moiré cell, with a gap on the scale of 8–11 meV. Finding no quantized plateau anywhere in that window, or a trivial zero-Hall insulator instead, would falsify the central claim; a corroborating failure mode is the absence of the $C=3$ phase, which the paper's own robustness scans show shrinks or disappears when $\gamma_0$ is reduced or lattice relaxation is switched off.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that TRMG3+2 becomes an interaction-stabilized higher-Chern insulator: after self-consistent Hartree-Fock renormalization at one electron per moiré cell, the first valence band below the Fermi level is a spectrally isolated, narrow, isospin-polarized band with $C=2$ in the antiparallel stack and $C=2$ or $C=3$ in the parallel stack, for displacement fields around $\Delta \in [-21,-12]$ meV and angles near $1.1^\circ$–$1.4^\circ$. Crucially, the optimized bands exist at parameter values where the non-interacting spectrum shows no isolated Chern band at all — for the antiparallel $C=2$ state at $\theta=1.20^\circ$, $\Delta=-18$ meV the band is not even fully gapped without interactions — so the band isolation is attributed to Coulomb interactions themselves. Interactions also lower the trace-condition violation from its non-interacting value (for example $\lambda\approx1.7\to1.32$ in the antiparallel case), meaning the renormalized band's quantum geometry sits closer to the ideal Landau-level geometry. The $C=2$ band in the antiparallel stack is the most robust, surviving changes in dielectric constant, gate distance, hopping parameters, and the $\gamma_{AA}/\gamma_{AB}$ ratio, while the parallel-stack $C=3$ band reaches the smallest $\lambda\approx0.55$; the paper proposes these bands as the appropriate starting points for fractional Chern insulating states upon partial doping.

Load-bearing premise

The prediction rests on how the twisted interface actually relaxes: the model captures lattice relaxation only through the single ratio $\gamma_{AA}/\gamma_{AB}$ of the two interfacial tunneling strengths, and the paper states that a comprehensive relaxation analysis is beyond its scope; if full relaxation reshapes the moiré stack differently, the isolated Chern bands and the phase diagram could shift or disappear.

Editorial extensions

If this is right

Partially doping the isolated $C=2$ or $C=3$ band should produce fractional anomalous Hall states, which would be the first fractionalized states to emerge from a higher-Chern band and would host quasiparticle statistics with no Landau-level counterpart.
No fine tuning is required: the $C=2$ and $C=3$ phases occupy contiguous regions of twist angle and displacement field, roughly $0.3^\circ$–$0.5^\circ$ wide in $\theta$ and about 10 meV wide in $\Delta$.
The optimal trace-condition violations ($\lambda \approx 0.55$–$0.87$) fall in the range computed for the two material families where fractional quantum anomalous Hall states have been observed, indicating the band geometry is as good as or better than those proven platforms.
Because the active band's charge sits mostly on the outer trilayer layer ($\gtrsim 70\%$), the mechanism resembles rhombohedral graphene–hBN devices, suggesting the same interaction-stabilization recipe carries over to the wider twisted rhombohedral multilayer family, possibly with even higher Chern numbers for thicker stacks.
An independent experimental report, noted in the paper, of a $C=3$ integer Chern insulator in this exact material at $\theta \approx 1.5^\circ$ lies inside the predicted parameter window and provides the phase diagram its first direct corroboration.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The dielectric constant is a direct experimental dial on the central mechanism: the paper's stability scans show the antiparallel $C=2$ phase gives way to a metal once $\epsilon_r \gtrsim 21$, so a device with stronger screening should lose the quantized plateau — a test the paper computes but does not itself propose as an experiment.
The $C=3$ phase is the most fragile result — it shrinks markedly when the nearest-neighbor hopping $\gamma_0$ is reduced or lattice relaxation is switched off in the model — so the decisive follow-up is a full atomistic relaxation calculation; the single ratio $\gamma_{AA}/\gamma_{AB}$ is unlikely to capture all reconstruction effects.
Motivated by the low mean-field values of $\lambda$, an exact-diagonalization study projected onto the isolated band is the natural next step to locate the fractional states and their fillings; the paper explicitly defers such a calculation.
The $C=2$ and $C=3$ bands concentrate charge at different Wyckoff positions in the moiré cell, suggesting that any fractional state formed from them would carry a lattice-scale charge modulation absent in Landau levels — a distinction a local probe could in principle resolve.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a solid mean-field study of a realistic moiré platform for |C|>1 Chern bands, and it does what it claims as long as you read "stabilized" as "self-consistent and lower in energy than one symmetric and one polarized ansatz." I would send it to referees. The weak point is not the numerics; it is the ground-state search, which the authors openly acknowledge.

What is genuinely new: the non-interacting C=2,3 bands in TRMG stacks were already known, but showing that Hartree-Fock renormalization produces spectrally isolated, narrow bands with favorable trace-condition values at parameters where the non-interacting bands are not isolated is a real step. The paper makes a testable prediction (C=3 near 1.5 degrees at negative displacement) and cites a recent experiment at that angle as consistent. The numerics are careful: multiple band numbers, grid sizes, dielectric constants, gate distances, hopping parameter sets, and a derived layer-dependent dual-gate Coulomb potential that checks against the classical capacitor model. The convergence checks on the three headline states are reassuring. The quantum-geometry diagnostics are given explicitly, and the authors do not overclaim about fractionality—they defer exact diagonalization to future work.

Soft spots, in proportion. The main one is the restricted ground-state search: only a symmetric ansatz and one isospin-polarized ansatz are compared. The supplementary text says randomized searches are "computationally prohibitive." Valley-coherent, nematic, or other broken-symmetry states could in principle be lower, and the phase boundaries in Figs. 2-3 are not fully settled. I do not think this kills the paper—the phases persist across many parameter sweeps, and the authors are transparent—but the word "stabilized" carries more weight than the search justifies. A referee should ask for the claim to be softened to "self-consistent within the studied ansatz family" or for a few random-seed checks at the optimal points. Second, full lattice relaxation is deferred; the gamma_AA/gamma_AB ratio is a crude proxy. The C=3 phase is more sensitive to parameters than C=2, especially at reduced gamma_0 (it essentially disappears without relaxation in the SM). This is a known boundary, not a hidden flaw.

Who this is for: the moiré and fractional Chern insulator community. It is a credible candidate-material paper with honest limitations. I would cite it for the HF phase diagrams and the interaction-stabilized higher-Chern claim, with the ansatz caveat. Send it to a serious referee; the numerics deserve scrutiny but the work is clearly above the bar.

Referee Report

2 major / 5 minor

Summary. The manuscript proposes twisted rhombohedral trilayer-bilayer graphene (TRMG3+2) as a platform for higher-Chern isolated narrow bands, reporting mean-field Hartree-Fock evidence for Coulomb-interaction-stabilized bands with Chern numbers C=2 (antiparallel and parallel stacking) and C=3 (parallel stacking) at fillings of one electron per moiré cell. The authors use a continuum model with all known rhombohedral-graphite hoppings, a layer-dependent dual-gate Coulomb potential, and two values of the lattice-relaxation ratio γ_AA/γ_AB, and they characterize the resulting bands by their gaps, bandwidths, Berry curvature, quantum metric, and trace-condition violation λ. The central claim is that Coulomb interactions are simultaneously responsible for spectrally isolating these narrow bands and for improving their quantum geometry, making them promising candidates for fractional Chern insulating states at partial filling. A note added cites a recent experimental observation of C=3 at θ≈1.5° as supportive.

Significance. If the mean-field results hold, this is a valuable step toward realizing fractional Chern insulators from higher-Chern bands: it identifies a specific, field-tunable material with isolated C=2 and C=3 bands whose quantum geometry approaches the ideal Chern-band limit. The numerical study is extensive and carefully cross-checked: convergence in band number and k-grid is documented (SM Figs. S41–S43), results are compared with and without lattice relaxation, with reduced γ0, with layer-independent versus layer-dependent Coulomb potentials, and for two dielectric constants. The layer-dependent Coulomb potential is derived from first principles and checked against the classical capacitor model. The prediction of C=3 at intermediate twist angles is falsifiable and has received preliminary experimental support. The main caveat is the restricted mean-field ansatz space, which limits the strength of the word 'stabilized' as applied to the ground state.

major comments (2)

[SM 'Initial Ansatz for Density Matrix'; main-text footnote after Eq. (2)] The ground-state search compares only a symmetric ansatz and a one-flavor-polarized ansatz. Because the paper's central claim is that the C=2 and C=3 bands are 'stabilized by Coulomb interactions,' the possibility that valley-coherent, spin-valley entangled, nematic, or charge-ordered states are lower in energy at the quoted (θ,Δ) is load-bearing: if such a state wins, the predicted phase is displaced. The SM states that randomized searches are 'computationally prohibitive,' which is an honest limitation, but the abstract and conclusion wording 'stabilized by Coulomb interactions' overstates what has been established. I recommend either performing additional biased searches (e.g., valley-coherent or nematic initial density matrices) at the optimal f.o.m. points and along phase boundaries, or explicitly restating the claim as 'lowest among the symmetric and isospin-polarized ansätze' throughout the abstract and conclusions.
[Main text, discussion of Figs. 2–3 and footnote 10; abstract] The statement that 'Coulomb interactions have the tendency to lower the trace condition violation' and the abstract's claim that Coulomb interactions are 'simultaneously crucial to the ... stabilization of their favorable quantum geometry' are supported by comparing optimal λ values at different (θ,Δ) points before and after HF renormalization (e.g., non-interacting λ≈1.7 versus HF λ≈1.32 for the antiparallel C=2 phase). Because the comparison is not at fixed parameters, the data do not directly establish that interactions reduce λ at a given point; they establish that interactions produce isolated bands with favorable λ at nearby but different parameters. Please provide same-(θ,Δ) comparisons for representative points where both non-interacting and HF bands are gapped, or qualify the claim accordingly.

minor comments (5)

[Main text, 'Self-consistent mean field phase diagram of the antiparallel configuration'] The sentence 'Focusing on the C=2 phase, we further characterize its spectral and quantum geometric properties in Fig. 1(b-d)' should refer to Fig. 2(b-d), since Figs. 1(b-d) show the non-interacting Chern characterization while Figs. 2(b-d) show the HF gap, bandwidth, and λ.
[Note added] The note states that the experimental C=3 observation at θ≈1.5° is 'entirely consistent' with the predicted C=3 window θ∈[1.05°,1.45°], but 1.5° lies just outside this window; please clarify the intended consistency (e.g., parameter uncertainty or a modest extension of the predicted window).
[SM, 'Continuum Hamiltonian'] The sentence 'For Param three and Param three, we reduce γ0...' contains a typo; it should read 'Param three and Param four.'
[SM, figure captions S37 and S38] The captions contain the typo 'without lattice relaxation amd' instead of 'and.'
[Main text, figure-of-merit definition] The phrase 'A large f.o.m. means that the band is susceptible to further renormalization by residual Coulomb interactions' appears to have the direction reversed: a large gap-to-bandwidth ratio usually indicates robustness to perturbation rather than susceptibility; please reword.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the C=2/C=3 phases emerge from self-consistent mean-field calculations using externally fixed inputs, not from fitted parameters or self-referential definitions.

full rationale

The paper's derivation chain is self-contained and does not reduce its predictions to its inputs by construction. The single-particle Hamiltonian (Eq. 1) uses standard continuum-model hoppings {gamma0,...,gamma4} taken from experimental literature and literature-based interlayer tunnel amplitudes (gamma_AA/gamma_AB = 0.1/0.12 eV), none of which are fitted to the target Chern numbers or to the reported C=2/C=3 phase regions. The Hartree-Fock calculation (Eqs. 2-4) solves the Coulomb interaction self-consistently starting from physically motivated isospin-polarized and symmetric ansatze; the resulting isolated Chern bands are outputs of the self-consistency loop, and the paper explicitly demonstrates that they appear at (theta,Delta) points where the non-interacting bands are not isolated, so the central 'stabilized by Coulomb interactions' claim is an emergent numerical result rather than a restatement of the non-interacting Chern maps. The quantum-geometry diagnostic lambda is computed from the HF bands, not imposed as a target. The only self-citation that could invite scrutiny is Ref. [98], a companion paper used for robustness checks and for the layer-dependent Coulomb potential derivation; however, the Coulomb potential derivation is reproduced in the Supplementary Material, and the robustness checks are parameter sweeps rather than fitted inputs, so the self-citation is not load-bearing. The acknowledged restriction of the ground-state search to two initial ansatze (with valley-coherent or nematic states left for future study) is an honest limitation affecting the definitiveness of the ground-state claim, but it is a correctness/completeness caveat, not a circularity. No quoted step equates a prediction with a fitted parameter or defines the output in terms of the input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard continuum-model and mean-field assumptions, plus an ad hoc treatment of lattice relaxation and a limited ground-state search. No free parameters are fitted to the target result; all model parameters come from prior experiments or are scanned environmental/tuning parameters. No new particles or entities are introduced.

assumptions (5)

domain assumption The continuum model with the specified hoppings describes the low-energy electronic structure of twisted rhombohedral multilayer graphene.
The central results are computed from Eq. 1, a standard valley-projected continuum Hamiltonian with parameters from Ref. [93]; deficiencies in the model would propagate to the predicted Chern bands.
domain assumption Electron-electron interactions are represented at the Hartree-Fock mean-field level.
The phase diagrams in Figs. 2-4 are obtained from self-consistent HF (Eqs. 2-4). Fluctuations beyond mean-field are not computed for the topological phases, so the stabilization may be over- or under-estimated.
ad hoc to paper The ratio gamma_AA/gamma_AB captures the relevant lattice relaxation at the twist interface.
The paper varies this ratio for robustness but does not perform a full relaxation calculation; it states a comprehensive analysis is beyond scope. If relaxation reconstructs the interface differently, the predicted isolation of Chern bands could change.
ad hoc to paper The two initial density matrix ansatzes (symmetric and one-flavor polarized) suffice to find the relevant ground state.
The supplementary notes randomized searches are computationally prohibitive. Other symmetry-broken states (e.g., valley-coherent, nematic) are not explored, potentially missing lower-energy states.
domain assumption The trace condition violation lambda is a valid diagnostic for the propensity of a band to host a fractional Chern insulator.
lambda measures closeness to the zeroth Landau level and is used in the literature for twisted MoTe2 and graphene-hBN; it is a heuristic, not a proof of FCI stability.

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Cite this review

Pith. "Pith review of Coulomb Interaction-Stabilized Isolated Narrow Bands with Chern Numbers $\mathcal{C} > 1$ in Twisted Rhombohedral Trilayer-Bilayer Graphene." pith.science (2026). https://pith.science/paper/B7EZLRWV

@misc{pith2026250507981,
  author       = {Pith},
  title        = {Pith review of: Coulomb Interaction-Stabilized Isolated Narrow Bands with Chern Numbers $\mathcalC > 1$ in Twisted Rhombohedral Trilayer-Bilayer Graphene},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B7EZLRWV}},
  note         = {Machine review of arXiv:2505.07981}
}

abstract

Recently, fractional quantum anomalous Hall effects have been discovered in two-dimensional moir\'{e} materials when a topologically nontrivial band with Chern number $\mathcal{C}=1$ is partially doped. Remarkably, superlattice Bloch bands can carry higher Chern numbers that defy the Landau-level paradigm and may even host exotic fractionalized states with non-Abelian quasiparticles. Inspired by this exciting possibility, we propose twisted \textit{rhombohedral} trilayer-bilayer graphene at $\theta \sim 1.2^\circ$ as a field-tunable quantum anomalous Chern insulator that features spectrally-isolated, kinetically-quenched, and topologically-nontrivial bands with $\mathcal{C} = 2,3$ favorable for fractional phases once fractionally doped, as characterized by their quantum geometry. Based on extensive self-consistent mean-field calculations, we show that these phases are stabilized by Coulomb interactions and are robust against variations in dielectric environment, tight-binding hopping parameters, and lattice relaxation.

Figures

Figures reproduced from arXiv: 2505.07981 by the authors.

**Figure 1.** Moire heterostructure schematic and topological char- ´ acterization of the non-interacting band structures. (a) Construction of a twisted 3 + 2 rhombohedral stack by rotating the bottom trilayer (top bilayer) by +θ/2 (−θ/2). There are two inequivalent configurations, antiparallel and parallel, depending on the relative stacking orders of the two subsystems. (b) Single-particle (non-interacting) Chern-number chara… view at source ↗

**Figure 2.** Self-consistent mean field phase diagram of the antiparallel configuration and characterization of the C = 2 phase. (a) Phase diagram in (θ, ∆) space showing several distinct metallic and insulating phases. Of particular interest is the presence of a C = 2 phase at negative ∆ and θ ∈ [1.1 ◦ , 1.4 ◦ ] featuring a spectrally-isolated, isospin-polarized, topologically-nontrivial band right below the Fermi level. The ba… view at source ↗

**Figure 3.** Self-consistent mean field phase diagram of the parallel configuration and characterization of the C = 2 and C = 3 phases. (a) Phase diagram in (θ, ∆) space showing several distinct metallic and insulating phases. Of importance is the presence of a contiguous region showing C = 2 and C = 3 phases at negative ∆ and θ ∈ [0.95◦ , 1.45◦ ] featuring a spectrally-isolated, isospinpolarized, topologically-nontrivial band… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Stability of topologically nontrivial phases. We show the evolution of states at fixed (θ, ∆) as a function of (D, ϵr). In the first column, we classify the states using the same method as in Figs. 2 and 3. In the second to fourth columns, we characterize the band gap,…

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Forward citations

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Reference graph

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The optimal angles are aroundθ∼1.2−1.3 ◦.The range of optimal displacement fields is∆∈[−27,−18]meV
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Forγ=−3100meV , increasing Coulomb interactions increases the phase-space area of this phase, decreases the optimal λ,and increases the bandwidths and band gaps
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The dependence of band gaps on lattice relaxation appears more complicated: it increasesδabove but decreases δbelow

Lattice relaxation appears to be beneficial to this phase as it decreases the optimalλwhile simultaneously lowers the bandwidths. The dependence of band gaps on lattice relaxation appears more complicated: it increasesδabove but decreases δbelow
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Here, we also observe that adding lattice relaxation decreases λand decreasesW,consistent with the other simulations

With reducedγ 0 =−2700meV , this phase remains robust. Here, we also observe that adding lattice relaxation decreases λand decreasesW,consistent with the other simulations. For theC= 2phase of the parallel configuration, we observe the following:
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The bandwidths are remarkably narrower in this phase compared to the other nontrivial Chern phases in both configura- tions. 49 Parameter set ϵr # of points θ(degrees) ∆(meV) δabove (meV) δbelow (meV) W(meV) λ Param one 12 22 1.10◦ −21 7.85 27.04 3.75 0.87 Param one 6 16 1.15◦ −30 16.33 40.88 6.51 0.74 Param two 12 27 1.10◦ −18 8.60 14.32 3.92 0.89 Param ...
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Forγ=−3100meV , lattice relaxation does not appear to affect the phase much. While including lattice does decrease optimalλ,it also decrease the phase areas
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true” values at such a smaller mesh. For a larger mesh ofN k×N k = 12×12,we observe in all three cases that the violations are within6%of the “true

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

The optimal angles are aroundθ∼1.2−1.3 ◦.The range of optimal displacement fields is∆∈[−27,−18]meV

[2] [2]

Forγ=−3100meV , increasing Coulomb interactions increases the phase-space area of this phase, decreases the optimal λ,and increases the bandwidths and band gaps

[3] [3]

The dependence of band gaps on lattice relaxation appears more complicated: it increasesδabove but decreases δbelow

Lattice relaxation appears to be beneficial to this phase as it decreases the optimalλwhile simultaneously lowers the bandwidths. The dependence of band gaps on lattice relaxation appears more complicated: it increasesδabove but decreases δbelow

[4] [4]

Here, we also observe that adding lattice relaxation decreases λand decreasesW,consistent with the other simulations

With reducedγ 0 =−2700meV , this phase remains robust. Here, we also observe that adding lattice relaxation decreases λand decreasesW,consistent with the other simulations. For theC= 2phase of the parallel configuration, we observe the following:

[5] [5]

The optimal angles are aroundθ∼1.1−1.3 ◦.The range of optimal displacement fields is∆∈[−30,−18]meV

[6] [6]

The bandwidths are remarkably narrower in this phase compared to the other nontrivial Chern phases in both configura- tions. 49 Parameter set ϵr # of points θ(degrees) ∆(meV) δabove (meV) δbelow (meV) W(meV) λ Param one 12 22 1.10◦ −21 7.85 27.04 3.75 0.87 Param one 6 16 1.15◦ −30 16.33 40.88 6.51 0.74 Param two 12 27 1.10◦ −18 8.60 14.32 3.92 0.89 Param ...

[7] [7]

The trace condition violations for this phase are generally smaller than theC= 2phase of the antiparallel configuration but are generally larger than theC= 3phase

[8] [8]

However, increasing Coulomb interactions increases band gaps and bandwidths whether or not lattice relaxation is accounted for

Strengthening Coulomb interactions decreasesλwhen lattice relaxation is kept but increasesλwhen lattice relaxation is ignored. However, increasing Coulomb interactions increases band gaps and bandwidths whether or not lattice relaxation is accounted for. Therefore, increasing Coulomb interactions is helpful to this phase. However, the phase areas have a c...