Quantum anomalous Hall phases in gated rhombohedral graphene

Guillaume Bal; Matthew Frazier

arxiv: 2509.05439 · v2 · submitted 2025-09-05 · ❄️ cond-mat.mes-hall · math-ph· math.MP

Quantum anomalous Hall phases in gated rhombohedral graphene

Matthew Frazier , Guillaume Bal This is my paper

Pith reviewed 2026-05-18 18:11 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall math-phmath.MP

keywords quantum anomalous Hall effectrhombohedral grapheneDirac operatorstopological phasesbulk-edge correspondencedisplacement fieldchiral edge statesmultilayer graphene

0 comments

The pith

A coupled Dirac operator model classifies all quantum anomalous Hall phases in gated rhombohedral graphene and links them to chiral edge states with quantized current.

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

The paper constructs a model of coupled Dirac operators for spin- and valley-polarized gated rhombohedral graphene with any number of layers. It classifies every quantum anomalous Hall phase allowed inside this model and proves a bulk-edge correspondence in which each bulk phase produces chiral edge states that carry a quantized anomalous Hall current. When the displacement field stays small relative to interlayer coupling, the model recovers the familiar phases whose charge equals the layer count. Larger fields produce additional topological transitions that change the number of conducting channels to new quantized values. Numerical checks confirm that the predicted edge currents match the bulk invariants.

Core claim

We consider a coupled system of Dirac operators that models spin- and valley-polarized gated rhombohedral graphene with an arbitrary number of layers. We classify all quantum anomalous Hall phases that are compatible with the model and show that a bulk-edge correspondence exists between bulk phases and chiral edge states carrying a quantized anomalous Hall current. When the displacement field is sufficiently small compared to the interlayer coupling in the RHG application, we retrieve the known phases where the charge is given by the number of graphene layers. When the displacement field increases, we identify all possible topological phase transitions and corresponding quantized chiral edge

What carries the argument

The coupled system of Dirac operators that encodes the rhombohedral multilayer stacking together with the gate-induced displacement field.

If this is right

For small displacement fields the anomalous Hall charge equals the number of graphene layers.
Increasing the displacement field triggers topological phase transitions that change the number of quantized chiral edge charges.
The bulk-edge correspondence guarantees matching chiral edge states for every classified bulk phase.
Numerical simulations confirm the complete set of phases and transitions allowed by the model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Varying gate voltage in experiment could select among the predicted numbers of conducting edge channels.
The classification supplies a concrete way to predict how many edge channels appear for given layer count and field strength.
The same operator approach may classify phases in other multilayer 2D systems with similar stacking.

Load-bearing premise

The physical gated rhombohedral graphene system with spin and valley polarization is faithfully represented by the chosen coupled system of Dirac operators.

What would settle it

An experimental count of chiral edge modes or measured Hall conductivity in a sample with known layer number and displacement field that deviates from the quantized values predicted by the classification.

Figures

Figures reproduced from arXiv: 2509.05439 by Guillaume Bal, Matthew Frazier.

**Figure 2.** Figure 2: Bulk spectrum for 5-layer model with γ = 1 and u = (1, 2.5, 3.5, 7) from left to right, top to bottom. For the low-potential case we recover the “Mexican hat” shape described in [23, 32] for the bands nearest E = 0 and for the high-potential case we note 5 minima (maxima) in the positive (negative) band closest to E = 0 as described in [7]. Intermediate values of u produce an intermediate number of minima … view at source ↗

**Figure 3.** Figure 3: Bulk spectrum for m = 9 layers. As number of layers increases the global band gap near phase transitions becomes increasingly small even for intermediate values of kx. Middle frame shows the branch closest to E = 0 and right from shows detail of the minimum in said branch closest to kx = 0. Spectrum shown is for (γ, u) = (1, 2.25) Assuming that for large enough |y|, the coefficients of A(kx, E) are constan… view at source ↗

**Figure 4.** Figure 4: Numerically calculated spectra for m = {3, 4, 5, 6} for all phases. We assume γ = 1 for simplicity and calculate the spectrum of HI for u = 2 (m = 3), u = (5, 2) (m = 4), u = (4, 1.2) (m = 5), and u = (6, 4, 2) (m = 6). The approximate critical values from (13) are uc ≈ 3.46 (m = 4), uc ≈ 2.83 (m = 5), and uc ≈ (2.58, 5.77) (m = 6). The number of edge states agrees with [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗

**Figure 5.** Figure 5: Illustration of the transition through the critical value [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: Eigenvectors of edge states in the 3-layer case (( [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: Edge spectrum for 5-layer case with A → A∗ transition. Parameter values used are u = (4, 1.2) and γ = 1. Comparison with [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

read the original abstract

We consider a coupled system of Dirac operators that models spin- and valley-polarized gated rhombohedral graphene (RHG) with an arbitrary number of layers. We classify all quantum anomalous Hall phases that are compatible with the model and show that a bulk-edge correspondence exists between bulk phases and chiral edge states carrying a quantized anomalous Hall current. When the displacement field is sufficiently small compared to the interlayer coupling in the RHG application, we retrieve the known phases where the charge is given by the number of graphene layers. When the displacement field increases, we identify all possible topological phase transitions and corresponding quantized chiral edge charges. Numerical simulations confirm the theoretical findings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Classifies all QAH phases in arbitrary-layer gated rhombohedral graphene via coupled Dirac operators, recovers small-D results, and maps new transitions plus quantized edge charges at larger D.

read the letter

This paper classifies every quantum anomalous Hall phase allowed by its coupled Dirac-operator model for gated rhombohedral graphene of any layer number. It establishes bulk-edge correspondence and tracks how the quantized chiral edge charge changes as the displacement field increases, recovering the known layer-count result at small fields and locating additional transitions beyond that regime. Numerical checks inside the model support the transitions. The systematic organization for arbitrary layers and the explicit link to measurable edge currents are the clearest additions over prior work on small fields. The main limitation is that everything stays inside the low-energy Dirac description. When the displacement field grows large enough that the potential difference pushes bands away from the Dirac point, terms omitted from the model (trigonal warping, next-nearest-neighbor hopping, lattice-scale effects) could shift gaps or Chern numbers. The numerics confirm the classification within the stated model, but the paper does not appear to benchmark the large-D spectrum against a more complete tight-binding Hamiltonian. That leaves the new phases as predictions of the effective theory rather than direct statements about real samples. The work is aimed at theorists who build phase diagrams for multilayer graphene or design topological devices around edge currents. A reader who needs a compact map of possible quantized responses will get concrete value from the classification. It is worth sending to peer review; the claims are model-dependent but internally consistent and numerically supported enough for referees to evaluate the derivations and the range of validity.

Referee Report

2 major / 1 minor

Summary. The manuscript analyzes a coupled system of Dirac operators modeling spin- and valley-polarized gated rhombohedral graphene with an arbitrary number of layers. It classifies all quantum anomalous Hall phases compatible with the model, establishes a bulk-edge correspondence linking bulk topological invariants to chiral edge states carrying quantized anomalous Hall current, recovers the known layer-number-dependent phases at small displacement field D, and identifies additional topological transitions at larger D, with numerical simulations presented as confirmation.

Significance. If the effective Dirac model remains faithful in the large-D regime, the complete classification and bulk-edge correspondence would provide a predictive framework for locating new QAH states in multilayer rhombohedral graphene, extending existing results and offering concrete guidance for transport experiments targeting quantized chiral currents.

major comments (2)

[Model description] Model description (opening paragraphs): the central claim that new QAH phases and transitions appear for D comparable to or larger than the interlayer coupling rests on the unverified assumption that the low-energy Dirac approximation continues to capture the correct gap structure and Chern numbers; no comparison is shown to a tight-binding Hamiltonian that includes trigonal warping or next-nearest-neighbor terms in this regime.
[Numerical simulations] Numerical simulations section: the statement that simulations confirm the theoretical phase diagram lacks explicit details on how Chern numbers or edge-state spectra are extracted for the large-D cases, making it impossible to judge whether the numerics independently test the extrapolation beyond the Dirac-point regime.

minor comments (1)

[Abstract] The abstract and introduction could clarify whether the bulk-edge correspondence is proved for the full family of coupled Dirac operators or only for the RHG parameter choices.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the positive assessment of the classification and bulk-edge correspondence. We address each major comment below and outline the revisions we will make.

read point-by-point responses

Referee: [Model description] Model description (opening paragraphs): the central claim that new QAH phases and transitions appear for D comparable to or larger than the interlayer coupling rests on the unverified assumption that the low-energy Dirac approximation continues to capture the correct gap structure and Chern numbers; no comparison is shown to a tight-binding Hamiltonian that includes trigonal warping or next-nearest-neighbor terms in this regime.

Authors: We agree that the range of validity of the effective Dirac model for large displacement fields is an important consideration. Our analysis is performed entirely within the coupled Dirac-operator model, which is the standard low-energy effective theory for gated rhombohedral graphene. Within this framework the topological classification, bulk invariants, and bulk-edge correspondence are exact. We will add a dedicated paragraph in the revised manuscript discussing the regime of applicability, noting that trigonal warping and next-nearest-neighbor terms become relevant only when D greatly exceeds the interlayer coupling scale; the topological transitions we identify occur at intermediate D where the Dirac approximation remains faithful. We will also reference existing tight-binding studies that support the persistence of the Dirac-like gap structure in this window. revision: partial
Referee: [Numerical simulations] Numerical simulations section: the statement that simulations confirm the theoretical phase diagram lacks explicit details on how Chern numbers or edge-state spectra are extracted for the large-D cases, making it impossible to judge whether the numerics independently test the extrapolation beyond the Dirac-point regime.

Authors: We apologize for the insufficient methodological detail. In the revised manuscript we will expand the numerical section to specify that Chern numbers are obtained by integrating the Berry curvature over a dense k-space mesh using the standard discretized formula, while edge-state spectra are computed for finite-width ribbons with open boundaries in the transverse direction. These calculations are performed with the identical effective Hamiltonian employed in the analytic part of the work, thereby providing an independent numerical confirmation of the phase diagram, including the large-D transitions. revision: yes

Circularity Check

0 steps flagged

No circularity: classification derived directly from Dirac-operator model analysis

full rationale

The paper defines a coupled system of Dirac operators as the model for spin- and valley-polarized gated rhombohedral graphene, then performs a mathematical classification of all quantum anomalous Hall phases compatible with that model and establishes bulk-edge correspondence via analysis of the operator spectrum and topological invariants. Known phases for small displacement field are recovered as special cases of the same classification, while new transitions at larger fields follow from the model's parameter dependence; numerical simulations serve as independent verification rather than input fitting. No step reduces a prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness claim, or renames an external result. The derivation remains self-contained within the stated model assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The classification depends on the domain assumption that the physical system is captured by the coupled Dirac operators and on standard properties of Dirac operators; no free parameters are fitted to data and no new entities are postulated.

axioms (1)

domain assumption Gated rhombohedral graphene with spin and valley polarization is modeled by a coupled system of Dirac operators.
This modeling choice is stated at the start of the abstract as the basis for the entire analysis.

pith-pipeline@v0.9.0 · 5630 in / 1218 out tokens · 60430 ms · 2026-05-18T18:11:05.390919+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Relation between the paper passage and the cited Recognition theorem.

We consider a coupled system of Dirac operators that models spin- and valley-polarized gated rhombohedral graphene (RHG) with an arbitrary number of layers... classify all quantum anomalous Hall phases... bulk-difference invariant (BDI)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages · 2 internal anchors

[1]

J. E. Avron, R. Seiler, and B. Simon , Charge deficiency, charge transport and comparison of dimensions , Comm. Math. Phys., 159 (1994), pp. 399--422

work page 1994
[2]

Bal , Continuous bulk and interface description of topological insulators , Journal of Mathematical Physics, 60 (2019)

G. Bal , Continuous bulk and interface description of topological insulators , Journal of Mathematical Physics, 60 (2019)

work page 2019
[3]

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height 2pt depth -1.6pt width 23pt, Topological invariants for interface modes , Communications in Partial Differential Equations, 47 (2022), pp. 1636--1679

work page 2022
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height 2pt depth -1.6pt width 23pt, Topological charge conservation for continuous insulators , Journal of Mathematical Physics, 64 (2023), p. 031508

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height 2pt depth -1.6pt width 23pt, Continuous topological insulator, classification and bulk edge correspondence , arXiv preprint arXiv:2412.00919, (2024)

work page arXiv 2024
[6]

G. Bal, P. Cazeaux, D. Massatt, and S. Quinn , Mathematical models of topologically protected transport in twisted bilayer graphene , Multiscale Modeling & Simulation, 21 (2023), pp. 1081--1121

work page 2023
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Bal and D

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Bal and J

G. Bal and J. Yu , Topological equatorial waves and violation (or not) of the bulk edge correspondence , Journal of Physics A: Mathematical and Theoretical, 57 (2024), p. 405204

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B. A. Bernevig and T. L. Hughes , Topological insulators and topological superconductors , Princeton university press, 2013

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Delplace, J

P. Delplace, J. Marston, and A. Venaille , Topological origin of equatorial waves , Science, 358 (2017), pp. 1075--1077

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Drouot , Microlocal analysis of the bulk-edge correspondence , Communications in Mathematical Physics, 383 (2021), p

A. Drouot , Microlocal analysis of the bulk-edge correspondence , Communications in Mathematical Physics, 383 (2021), p. 2069–2112

work page 2021
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Topological edge states of continuous Hamiltonians

M. Frazier and G. Bal , Topological edge states of continuous hamiltonians , arXiv preprint arXiv:2503.11811, (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025
[13]

Fu and H

Y. Fu and H. Qin , Topological phases and bulk-edge correspondence of magnetized cold plasmas , Nature Communications, 12 (2021), p. 3924

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Parity Anomaly

F. D. M. Haldane , Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly" , Phys. Rev. Lett., 61 (1988), pp. 2015--2018

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F. D. M. Haldane and S. Raghu , Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry , Phys. Rev. Lett., 100 (2008), p. 013904

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G. W. Hanson, S. Gangaraj, and A. Nemilentsau , Notes on photonic topological insulators and scattering-protected edge states-a brief introduction , arXiv preprint arXiv:1602.02425, (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016
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Kato , Perturbation theory for linear operators , vol

T. Kato , Perturbation theory for linear operators , vol. 132, Springer Science & Business Media, 2013

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K. v. Klitzing, G. Dorda, and M. Pepper , New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance , Physical review letters, 45 (1980), p. 494

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L. Lu, J. D. Joannopoulos, and M. Solja c i \'c , Topological photonics , Nature Photonics, 8 (2014), pp. 821--829

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J. W. McIver, B. Schulte, F.-U. Stein, T. Matsuyama, G. Jotzu, G. Meier, and A. Cavalleri , Light-induced anomalous hall effect in graphene , Nature physics, 16 (2020), pp. 38--41

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[23]

Min and A

H. Min and A. H. MacDonald , Electronic structure of multilayer graphene , Progress of Theoretical Physics Supplement, 176 (2008), pp. 227--252

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P. M. Perez-Piskunow, L. E. F. Foa Torres, and G. Usaj , Hierarchy of Floquet gaps and edge states for driven honeycomb lattices , Phys. Rev. A, 91 (2015), p. 043625

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Prodan and H

E. Prodan and H. Schulz-Baldes , Bulk and boundary invariants for complex topological insulators: From K-Theory to Physics , Springer Verlag, Berlin, 2016

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[26]

Quinn and G

S. Quinn and G. Bal , Approximations of interface topological invariants , SIAM Journal on Mathematical Analysis, 56 (2024), pp. 5521--5582

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[27]

Rossi and A

S. Rossi and A. Tarantola , Topology of 2D Dirac operators with variable mass and an application to shallow-water waves , Journal of Physics A: Mathematical and Theoretical, 57 (2024), p. 065201

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[28]

M. G. Silveirinha , Chern invariants for continuous media , Physical Review B, 92 (2015), p. 125153

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[29]

height 2pt depth -1.6pt width 23pt, Bulk-edge correspondence for topological photonic continua , Physical Review B, 94 (2016), p. 205105

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[30]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs , Quantized Hall Conductance in a Two-Dimensional Periodic Potential , Phys. Rev. Lett., 49 (1982), pp. 405--408

work page 1982
[31]

Zhang, J

F. Zhang, J. Jung, G. A. Fiete, Q. Niu, and A. H. MacDonald , Spontaneous quantum hall states in chirally stacked few-layer graphene systems , Physical review letters, 106 (2011), p. 156801

work page 2011
[32]

Zhang, B

F. Zhang, B. Sahu, H. Min, and A. H. MacDonald , Band structure of ABC-stacked graphene trilayers , Physical Review B—Condensed Matter and Materials Physics, 82 (2010), p. 035409

work page 2010

[1] [1]

J. E. Avron, R. Seiler, and B. Simon , Charge deficiency, charge transport and comparison of dimensions , Comm. Math. Phys., 159 (1994), pp. 399--422

work page 1994

[2] [2]

Bal , Continuous bulk and interface description of topological insulators , Journal of Mathematical Physics, 60 (2019)

G. Bal , Continuous bulk and interface description of topological insulators , Journal of Mathematical Physics, 60 (2019)

work page 2019

[3] [3]

1636--1679

height 2pt depth -1.6pt width 23pt, Topological invariants for interface modes , Communications in Partial Differential Equations, 47 (2022), pp. 1636--1679

work page 2022

[4] [4]

height 2pt depth -1.6pt width 23pt, Topological charge conservation for continuous insulators , Journal of Mathematical Physics, 64 (2023), p. 031508

work page 2023

[5] [5]

height 2pt depth -1.6pt width 23pt, Continuous topological insulator, classification and bulk edge correspondence , arXiv preprint arXiv:2412.00919, (2024)

work page arXiv 2024

[6] [6]

G. Bal, P. Cazeaux, D. Massatt, and S. Quinn , Mathematical models of topologically protected transport in twisted bilayer graphene , Multiscale Modeling & Simulation, 21 (2023), pp. 1081--1121

work page 2023

[7] [7]

Bal and D

G. Bal and D. Massatt , Multiscale invariants of Floquet topological insulators , Multiscale Modeling & Simulation, 20 (2022), pp. 493--523

work page 2022

[8] [8]

Bal and J

G. Bal and J. Yu , Topological equatorial waves and violation (or not) of the bulk edge correspondence , Journal of Physics A: Mathematical and Theoretical, 57 (2024), p. 405204

work page 2024

[9] [9]

B. A. Bernevig and T. L. Hughes , Topological insulators and topological superconductors , Princeton university press, 2013

work page 2013

[10] [10]

Delplace, J

P. Delplace, J. Marston, and A. Venaille , Topological origin of equatorial waves , Science, 358 (2017), pp. 1075--1077

work page 2017

[11] [11]

Drouot , Microlocal analysis of the bulk-edge correspondence , Communications in Mathematical Physics, 383 (2021), p

A. Drouot , Microlocal analysis of the bulk-edge correspondence , Communications in Mathematical Physics, 383 (2021), p. 2069–2112

work page 2021

[12] [12]

Topological edge states of continuous Hamiltonians

M. Frazier and G. Bal , Topological edge states of continuous hamiltonians , arXiv preprint arXiv:2503.11811, (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[13] [13]

Fu and H

Y. Fu and H. Qin , Topological phases and bulk-edge correspondence of magnetized cold plasmas , Nature Communications, 12 (2021), p. 3924

work page 2021

[14] [14]

835880401

height 2pt depth -1.6pt width 23pt, The dispersion and propagation of topological Langmuir-cyclotron waves in cold magnetized plasmas , Journal of Plasma Physics, 88 (2022), p. 835880401

work page 2022

[15] [15]

Parity Anomaly

F. D. M. Haldane , Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the "Parity Anomaly" , Phys. Rev. Lett., 61 (1988), pp. 2015--2018

work page 1988

[16] [16]

F. D. M. Haldane and S. Raghu , Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry , Phys. Rev. Lett., 100 (2008), p. 013904

work page 2008

[17] [17]

T. Han, Z. Lu, Y. Yao, J. Yang, J. Seo, C. Yoon, K. Watanabe, T. Taniguchi, L. Fu, F. Zhang, and L. Ju , Large quantum anomalous Hall effect in spin-orbit proximitized rhombohedral graphene , Science, 384 (2024), pp. 647--651

work page 2024

[18] [18]

G. W. Hanson, S. Gangaraj, and A. Nemilentsau , Notes on photonic topological insulators and scattering-protected edge states-a brief introduction , arXiv preprint arXiv:1602.02425, (2016)

work page internal anchor Pith review Pith/arXiv arXiv 2016

[19] [19]

Kato , Perturbation theory for linear operators , vol

T. Kato , Perturbation theory for linear operators , vol. 132, Springer Science & Business Media, 2013

work page 2013

[20] [20]

K. v. Klitzing, G. Dorda, and M. Pepper , New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance , Physical review letters, 45 (1980), p. 494

work page 1980

[21] [21]

L. Lu, J. D. Joannopoulos, and M. Solja c i \'c , Topological photonics , Nature Photonics, 8 (2014), pp. 821--829

work page 2014

[22] [22]

J. W. McIver, B. Schulte, F.-U. Stein, T. Matsuyama, G. Jotzu, G. Meier, and A. Cavalleri , Light-induced anomalous hall effect in graphene , Nature physics, 16 (2020), pp. 38--41

work page 2020

[23] [23]

Min and A

H. Min and A. H. MacDonald , Electronic structure of multilayer graphene , Progress of Theoretical Physics Supplement, 176 (2008), pp. 227--252

work page 2008

[24] [24]

P. M. Perez-Piskunow, L. E. F. Foa Torres, and G. Usaj , Hierarchy of Floquet gaps and edge states for driven honeycomb lattices , Phys. Rev. A, 91 (2015), p. 043625

work page 2015

[25] [25]

Prodan and H

E. Prodan and H. Schulz-Baldes , Bulk and boundary invariants for complex topological insulators: From K-Theory to Physics , Springer Verlag, Berlin, 2016

work page 2016

[26] [26]

Quinn and G

S. Quinn and G. Bal , Approximations of interface topological invariants , SIAM Journal on Mathematical Analysis, 56 (2024), pp. 5521--5582

work page 2024

[27] [27]

Rossi and A

S. Rossi and A. Tarantola , Topology of 2D Dirac operators with variable mass and an application to shallow-water waves , Journal of Physics A: Mathematical and Theoretical, 57 (2024), p. 065201

work page 2024

[28] [28]

M. G. Silveirinha , Chern invariants for continuous media , Physical Review B, 92 (2015), p. 125153

work page 2015

[29] [29]

height 2pt depth -1.6pt width 23pt, Bulk-edge correspondence for topological photonic continua , Physical Review B, 94 (2016), p. 205105

work page 2016

[30] [30]

D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs , Quantized Hall Conductance in a Two-Dimensional Periodic Potential , Phys. Rev. Lett., 49 (1982), pp. 405--408

work page 1982

[31] [31]

Zhang, J

F. Zhang, J. Jung, G. A. Fiete, Q. Niu, and A. H. MacDonald , Spontaneous quantum hall states in chirally stacked few-layer graphene systems , Physical review letters, 106 (2011), p. 156801

work page 2011

[32] [32]

Zhang, B

F. Zhang, B. Sahu, H. Min, and A. H. MacDonald , Band structure of ABC-stacked graphene trilayers , Physical Review B—Condensed Matter and Materials Physics, 82 (2010), p. 035409

work page 2010