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arxiv: 2511.18332 · v3 · submitted 2025-11-23 · ❄️ cond-mat.mes-hall

Real-space formulation of the Chern invariant and topological phases in a disordered Chern insulator

Kiminori Hattori , Shinji Nakata This is my paper

Pith reviewed 2026-05-17 06:32 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall
keywords real-space Chern numberWilson loopBott indexdisordered Chern insulatortopological phase transitionRice-Mele modelsupercell formulation
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The pith

A real-space Chern number computed from supercell Wilson loops is quantized to integers and matches the Bott index.

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

The paper formulates the Chern number in real space within a supercell setup. It derives overlap matrices from the real-space Hamiltonian and constructs a Wilson loop to obtain the invariant. This real-space Chern number is proven to be an integer for large systems and to equal the Bott index. The method is then used to examine how disorder influences topological phases in a Chern insulator extended from the Rice-Mele model. Sympathetic readers would value this because it simplifies calculations for disordered systems where traditional momentum-space methods fail.

Core claim

In the supercell framework the overlap matrix between two corners of the Brillouin zone is derived from diagonalizing the real-space Hamiltonian with periodic boundary conditions. The path-ordered product of these overlap matrices around the BZ boundary forms a Wilson loop that defines the real-space Chern number. Analytically, this Chern number is quantized to integers for large enough systems and coincides with the Bott index. The formulation simplifies numerical work and is applied to show that normal disorder induces a topological phase transition while polarized disorder does not affect the phase diagram.

What carries the argument

Wilson loop of overlap matrices obtained from the real-space supercell Hamiltonian diagonalization

If this is right

  • Numerical computations of the Chern number become more efficient.
  • In the disordered Rice-Mele Chern insulator with normal random onsite potential, increasing disorder strength causes a transition from nontrivial to trivial topology.
  • Polarized disorder leaves the nontrivial topological phase largely intact.
  • Linear conductance and bulk density of states confirm the topological character.

Where Pith is reading between the lines

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

  • The method opens the door to investigating topological protection in other disordered lattice models beyond the Chern insulator.
  • One could test the robustness by applying the formula to systems with different disorder distributions or correlations.
  • This real-space approach may prove useful for simulating topological materials in the presence of defects or impurities that break translation symmetry.

Load-bearing premise

The supercell is large enough that finite-size effects do not destroy the integer quantization of the Wilson-loop phase.

What would settle it

A direct numerical calculation showing that the phase of the Wilson loop approaches an integer multiple of 2π as the supercell size increases, or an exact match between the real-space Chern number and the Bott index for a specific disordered configuration.

Figures

Figures reproduced from arXiv: 2511.18332 by Kiminori Hattori, Shinji Nakata.

Figure 2
Figure 2. Figure 2: FIG. 2. The upper two panels show energy eigenvalues of (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Phase diagrams of Chern numbers [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Real-space Chern numbers [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Numerical evaluations of Eqs. (14), (15) and (17). [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) shows the linear conductance G as a func￾tion of W and µ for normal disorder. When W is small enough, G remains quantized to be e 2/h due to the 1D conduction through edge states in the gap. The 2D con￾duction via bulk bands out of the gap is greatly sup￾pressed for W & 1, whereas edge conduction survives against disorder up to W ≃ 3. As W further increases, G totally vanishes, indicating that all the … view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Real-space Chern number [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Real-space Chern numbers [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Disorder-averaged [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Density of states [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
read the original abstract

In this paper, we formulate the real-space Chern number in a supercell framework. In this framework, the overlap matrix between two corners of the Brillouin zone (BZ) is derived from diagonalizing the real-space Hamiltonian with periodic boundary conditions. The path-ordered product of overlap matrices around the BZ boundary forms a Wilson loop, and defines the Chern number in real space. It is analytically shown that the real-space Chern number is quantized at integers for large enough systems and coincides with the Bott index used in the previous studies. The formulation is greatly simplified for the former so that it makes numerical computations more efficient. The real-space formula is used to numerically elucidate topological phases in a disordered Chern insulator. The Chern insulator is modeled by dimensional extension of the Rice-Mele model consisting of two sublattices, and is disordered by including a random onsite potential. As disorder strength increases, the nontrivial-to-trivial phase transition takes place for normal disorder with no sublattice polarization. By contrast, the phase diagram is almost unaffected by polarized disorder, indicating that nontrivial topology persists against disorder. These observations are supported by the linear conductance and the density of bulk states.

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.

Referee Report

2 major / 2 minor

Summary. The paper formulates the real-space Chern number in a supercell framework by constructing overlap matrices from the eigenvectors of the real-space Hamiltonian under periodic boundary conditions. The Chern number is obtained from the phase of the Wilson loop formed by the path-ordered product of these overlap matrices around the Brillouin zone boundary. An analytical argument is given that this quantity is quantized to integers for sufficiently large systems and is equivalent to the Bott index. The method is applied to a disordered Chern insulator obtained by dimensional extension of the Rice-Mele model with random onsite potentials; numerical results show a nontrivial-to-trivial transition under normal (unpolarized) disorder but persistence of topology under polarized disorder, corroborated by linear conductance and bulk density-of-states calculations.

Significance. If the analytical quantization and equivalence to the Bott index hold with controlled finite-size errors, the simplified real-space formulation would offer an efficient computational route to topological invariants in disordered systems without requiring momentum-space diagonalization. The numerical exploration of disorder-type dependence in the Rice-Mele Chern insulator adds concrete insight into the robustness of topology, which is relevant for mesoscopic and materials applications.

major comments (2)
  1. [Analytical derivation of real-space Chern number (abstract and corresponding section)] The analytical demonstration that the real-space Chern number (defined via the Wilson-loop phase of overlap matrices) is exactly quantized for large enough supercells lacks a quantitative bound on finite-size corrections or an explicit criterion for the minimal linear dimension relative to the localization length in the disordered case. This assumption is load-bearing for both the quantization claim and the subsequent numerical phase diagram, yet no error estimate or convergence test with system size is supplied.
  2. [Section establishing equivalence to Bott index] The equivalence to the Bott index is stated to follow from the real-space formulation, but the manuscript does not provide an explicit step-by-step reduction showing that the Wilson-loop phase coincides with the Bott index expression used in prior literature; a direct comparison of the two formulas would strengthen the claim.
minor comments (2)
  1. [Formulation of overlap matrices] The definition of the overlap matrix elements (in terms of the eigenvectors of the supercell Hamiltonian) should be written explicitly with indices to avoid ambiguity in the path-ordered product.
  2. [Numerical results section] Figure captions for the phase diagrams and conductance plots should include the supercell sizes used and the disorder averaging procedure for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Analytical derivation of real-space Chern number (abstract and corresponding section)] The analytical demonstration that the real-space Chern number (defined via the Wilson-loop phase of overlap matrices) is exactly quantized for large enough supercells lacks a quantitative bound on finite-size corrections or an explicit criterion for the minimal linear dimension relative to the localization length in the disordered case. This assumption is load-bearing for both the quantization claim and the subsequent numerical phase diagram, yet no error estimate or convergence test with system size is supplied.

    Authors: We agree that an explicit quantitative bound and convergence tests would strengthen the manuscript. The analytical argument establishes that the real-space Chern number is integer-quantized for sufficiently large supercells because the overlap matrices approach unitarity in the large-system limit. To address the concern, we have added numerical convergence tests with system size in the revised manuscript, along with a discussion relating the required supercell dimension to the localization length extracted from the disordered eigenstates. These additions provide concrete error estimates for the numerical phase diagram. revision: yes

  2. Referee: [Section establishing equivalence to Bott index] The equivalence to the Bott index is stated to follow from the real-space formulation, but the manuscript does not provide an explicit step-by-step reduction showing that the Wilson-loop phase coincides with the Bott index expression used in prior literature; a direct comparison of the two formulas would strengthen the claim.

    Authors: We thank the referee for this observation. While the manuscript notes the equivalence arising from the shared real-space construction, we acknowledge that an explicit reduction was not detailed. We have added a new appendix in the revised manuscript that provides a step-by-step derivation showing how the phase of our Wilson loop reduces to the standard Bott-index expression, including a side-by-side comparison of the formulas. revision: yes

Circularity Check

0 steps flagged

No significant circularity; real-space Chern formulation is derived independently and shown equivalent to Bott index

full rationale

The paper defines the real-space Chern number via overlap matrices extracted from diagonalization of the real-space Hamiltonian under periodic boundary conditions, constructs the Wilson loop around the BZ, and analytically proves both integer quantization for large supercells and exact coincidence with the Bott index. This equivalence is a mathematical demonstration rather than a definitional identity or parameter fit, and the Bott index is an external reference from prior literature. Subsequent numerical application to the disordered Rice-Mele model uses the formulation but introduces no feedback loop into the derivation. No self-citation is load-bearing for the quantization claim, and the steps remain self-contained against the independent Bott-index benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of the Chern number and the assumption that the supercell Hamiltonian with periodic boundaries captures the topological invariant for large cells. No new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The path-ordered product of overlap matrices around the Brillouin zone boundary yields an integer Chern number for sufficiently large supercells.
    Invoked when stating that the real-space Chern number is quantized at integers.

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