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arxiv: 2511.19596 · v2 · submitted 2025-11-24 · ❄️ cond-mat.mes-hall · cond-mat.dis-nn

Topological surface-state destruction via trivializing proximity effect: Lattice localization despite continuum criticality

Arthur Niwazuki , Matthew S. Foster This is my paper

Pith reviewed 2026-05-17 04:48 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.dis-nn
keywords topological insulatorsAnderson localizationclass-CIsurface statestrivializing proximity effectquantum criticalitylattice modelscontinuum Dirac theories
0
0 comments X

The pith

Hybridization with a trivial 2D band localizes surface states of class-CI topological lattice models under weak disorder.

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

The paper examines the stability of surface states in three-dimensional class-CI topological phases, which belong to the localizable classes where bulk states can be Wannier localized and surface states are typically Anderson localizable by weak disorder. It demonstrates that these surface states become fragile when the lattice model hybridizes with an additional trivial 2D band, leading to localization even with weak quenched randomness. This trivializing proximity effect makes stronger disorder more effective at destroying the surface states, in contrast to the pure topological lattice model where states remain extended with robust spectrum-wide quantum criticality. The work also contrasts this lattice behavior with effective continuum Dirac theories, including models for twisted bilayer graphene, where stronger disorder instead tends to restore criticality.

Core claim

Surface states of a bulk class-CI topological lattice model can be Anderson localized by the combination of weak quenched randomness and hybridization with an additional trivial 2D band via the trivializing proximity effect. Without the additional band the surface states remain extended and exhibit robust spectrum-wide quantum criticality. Continuum models show the opposite trend, with localization near gap edges at weak disorder but restored criticality at stronger disorder that fills spectral gaps.

What carries the argument

The trivializing proximity effect, defined as hybridization between topological surface states and an added trivial 2D band, which enables Anderson localization when combined with quenched disorder.

If this is right

  • Stronger disorder becomes more effective at localizing surface states once the trivializing proximity effect is present.
  • Pure class-CI lattice models without extra bands support extended surface states with spectrum-wide quantum criticality.
  • Effective continuum Dirac theories miss the localization induced by the trivializing proximity effect that appears in lattice models.

Where Pith is reading between the lines

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

  • Real materials with inevitable proximity to trivial bands may exhibit localized rather than conducting surface states under weak disorder.
  • Lattice-specific effects appear necessary to explain fragility that continuum approximations overlook.
  • Similar trivializing effects could be tested numerically in other localizable topological classes to check for general fragility of surface states.

Load-bearing premise

The chosen numerical lattice model and disorder implementation capture the essential symmetries and physics of real class-CI materials without missing longer-range effects that could change the localization outcome.

What would settle it

A calculation or measurement showing that surface states of the class-CI lattice model remain extended and critical even after adding the trivial 2D band and applying weak disorder.

Figures

Figures reproduced from arXiv: 2511.19596 by Arthur Niwazuki, Matthew S. Foster.

Figure 1
Figure 1. Figure 1: Contrasting the behavior of lattice and continuum [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Slab spectrum of the cubic-lattice class-CI topolog [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Splitting of the quadratic surface-state touching by [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Slab energy spectrum versus ky of the topological CI lattice model (as in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spectrum of the CI 2D continuum surface-Dirac [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic diagram of the momentum lattice used [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effects of disorder on the surface states of the CI [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The same as Fig [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Benchmark results for the CI continuum (2D) [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Robust surface states of the CI continuum Dirac [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Effects of chiral “twist” disorder [Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p012_13.png] view at source ↗
read the original abstract

In a significant conceptual revision to the tenfold classification scheme for topological insulators and superconductors, it was recently demonstrated that most three-dimensional (3D) classes are simultaneously "localizable" in two distinct, but intricately connected ways: (1) There is no obstruction to Wannier localization of all bulk eigenstates, and (2) Almost all surface states can be Anderson localized by arbitrarily weak symmetry-preserving quenched disorder. Here we consider the localizable class CI in 3D, and numerically investigate the stability of surface states. We demonstrate that surface states of a bulk class-CI topological lattice model are fragile in that they can be Anderson localized by the combination of weak quenched randomness and hybridization with an additional trivial 2D band (a trivializing proximity effect, TPE). With the TPE, stronger disorder is more destructive to the surface states of the bulk lattice model. By contrast, without additional bands the surface states remain extended, exhibiting robust spectrum-wide quantum criticality. We also investigate the fragility of surface states in effective 2D class-CI continuum Dirac theories, including the chiral limit of the Bistritzer-MacDonald model for twisted bilayer graphene. Although the continuum models exhibit signs of Anderson localization near gap edges for weak disorder, stronger disorder instead appears to heal the surface, restoring criticality whilst filling in spectral energy gaps. Our results provide further evidence that effective continuum field theories fail to capture key aspects of surface-state physics in localizable topological phases.

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 manuscript examines the stability of surface states in three-dimensional class-CI topological insulators using lattice models. It demonstrates that these surface states can be Anderson localized through the combination of weak symmetry-preserving quenched disorder and hybridization with an additional trivial 2D band, referred to as the trivializing proximity effect (TPE). In the absence of this hybridization, the surface states remain extended and exhibit robust spectrum-wide quantum criticality. The study contrasts this with effective 2D continuum Dirac models, including the chiral limit of the Bistritzer-MacDonald model, where stronger disorder appears to restore criticality. The results suggest that continuum field theories inadequately capture surface-state physics in localizable topological phases.

Significance. If substantiated, this work offers valuable numerical support for the fragility of surface states in localizable topological classes like CI, highlighting the role of lattice-specific effects and proximity to trivial bands. It strengthens the case for rethinking aspects of the tenfold classification regarding localizability and provides a contrast between lattice and continuum descriptions that could guide future theoretical and experimental studies in topological materials.

major comments (2)
  1. [Numerical Simulations] The description of the numerical results lacks essential details such as the system sizes used, the number of disorder realizations for averaging, error bars on the localization measures, and the precise diagnostics employed to identify Anderson localization versus criticality (e.g., inverse participation ratio or conductance). This information is necessary to assess whether the observation that stronger disorder is more destructive is robust, particularly given the central claim about TPE.
  2. [Lattice Model and Hybridization] The explicit form of the trivial 2D band and the hybridization term with the bulk class-CI model is not detailed, nor is there verification that this hybridization commutes with the time-reversal, particle-hole, and spin-rotation symmetries defining class CI. Since the central claim attributes localization specifically to the TPE rather than symmetry reduction, this verification is load-bearing and should be provided, for example through commutation checks or explicit matrix forms in the model section.
minor comments (2)
  1. [Abstract] The abstract mentions 'numerically investigate' but could briefly note the key observables or system parameters for better context.
  2. [References] Ensure all relevant prior works on class-CI models and Anderson localization in topological systems are cited, particularly recent revisions to the tenfold scheme.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Numerical Simulations] The description of the numerical results lacks essential details such as the system sizes used, the number of disorder realizations for averaging, error bars on the localization measures, and the precise diagnostics employed to identify Anderson localization versus criticality (e.g., inverse participation ratio or conductance). This information is necessary to assess whether the observation that stronger disorder is more destructive is robust, particularly given the central claim about TPE.

    Authors: We agree that these methodological details are essential for assessing the robustness of the results. In the revised manuscript we will add a dedicated subsection in the methods or numerical details section specifying the lattice sizes used (e.g., linear dimensions and total sites), the number of independent disorder realizations for ensemble averaging, the presence of error bars on all plotted localization measures, and the precise diagnostics employed, including the inverse participation ratio to quantify localization and conductance or level-spacing statistics to identify criticality. These additions will directly support the claim that stronger disorder is more destructive in the presence of the TPE. revision: yes

  2. Referee: [Lattice Model and Hybridization] The explicit form of the trivial 2D band and the hybridization term with the bulk class-CI model is not detailed, nor is there verification that this hybridization commutes with the time-reversal, particle-hole, and spin-rotation symmetries defining class CI. Since the central claim attributes localization specifically to the TPE rather than symmetry reduction, this verification is load-bearing and should be provided, for example through commutation checks or explicit matrix forms in the model section.

    Authors: We agree that explicit model details and symmetry verification are necessary to substantiate that localization is due to the TPE and not to any unintended symmetry breaking. In the revised manuscript we will include the explicit Hamiltonian for the additional trivial 2D band and the hybridization term coupling it to the bulk class-CI lattice model. We will also add a verification subsection showing that the hybridization commutes with the time-reversal, particle-hole, and spin-rotation symmetry operators of class CI, either via explicit commutation relations or by presenting the matrix representations in the appropriate basis. This will confirm that the symmetry class remains CI. revision: yes

Circularity Check

0 steps flagged

Numerical simulations establish independent results on surface-state fragility

full rationale

The paper's central results come from direct numerical diagonalization and localization diagnostics on explicit lattice Hamiltonians and continuum Dirac models. No derivation chain reduces a claimed prediction or first-principles outcome to a fitted parameter, self-defined quantity, or self-citation by construction. The contrast between lattice localization under TPE and continuum criticality is obtained by explicit computation of eigenstates and participation ratios rather than by renaming or tautological re-expression of inputs. Self-citations, if present for the prior tenfold-classification revision, are not load-bearing for the new numerical claims here.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Limited information available from abstract only; the central claim relies on the assumption that the simulated lattice model belongs to class CI and that the added 2D band is truly trivial without introducing new topological features.

free parameters (1)
  • disorder strength
    Quenched randomness amplitude is varied to demonstrate localization transition; exact distribution and range not specified in abstract.
axioms (1)
  • domain assumption The bulk lattice model realizes the 3D class-CI topological phase with no Wannier obstruction.
    Invoked to set up the surface-state problem in the localizable class.

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Reference graph

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