arxiv: 2604.10190 · v1 · submitted 2026-04-11 · ❄️ cond-mat.mes-hall

Recognition: unknown

Local topological markers for Chern insulators in ribbon geometry

Maks Rep\v{s}e , Toma\v{z} Rejec , Jernej Mravlje

Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:41 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords Chern insulatorslocal Chern markerlocal Středa markerribbon geometryHaldane modeldisorderKibble-Zurek mechanismscaling exponents

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The pith

In ribbon geometries, the local Chern marker for Chern insulators agrees with the local Středa marker throughout the bulk, with boundary mismatches shrinking as the system enlarges.

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

This work focuses on local topological markers to probe Chern insulators that have spatial inhomogeneities like edges or disorder. By using a hybrid basis that takes advantage of the ribbon's translational symmetry in one direction, the authors compute the local Chern marker for Haldane ribbons and compare it directly to the local Středa marker. The two markers align well in the interior for clean systems and remain consistent under weak disorder. Near the boundaries the markers differ, but the difference lessens with bigger systems. The same efficiency lets them extract critical scaling exponents from the local marker during a quench in a disordered Qi-Wu-Zhang model, matching theoretical expectations.

Core claim

We express the local Chern marker in the hybrid position-momentum basis for ribbon geometries with partial translational symmetry. For the Haldane model, this marker shows qualitatively different behavior at the two boundaries compared to fully open systems. It agrees with the local Středa marker in the bulk, with small boundary deviations that diminish as the system size increases. This agreement holds for weakly disordered cases as long as disorder does not substantially change the Chern number. Using this setup, scaling exponents extracted from the local Chern marker in a weakly disordered Qi-Wu-Zhang Chern insulator converge to analytically predicted values with increasing system size.

What carries the argument

The local Chern marker formulated in the hybrid position-momentum basis for systems with partial translational symmetry, which enables efficient computation and direct comparison to the Středa marker.

If this is right

The markers agree in the bulk even when weak disorder is present, as long as it does not induce large changes in the Chern number.
Deviations at the ribbon boundaries decrease with increasing transverse system size.
Extracted scaling exponents from local marker profiles approach analytic predictions for larger systems.
Boundary effects differ between ribbon and fully open boundary conditions.

Where Pith is reading between the lines

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

Such local markers could be used to map topological properties in mesoscopic devices with natural ribbon-like geometries.
The hybrid basis method might reduce computational cost for studying dynamics in other partially symmetric topological systems.
If the agreement holds more generally, it could validate local markers as reliable probes for experimental signatures of Chern phases in finite samples.

Load-bearing premise

The hybrid-basis expression for the local marker, enabled by partial translational symmetry, does not introduce systematic errors that would invalidate its comparison to the Středa marker or to fully open geometries.

What would settle it

Observation that the bulk values of the local Chern marker and local Středa marker differ by an amount that remains finite as the ribbon width increases to infinity would contradict the claimed agreement.

Figures

Figures reproduced from arXiv: 2604.10190 by Jernej Mravlje, Maks Rep\v{s}e, Toma\v{z} Rejec.

**Figure 2.** Figure 2: LCM configurations (a) of the system in the trivial [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 4.** Figure 4: Configuration of LCM c (blue line) and local Stˇreda marker cS (orange line) in the topological phase with on-site disorder. Disorder amplitude is δ = 1, size of the system is Nx = Ny = 300 and δϕ = 30 ϕ0/Ny. IV. KIBBLE-ZUREK MECHANISM IN A DISORDERED CHERN INSULATOR Local Chern marker has also been used to characterize non-equilibrium states after quantum quenches across topological phase transitions [20… view at source ↗

**Figure 5.** Figure 5: (a) LCM in the ground state of an instantaneous Hamiltonian at [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Left: Correlation length in the ground state at [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: shows LCM profiles at the end of quenches with different τ . The slower the quench, the larger the LCM inhomogeneities in the final state. Note that the average LCM is not conserved exactly during the evolution for quenches with large τ , despite C being conserved under unitary evolution [27]. This is due to finite-size effects. (a) −0.002 0.000 0.002 (b) −0.005 0.000 0.005 0.010 (c) 0.11 0.12 0.13 0.14 (d… view at source ↗

**Figure 8.** Figure 8: Left: Correlation length after slow linear quenches [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗

**Figure 9.** Figure 9: Configuration of LCM and local Stˇreda markers in [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

**Figure 10.** Figure 10: LCM (blue) and local Stˇreda marker (orange) [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗

**Figure 11.** Figure 11: LCM (blue) and local Stˇreda marker (orange) for a [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗

**Figure 12.** Figure 12: LCM average over the sites in the bulk at different [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗

read the original abstract

Local topological markers are used to characterize Chern insulators in the presence of spatial inhomogeneities, such as boundaries and disorder. In this paper, we study the local Chern marker in systems with partial translational symmetry. We express the local Chern marker in the hybrid position-momentum basis for both open and periodic boundary conditions. We calculate the local Chern marker for a Haldane model ribbon. We show that the behavior at the two boundaries is qualitatively different from fully open geometries. We further compare the local Chern marker with the local St\v{r}eda marker and show agreement in the bulk and small deviations at the boundaries that diminish with increasing system size. The correspondence between the two markers remains good if disorder is introduced, provided its magnitude remains below large values that cause substantial change of the Chern number due to Anderson physics. Finally, by exploiting the numerical efficiency due to partial translational symmetry, we study equilibrium critical behavior and the Kibble-Zurek mechanism in a weakly disordered Qi-Wu-Zhang Chern insulator. We extract relevant scaling exponents from the local Chern marker configuration and show that they converge to the analytically predicted values with increasing system size.

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

The hybrid-basis local Chern marker works in the bulk for ribbons and yields KZ exponents that approach theory, but the boundary differences from open geometries need a clearer limit check to rule out definitional artifacts.

read the letter

The paper gives a hybrid position-momentum formula for the local Chern marker that exploits partial translational symmetry in ribbons. It tests this on the Haldane model, compares it directly to the local Středa marker, and then applies the marker to track Kibble-Zurek scaling in a weakly disordered Qi-Wu-Zhang insulator. The numerical efficiency is the practical gain: larger systems become feasible, and the extracted scaling exponents converge toward the expected analytic values with increasing size. Bulk agreement between the two markers holds for both clean and weakly disordered cases, with boundary deviations shrinking as system size grows. That part is useful for anyone who needs local topology diagnostics in ribbon or strip geometries without paying the full cost of open boundaries everywhere. The soft spot sits at the definition. The abstract itself flags qualitatively different boundary behavior compared with fully open geometries, yet it does not show an explicit wide-ribbon limit in which the hybrid marker recovers the standard real-space expression in the interior. If the hybrid projector or Berry-phase integral leaves behind O(1) corrections that do not vanish uniformly, the reported bulk agreement and the boundary deviations could partly be artifacts of the formulation rather than physical. The KZ extraction would also benefit from more detail on how the local marker configuration is turned into exponents, including fitting windows and error estimates. The work is aimed at condensed-matter theorists who run numerics on topological ribbons or disordered Chern insulators and want a cheaper local probe. It is an incremental but usable extension of existing marker techniques rather than a conceptual shift. I would send it to peer review. The numerical consistency is there, the method is straightforward to implement, and the boundary question is fixable with one additional check.

Referee Report

2 major / 2 minor

Summary. The manuscript develops expressions for the local Chern marker in the hybrid position-momentum basis for ribbon geometries that retain partial translational symmetry along one direction. For the Haldane model ribbon it reports bulk agreement between this local Chern marker and the local Středa marker, with boundary deviations that decrease as ribbon width increases; the same markers remain consistent under weak disorder. The numerical efficiency of the hybrid formulation is then used to extract Kibble-Zurek scaling exponents from the local Chern marker configuration in a weakly disordered Qi-Wu-Zhang Chern insulator, showing convergence to analytically predicted values with increasing system size.

Significance. If the hybrid-basis definition is shown to be free of spurious boundary artifacts, the work supplies an efficient route to local topological diagnostics in systems with one periodic direction. This would be useful for studying inhomogeneities, disorder, and non-equilibrium dynamics in Chern insulators, and the reported marker agreement plus exponent convergence would strengthen the case for using local markers in ribbon geometries.

major comments (2)

[Section on hybrid-basis definition (near the start of the results)] The hybrid position-momentum expression for the local Chern marker (introduced for both open and periodic boundary conditions) is load-bearing for all subsequent claims. The manuscript does not supply an explicit analytic or numerical limit demonstrating that, as ribbon width tends to infinity, the bulk value of this hybrid marker recovers the standard real-space local Chern marker of a fully open system. Without such a check, the reported qualitative differences in boundary behavior relative to open geometries and the bulk agreement with the Středa marker could contain definitional contributions rather than purely physical ones.
[Section on equilibrium critical behavior and Kibble-Zurek mechanism] In the Qi-Wu-Zhang scaling analysis, the extraction of Kibble-Zurek exponents from the local Chern marker configuration is presented without a description of the fitting procedure, the precise range of system sizes and disorder strengths employed, or uncertainty estimates on the extracted exponents. This omission makes it impossible to judge whether the observed convergence to analytical values is robust against post-hoc choices of fitting windows or finite-size cutoffs.

minor comments (2)

[Abstract and associated figures] The abstract states that boundary deviations 'diminish with increasing system size' but does not quantify the rate or show the corresponding data collapse; a brief statement or inset in the relevant figure would clarify the scaling.
[Methods section] Notation for the hybrid-basis projector and Berry-phase integral should be cross-referenced to the standard real-space definitions to facilitate comparison with the literature on local markers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript accordingly to improve clarity and rigor.

read point-by-point responses

Referee: [Section on hybrid-basis definition (near the start of the results)] The hybrid position-momentum expression for the local Chern marker (introduced for both open and periodic boundary conditions) is load-bearing for all subsequent claims. The manuscript does not supply an explicit analytic or numerical limit demonstrating that, as ribbon width tends to infinity, the bulk value of this hybrid marker recovers the standard real-space local Chern marker of a fully open system. Without such a check, the reported qualitative differences in boundary behavior relative to open geometries and the bulk agreement with the Středa marker could contain definitional contributions rather than purely physical ones.

Authors: We agree that an explicit check of the bulk limit is valuable for confirming that the hybrid marker is consistent with the standard real-space definition. Although the manuscript already demonstrates bulk agreement with the local Středa marker (which itself recovers the Chern number in the thermodynamic limit), we will add a numerical demonstration in the revised manuscript: we will compute the hybrid marker for progressively wider ribbons and show convergence of the bulk value to the known real-space local Chern marker obtained from fully open geometries. This will clarify that the reported boundary differences arise from the ribbon geometry itself rather than from the hybrid formulation. revision: yes
Referee: [Section on equilibrium critical behavior and Kibble-Zurek mechanism] In the Qi-Wu-Zhang scaling analysis, the extraction of Kibble-Zurek exponents from the local Chern marker configuration is presented without a description of the fitting procedure, the precise range of system sizes and disorder strengths employed, or uncertainty estimates on the extracted exponents. This omission makes it impossible to judge whether the observed convergence to analytical values is robust against post-hoc choices of fitting windows or finite-size cutoffs.

Authors: We acknowledge that the details of the fitting procedure were insufficiently described. In the revised manuscript we will expand this section to include: (i) a precise description of the fitting procedure (including the functional form fitted to the local Chern marker profiles and the criteria for choosing fitting windows), (ii) the exact ranges of system sizes and disorder strengths used in the scaling analysis, and (iii) uncertainty estimates on the extracted exponents, obtained via ensemble averaging over multiple disorder realizations together with standard-error bars. These additions will allow readers to assess the robustness of the convergence to the analytically predicted values. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical markers compared to independent analytical predictions

full rationale

The paper adapts the standard local Chern marker to ribbon geometry via a hybrid position-momentum basis enabled by partial translational symmetry, then computes it numerically for clean and disordered Haldane and Qi-Wu-Zhang models. Bulk agreement with the separately defined local Středa marker, diminishing boundary deviations with system size, and convergence of extracted scaling exponents to analytically predicted Kibble-Zurek values are all external benchmarks, not tautological by construction. No self-citations are load-bearing for the central claims, no parameters are fitted to the target quantities, and the hybrid expression is presented as a calculational tool rather than a redefinition that forces the reported agreements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on standard non-interacting band theory for Chern insulators and the established definitions of local topological markers; no new entities are postulated and no parameters are fitted to produce the reported agreements.

axioms (2)

domain assumption Non-interacting electrons in a lattice with well-defined Chern bands remain valid under weak disorder that does not induce Anderson localization strong enough to change the global Chern number.
Invoked when stating that correspondence between markers holds provided disorder magnitude remains below values that cause substantial change of the Chern number.
domain assumption The hybrid position-momentum representation correctly captures the local Chern marker for systems with partial translational symmetry.
Central to the expression of the marker for open and periodic boundary conditions in ribbons.

pith-pipeline@v0.9.0 · 5501 in / 1565 out tokens · 54529 ms · 2026-05-10T15:41:21.974143+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Energy-Resolved Quantum Geometry from St\v{r}eda Response: Driven-Dissipative Bosonic Lattices and Disordered Systems
cond-mat.mes-hall 2026-05 unverdicted novelty 6.0

Driven-dissipative bosonic lattices enable reconstruction of a coarse-grained energy-resolved Středa marker that reveals quantum-geometric features of topological bands even under strong disorder.

Reference graph

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