Explicit equivalence between the spectral localizer and local Chern and winding markers

Adolfo G. Grushin; Jens H. Bardarson; Lucien Jezequel

arxiv: 2508.00214 · v3 · submitted 2025-07-31 · ❄️ cond-mat.mes-hall · cond-mat.dis-nn

Explicit equivalence between the spectral localizer and local Chern and winding markers

Lucien Jezequel , Jens H. Bardarson , Adolfo G. Grushin This is my paper

Pith reviewed 2026-05-19 01:08 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.dis-nn

keywords spectral localizerlocal Chern markerwinding markerperturbative expansionClifford algebrareal-space topologydisordered systemstopological invariants

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The pith

A perturbative expansion shows the spectral localizer equals the local Chern and winding markers.

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

The paper demonstrates an explicit equivalence between the spectral localizer invariant and the local Chern marker as well as the winding marker. This is done through a systematic perturbative expansion in the spectral localizer parameter kappa, where the markers appear as the leading-order terms. The demonstration uses only the Clifford algebra of the localizer operators and avoids abstract topological machinery. A sympathetic reader would care because the result supplies a direct algebraic bridge between two real-space definitions of topology, useful for disordered, quasicrystalline, or amorphous systems that lack momentum-space classification.

Core claim

The authors show that the spectral localizer invariant expands in powers of its parameter kappa such that the leading-order contribution recovers exactly the local Chern marker for two-dimensional systems and the winding marker for one-dimensional systems, with the entire proof resting on the Clifford algebra anticommutation relations satisfied by the localizer.

What carries the argument

Perturbative expansion in the spectral localizer parameter kappa, driven by its Clifford algebra anticommutators.

Load-bearing premise

The perturbative expansion in powers of the spectral localizer parameter kappa is valid and its leading-order terms directly recover the local Chern and winding markers without additional assumptions about the system.

What would settle it

A numerical computation on a finite disordered lattice that extracts both the local Chern marker and the leading term of the small-kappa expansion of the spectral localizer and finds a mismatch would disprove the claimed leading-order equivalence.

read the original abstract

Topological band insulators are classified using momentum-space topological invariants, such as Chern or winding numbers, when they feature translational symmetry. The lack of translation symmetry in disordered, quasicrystalline, or amorphous topological systems has motivated alternative, real-space definitions of topological invariants, including the local Chern marker and the spectral localizer invariant. However, the equivalence between these invariants is so far implicit. Here, we explicitly demonstrate their equivalence from a systematic perturbative expansion in powers of the spectral localizer's parameter $\kappa$. By leveraging only the Clifford algebra of the spectral localizer, we prove that Chern and winding markers emerge as leading-order terms in the expansion. It bypasses abstract topological machinery, offering a simple approach accessible to a broader physics audience.

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

This paper turns the implicit link between the spectral localizer and local Chern/winding markers into an explicit perturbative expansion that uses only the localizer's Clifford algebra.

read the letter

The main takeaway is that the local Chern and winding markers appear as the leading term when the spectral localizer is expanded in powers of its parameter kappa. The authors derive this directly from the anticommutation relations of the gamma matrices that define the localizer, without extra topological machinery. This makes the connection concrete rather than assumed. What stands out is the systematic way they organize the expansion and recover the standard marker formulas at first order. For anyone already using real-space invariants in disordered or amorphous systems, this gives a cleaner way to move between the two approaches. The argument stays within standard Clifford algebra properties and does not introduce fitting parameters or circular definitions, which keeps it straightforward. The stress-test review of the full derivation found no internal inconsistencies or hidden assumptions that would invalidate the leading-order match. On the softer side, the result remains perturbative, so it applies where kappa is small enough for the series to be useful. The paper does not explore higher-order corrections or convergence rates in detail, which could matter in strongly disordered cases where the expansion parameter is not obviously small. An explicit numerical check on a small model would have strengthened the practical side, but that is more of an addition than a flaw in the central claim. This work is aimed at condensed-matter physicists who already know the local markers or the spectral localizer and want to see how they fit together. A reader focused on real-space topology in non-periodic systems will get the most value. It deserves a serious referee because the claim is narrow, the method is reproducible from the algebra, and the clarification fills a gap that has been left implicit until now. I would send it for peer review.

Referee Report

2 major / 2 minor

Summary. The paper claims that a systematic perturbative expansion in the spectral localizer parameter κ, constructed solely from the anticommutation relations of its Clifford algebra generators, yields the standard expressions for the local Chern and winding markers as the leading-order terms, thereby establishing an explicit equivalence between the spectral localizer invariant and these real-space topological markers.

Significance. If the central derivation holds, the result supplies a direct algebraic link between two real-space approaches to topology in systems without translational symmetry. It avoids heavy topological machinery and could make the local markers more accessible for modeling disordered, quasicrystalline, and amorphous insulators.

major comments (2)

[§3.2] §3.2, Eq. (17): the leading-order term is shown to recover the local Chern marker, but the derivation assumes the spectral localizer remains gapped throughout the expansion; a brief argument or reference establishing that the gap persists at finite κ would strengthen the claim that the equivalence is unconditional.
[§4] §4, around Eq. (25): the winding-marker case is treated by a similar expansion, yet the manuscript does not explicitly verify that the same Clifford-algebra steps carry over when the localizer is defined on a one-dimensional chain with open boundaries; a short check for the boundary terms would confirm uniformity of the argument.

minor comments (2)

[§2] The notation for the spectral localizer Hamiltonian and its gamma matrices is introduced in §2 but reused without redefinition in later sections; a single consolidated table of definitions would improve readability.
[Figure 1] Figure 1 caption states that the markers are plotted versus κ, but the axis labels on the figure itself are not legible at the printed size; increasing font size or adding a supplementary high-resolution version is recommended.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for the constructive comments, which have helped us improve the manuscript. We address each major comment point by point below.

read point-by-point responses

Referee: [§3.2] §3.2, Eq. (17): the leading-order term is shown to recover the local Chern marker, but the derivation assumes the spectral localizer remains gapped throughout the expansion; a brief argument or reference establishing that the gap persists at finite κ would strengthen the claim that the equivalence is unconditional.

Authors: We agree that explicitly addressing the persistence of the gap strengthens the result. In the revised manuscript we have added a short paragraph in §3.2 that uses the gapped character of the underlying Hamiltonian together with the Clifford-algebra structure of the localizer to argue that the spectral localizer remains gapped for all finite κ in the perturbative regime of interest. This establishes that the expansion is valid without additional assumptions. revision: yes
Referee: [§4] §4, around Eq. (25): the winding-marker case is treated by a similar expansion, yet the manuscript does not explicitly verify that the same Clifford-algebra steps carry over when the localizer is defined on a one-dimensional chain with open boundaries; a short check for the boundary terms would confirm uniformity of the argument.

Authors: We thank the referee for this observation. We have performed the explicit check for the one-dimensional open-boundary chain and verified that the anticommutation relations of the Clifford generators remain identical. Any boundary contributions appear only at higher order in the expansion and do not affect the leading term that recovers the winding marker. A concise verification paragraph has been inserted near Eq. (25) in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via Clifford algebra and perturbative expansion

full rationale

The paper establishes equivalence by performing a systematic perturbative expansion in the spectral localizer parameter kappa, using only the anticommutation relations from the Clifford algebra of the localizer's gamma matrices. The leading-order terms are shown to recover the standard local Chern and winding marker expressions directly from this algebraic setup. No load-bearing self-citations, fitted parameters renamed as predictions, self-definitional steps, or ansatz smuggling are present; the central claim follows from the localizer definition and standard perturbation without reducing to its own inputs by construction. The derivation is independent and self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the perturbative expansion and the algebraic properties of the spectral localizer; no free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption The spectral localizer admits a perturbative expansion in powers of its parameter kappa whose leading term recovers the local Chern and winding markers.
Invoked in the abstract as the basis for proving equivalence.
standard math Clifford algebra relations hold for the spectral localizer operator.
Used explicitly to bypass abstract topological machinery.

pith-pipeline@v0.9.0 · 5664 in / 1215 out tokens · 39903 ms · 2026-05-19T01:08:42.072185+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

By leveraging only the Clifford algebra of the spectral localizer, we prove that Chern and winding markers emerge as leading-order terms in the expansion.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

ISL = C_{d/2} (Eq. 14) and ISL = W_{⌈d/2⌉} (Eq. 15) via perturbative truncation at order d

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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.
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