arxiv: 2512.00464 · v2 · submitted 2025-11-29 · ❄️ cond-mat.mes-hall

Recognition: 2 theorem links

· Lean Theorem

Characterizing topology at nonzero temperature: Topological invariants and indicators in the extended SSH model

Julia D. Hannukainen , Nigel R. Cooper

Authors on Pith no claims yet

Pith reviewed 2026-05-17 03:26 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords topological invariantsfinite temperaturemixed Gaussian statesSSH modellocal twist operatorschiral markerwinding numberpurity gap

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The pith

Local twist operators identify topological phases at nonzero temperature from a few local measurements in the extended SSH model.

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

This paper tries to establish practical ways to detect topological phases in the extended SSH model at nonzero temperature. Traditional approaches like the ensemble geometric phase lose practicality in large systems because their signals vanish. Local twist operators provide a solution by letting the phase be read from the relative size of just two local expectation values. The work also shows how to extend the chiral marker to mixed Gaussian states via the correlation matrix when a purity gap is present, recovering the winding number at zero temperature.

Core claim

The central claim is that in the extended SSH model for mixed Gaussian states at nonzero temperature, the topological phase is identified from the relative magnitude of local twist operator expectation values, which only requires measuring two local expectation values, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. The local chiral marker is generalized to mixed Gaussian states, fully determined by its single-particle correlation matrix with a nonzero purity gap, yielding a real-space topological invariant that coincides with the winding number in the zero-temperature limit.

What carries the argument

Local twist operators acting on neighboring sites and the band-flattened local chiral marker based on the single-particle correlation matrix.

Load-bearing premise

The states remain mixed Gaussian states possessing a nonzero purity gap in their effective single-particle Hamiltonian, allowing the correlation matrix to be flattened to an effective projector.

What would settle it

Measuring equal magnitudes for the local twist operator expectation values in a system known to be in the topological phase at finite temperature would contradict the proposed diagnostic.

Figures

Figures reproduced from arXiv: 2512.00464 by Julia D. Hannukainen, Nigel R. Cooper.

**Figure 2.** Figure 2: The amplitude |⟨T⟩| as a function of t1/t2 and inverse temperature β, where t1 is the intracell hopping energy, t2 = 2 is the intercell hopping energy in the SSH model. The number of unit cells is N = 150. The overlaid contour lines in brown indicate the loci where |⟨T⟩| = 0.01 for systems with N = 150 (solid), N = 100 (dashed), and N = 50 (dotted). εk = ±t1, with k independent eigenvectors |u ± k ⟩ = (1,… view at source ↗

**Figure 3.** Figure 3: The SSH chains with three cells labeled by [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: (a): The magnitude of the expectation value of the local density operator, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: (a): T3 as a function of the ratios t3/t1 and t2/t1,the hopping parameters of the next-nearest-neighbor SSH model, at zero temperature. The dotted line denotes T3 = 0. (b): I, as a function of t3/t1 and t2/t1 at zero temperature, where the difference in sign distinguishes between two topological phases. (c): T3 as a function of inverse temperature β and t2/t1. (d): I as a function of inverse temperature β … view at source ↗

**Figure 6.** Figure 6: (a): The gap ∆ of the spectrum of the correlation matrix f(HSSH) as a function of inverse temperature β and the ratio of the intra and inter hopping parameters t2/t1. (b): The chiral marker ν as a function of β and the ratio t2/t1 for the SSH model with only nearest-neighbor hopping. The stripes denote the region where ∆ < 0.1 (c): The chiral marker ν at zero temperature as a function of t3/t1 and t2/t1 fo… view at source ↗

read the original abstract

We compare three complementary diagnostics for mixed Gaussian states at nonzero temperature, focusing on the Su-Schrieffer-Heeger (SSH) chain and its inversion-symmetric extension. Whilst the ensemble geometric phase, a mixed-state generalization of the Zak phase, remains well defined at nonzero temperature, the modulus of the corresponding expectation value vanishes in the thermodynamic limit, limiting its practical use. To develop diagnostics suitable for large systems, we introduce local twist operators acting on neighboring sites, whose expectation values provide local indicators of the underlying topological phase. The topological phase is identified from the relative magnitude of these expectation values, which only requires measuring two local expectation values at nonzero temperature, together with one additional nonlocal expectation value when next-nearest-neighbor hopping is included. In addition, we generalize the local chiral marker to mixed Gaussian states, fully determined by its single-particle correlation matrix, with a nonzero purity gap in their effective single-particle Hamiltonian. The presence of a purity gap ensures that the correlation matrix can be flattened to an effective projector. Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real-space topological invariant that coincides with the winding number in the zero-temperature limit. The ensemble geometric phase, the local twist operators, and the local chiral marker provide complementary methods to characterize topology in the SSH chain beyond pure 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.

Desk Editor's Note private letter to a colleague

The paper gives concrete local twist operators and a purity-gap chiral marker for reading topology in finite-T SSH chains from a few measurements, but the gap assumption is a real soft spot.

read the letter

The main takeaway is that this work supplies local twist operators whose expectation values distinguish topological phases in the SSH chain and its inversion-symmetric extension at nonzero temperature, using only two local measurements plus one nonlocal one when next-nearest-neighbor terms are present. They also extend the local chiral marker to mixed Gaussian states by flattening the correlation matrix when a purity gap exists in the effective single-particle Hamiltonian, recovering the winding number at zero temperature. The ensemble geometric phase is shown to lose its signal in the thermodynamic limit, so these alternatives are positioned as more practical for large systems. The constructions are defined directly from the correlation matrix and local twists, which keeps them straightforward and avoids obvious circularity. This is useful because it points toward diagnostics that experiments on chains could actually perform without full-system access. The approach is coherent on its own terms and engages the literature on mixed-state invariants without obvious internal contradictions. The soft spot is the purity-gap requirement. Thermal states generically lower purity, and the paper does not appear to demonstrate that the gap remains open and sufficiently large throughout the topological region or relative to temperature. If the gap closes or becomes comparable to T, the flattening step fails and the local indicators lose their topological content. The abstract sketches the idea but leaves the range of validity and any supporting checks for the full text; without those the central claim rests on an assumption that may not hold generically. This paper is for condensed-matter theorists working on finite-temperature topology in one-dimensional models. A reader in that niche who wants concrete, potentially measurable invariants will get value from the specific constructions. It shows clear thinking and honest engagement with the problem, so it deserves a serious referee rather than a desk reject. I would send it for review and ask the referee to check the validity range of the purity gap and any numerical benchmarks that support the local indicators.

Referee Report

2 major / 1 minor

Summary. The manuscript develops three complementary diagnostics for characterizing topological phases in the extended Su-Schrieffer-Heeger (SSH) model at nonzero temperature for mixed Gaussian states. These include the ensemble geometric phase, local twist operators whose expectation values serve as local indicators of the topological phase based on their relative magnitudes, and a generalization of the local chiral marker that uses the single-particle correlation matrix under the assumption of a nonzero purity gap to flatten it to an effective projector, recovering the winding number at zero temperature.

Significance. If the proposed methods hold, they offer practical tools for identifying topology in thermal systems using local measurements, which is significant for experimental realizations in mesoscopic condensed matter systems where global invariants are hard to access.

major comments (2)

[Generalization of the local chiral marker] The construction of the local chiral marker relies on the presence of a nonzero purity gap to flatten the correlation matrix. However, as noted in the skeptic's concern, thermal states in the extended SSH model may not maintain this gap, potentially causing the marker to lose its topological significance. The paper should include analysis or simulations showing the gap's persistence in the relevant parameter regimes at finite temperature.
[Local twist operators] The identification of the topological phase from the relative magnitude of two (or three) local expectation values of twist operators is central, but the manuscript needs to provide more detail on why this relative magnitude remains a reliable indicator at nonzero temperature, particularly if it depends on the same Gaussian state assumptions.

minor comments (1)

Clarify the specific temperature values and system sizes used in any numerical examples to support the claims about the diagnostics.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which we address point by point below.

read point-by-point responses

Referee: [Generalization of the local chiral marker] The construction of the local chiral marker relies on the presence of a nonzero purity gap to flatten the correlation matrix. However, as noted in the skeptic's concern, thermal states in the extended SSH model may not maintain this gap, potentially causing the marker to lose its topological significance. The paper should include analysis or simulations showing the gap's persistence in the relevant parameter regimes at finite temperature.

Authors: We agree that demonstrating the persistence of the purity gap is necessary to establish the regime of validity for the local chiral marker. The manuscript states that the marker is defined under the assumption of a nonzero purity gap in the effective single-particle Hamiltonian, which permits flattening the correlation matrix to an effective projector. To address the concern, we will incorporate additional numerical results in the revised version showing the temperature dependence of the purity gap for the extended SSH model in the topologically nontrivial parameter regimes. These simulations confirm that the gap remains open at moderate temperatures before closing at higher temperatures where the state becomes too mixed. revision: yes
Referee: [Local twist operators] The identification of the topological phase from the relative magnitude of two (or three) local expectation values of twist operators is central, but the manuscript needs to provide more detail on why this relative magnitude remains a reliable indicator at nonzero temperature, particularly if it depends on the same Gaussian state assumptions.

Authors: The local twist operators are defined directly on the Gaussian mixed state via its single-particle correlation matrix, and their expectation values quantify local bond correlations. The relative magnitude between the two (or three) operators identifies the topological phase because it encodes the asymmetry in nearest-neighbor (and next-nearest-neighbor) correlations that survives thermal mixing, provided the temperature is below the scale set by the gap. We will add a dedicated paragraph in the revised manuscript with an explicit calculation for the standard SSH chain at finite temperature, showing how the correlation-matrix expression for the twist-operator expectations leads to the observed magnitude ordering without requiring additional assumptions beyond Gaussianity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; diagnostics derived from correlation matrix without reduction to inputs

full rationale

The derivation introduces local twist operators whose expectation values serve as indicators, with the topological phase read from their relative magnitudes, and generalizes the chiral marker to mixed Gaussian states via the single-particle correlation matrix under an explicit nonzero purity gap assumption that permits flattening to an effective projector. This flattened marker is then shown to coincide with the winding number at zero temperature. No quoted step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims follow from direct definitions on the correlation matrix and limiting behavior rather than tautological renaming or imported uniqueness. The purity gap is presented as a stated condition for validity, not derived from the diagnostic itself, keeping the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that the states are mixed Gaussian states with a nonzero purity gap; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption The system is described by mixed Gaussian states
Explicitly used for all three diagnostics in the abstract.
domain assumption Nonzero purity gap exists in the effective single-particle Hamiltonian
Required to flatten the correlation matrix to an effective projector.

pith-pipeline@v0.9.0 · 5538 in / 1223 out tokens · 57347 ms · 2026-05-17T03:26:43.956187+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

The presence of a purity gap ensures that the correlation matrix can be adiabatically flattened to an effective projector... Evaluating the chiral marker with respect to the band-flattened correlation matrix yields a real-space topological invariant
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

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

local twist operators acting on neighboring sites, whose expectation values provide local indicators of the underlying topological phase... only requires measuring two local expectation values

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.

Forward citations

Cited by 3 Pith papers

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

Anomalous Mixed-State Floquet Topology in One-Dimensional Open Quantum Systems
cond-mat.mes-hall 2026-04 unverdicted novelty 6.0

A driven-dissipative SSH chain has steady-state Z×Z Floquet topology via two ensemble geometric phase invariants, with protected edge modes in both 0 and π quasienergy gaps.
Anomalous Mixed-State Floquet Topology in One-Dimensional Open Quantum Systems
cond-mat.mes-hall 2026-04 unverdicted novelty 6.0

A periodically driven dissipative SSH chain exhibits a Z x Z topological classification in its mixed steady state via ensemble geometric phases in the 0 and pi gaps.
Topological markers for a one-dimensional fermionic chain coupled to a single-mode cavity
cond-mat.mes-hall 2026-04 unverdicted novelty 5.0

Cavity-mediated interactions in an effective SSH Hamiltonian produce consistent topological phase diagrams across three markers, confirming edge states via correlations.

Reference graph

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