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arxiv: 2508.15897 · v2 · submitted 2025-08-21 · ❄️ cond-mat.str-el · cond-mat.dis-nn· cond-mat.quant-gas· quant-ph

Entanglement entropy as a probe of topological phase transitions

Manish Kumar , Bharadwaj Vedula , Suhas Gangadharaiah , Auditya Sharma This is my paper

Pith reviewed 2026-05-18 21:26 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.dis-nncond-mat.quant-gasquant-ph
keywords entanglement entropytopological phase transitionsSu-Schrieffer-Heeger modeldisorderedge statestopological invariantsLyapunov exponents
0
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The pith

The difference in entanglement entropy between half-filled and near-half-filled states vanishes in the topological phase of SSH models but stays finite in the trivial phase.

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

This paper develops an entanglement entropy framework to detect topological phase transitions in variants of the Su-Schrieffer-Heeger model, including cases with disorder. It shows that the difference in entanglement entropy between half-filled and near-half-filled ground states goes to zero in the topological phase because of localized edge states, while it stays nonzero in the trivial phase. The distinction continues to hold when quasiperiodic or binary disorder is present. Subsystem tuning separates genuine topological zero-energy states from trivial localized states induced by disorder. Phase boundaries obtained this way match those from Lyapunov exponents and the topological invariant, and in some instances the entanglement measure performs better.

Core claim

For a class of Su-Schrieffer-Heeger model variants, the difference in entanglement entropy between half-filled and near-half-filled ground states vanishes in the topological phase but remains finite in the trivial phase, a direct consequence of edge-state localization. This behavior persists even in the presence of quasiperiodic or binary disorder. By analyzing domain-wall configurations, subsystem tuning distinguishes genuine topological zero-energy eigenstates from trivial localized states. Exact phase boundaries from Lyapunov exponents via transfer matrices agree closely with numerical results from the entanglement difference and the topological invariant, with cases where the entropy gap

What carries the argument

The difference ΔS^A in entanglement entropy between half-filled and near-half-filled ground states, which vanishes due to edge-state localization in the topological phase.

If this is right

  • The entanglement entropy difference provides a robust diagnostic of topological phases that continues to work in the presence of quasiperiodic or binary disorder.
  • Subsystem tuning offers a practical method to tell genuine topological zero modes apart from disorder-induced trivial localized states.
  • Phase boundaries extracted from the entropy difference match or exceed the accuracy of those from the topological invariant Q in some realizations.
  • The approach links quantum information quantities directly to condensed-matter diagnostics of topology.

Where Pith is reading between the lines

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

  • The same filling-dependent entropy difference could be tested in other one-dimensional symmetry-protected topological models to check whether the vanishing signature is general.
  • Experimental platforms that measure entanglement via quantum information techniques might use this difference as a practical indicator of topology in disordered samples.
  • Extending the framework to two-dimensional topological systems or weakly interacting cases would test whether the edge-localization effect on the entropy gap survives.

Load-bearing premise

That the vanishing of ΔS^A specifically tracks edge-state localization and that subsystem tuning can reliably separate genuine topological zero-energy states from trivial localized states created by disorder.

What would settle it

A controlled experiment on an SSH chain with tunable quasiperiodic or binary disorder that measures whether the entanglement entropy difference reaches zero precisely in the phase independently confirmed to host protected edge states via conductance or direct state imaging.

Figures

Figures reproduced from arXiv: 2508.15897 by Auditya Sharma, Bharadwaj Vedula, Manish Kumar, Suhas Gangadharaiah.

Figure 1
Figure 1. Figure 1: FIG. 1. SSH chain where the green dashed box shows subsys [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Phase diagrams of the SSH model variants. In (a)-(c) the color scale denotes ∆ [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Distinguishing topological and trivial phases via occupation number, topological invariants, and entanglement entropy. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Entanglement entropy (EE) provides a powerful probe of quantum phases, yet its role in identifying topological phase transitions in disordered systems remains underexplored. We introduce an exact EE-based framework that captures topological phase transitions even in the presence of disorder. Specifically, for a class of Su-Schrieffer-Heeger (SSH) model variants, we show that the difference in EE between half-filled and near-half-filled ground states, $\Delta S^{\mathcal{A}}$, vanishes in the topological phase but remains finite in the trivial phase, a direct consequence of edge-state localization. This behavior persists even in the presence of quasiperiodic or binary disorder. By analyzing domain-wall configurations in the SSH chain, we further show how subsystem tuning allows one to distinguish genuine topological zero-energy eigenstates from trivial localized states. Exact phase boundaries, derived from Lyapunov exponents via transfer matrices, agree closely with numerical results from $\Delta S^{\mathcal{A}}$ and the topological invariant $\mathcal{Q}$, with instances where $\Delta S^{\mathcal{A}}$ outperforms $\mathcal{Q}$. Our results highlight EE as a robust diagnostic tool and a potential bridge between quantum information and condensed matter approaches to topological matter.

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 introduces an entanglement-entropy-based diagnostic ΔS^A (difference between half-filled and near-half-filled ground states) for topological phase transitions in Su-Schrieffer-Heeger (SSH) model variants, including those with quasiperiodic or binary disorder. It claims that ΔS^A vanishes in the topological phase due to protected edge-state localization but remains finite in the trivial phase, that subsystem tuning in domain-wall configurations distinguishes genuine topological zero-energy states from trivial disorder-induced localized states, and that phase boundaries obtained from Lyapunov exponents via transfer matrices agree closely with numerical ΔS^A results (sometimes outperforming the invariant Q).

Significance. If the central claims are substantiated, the work supplies a practical EE-based probe for topological transitions in disordered systems where conventional invariants can be ambiguous. The anchoring of phase boundaries in an independent transfer-matrix Lyapunov-exponent calculation is a methodological strength that keeps circularity low and allows direct comparison with both ΔS^A and Q. The persistence of the signal under disorder and the reported instances where ΔS^A outperforms Q would be useful additions to the toolbox for diagnosing topology in one-dimensional chains.

major comments (2)
  1. [§4 (Subsystem Tuning)] §4 (Subsystem Tuning): The central interpretive step—that subsystem tuning reliably isolates protected topological zero-energy modes from trivial disorder-localized states near zero energy—is load-bearing for the claim that ΔS^A vanishes specifically in the topological phase. The manuscript appears to demonstrate this via numerical examples in selected configurations rather than a general argument or systematic counter-example search; if a trivial localized state can be engineered to produce an identical ΔS^A signature under some tuning, the phase distinction collapses.
  2. [§3.1 and §5 (Numerical Agreement and Error Analysis)] §3.1 and §5 (Numerical Agreement and Error Analysis): The abstract asserts close agreement between Lyapunov-exponent boundaries and numerical ΔS^A results, yet the manuscript does not provide explicit finite-size scaling, disorder-averaged error bars, or data-exclusion criteria for the disordered cases. Without these, it is difficult to quantify how robust the reported superiority over Q is when fluctuations are large.
minor comments (2)
  1. [Figures] Figure captions should explicitly label the topological versus trivial regimes and indicate the subsystem sizes used for each ΔS^A curve to facilitate direct visual comparison with the Lyapunov data.
  2. [Notation] Notation for the near-half-filled filling (e.g., the precise value of δN) should be defined once in the main text rather than only in figure legends.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments point by point below and indicate the revisions planned for the resubmitted manuscript.

read point-by-point responses

  1. Referee: §4 (Subsystem Tuning): The central interpretive step—that subsystem tuning reliably isolates protected topological zero-energy modes from trivial disorder-localized states near zero energy—is load-bearing for the claim that ΔS^A vanishes specifically in the topological phase. The manuscript appears to demonstrate this via numerical examples in selected configurations rather than a general argument or systematic counter-example search; if a trivial localized state can be engineered to produce an identical ΔS^A signature under some tuning, the phase distinction collapses.

    Authors: We agree that the distinction between protected topological zero modes and trivial disorder-induced states is central. Our argument rests on the fact that genuine topological edge states remain exponentially localized at the physical boundaries with a localization length fixed by the bulk gap, whereas trivial localized states have no such protection and their positions are uncorrelated with the subsystem boundaries. Subsystem tuning exploits this difference by varying the cut position and observing whether the entanglement contribution persists only when the cut isolates the protected edge. While the manuscript illustrates this with representative configurations, we acknowledge the absence of an exhaustive counter-example search. In the revised manuscript we will add a short analytic argument based on the transfer-matrix localization length and include additional numerical scans over a wider ensemble of disorder realizations to test for possible mimicry. revision: partial

  2. Referee: §3.1 and §5 (Numerical Agreement and Error Analysis): The abstract asserts close agreement between Lyapunov-exponent boundaries and numerical ΔS^A results, yet the manuscript does not provide explicit finite-size scaling, disorder-averaged error bars, or data-exclusion criteria for the disordered cases. Without these, it is difficult to quantify how robust the reported superiority over Q is when fluctuations are large.

    Authors: We accept that quantitative error analysis is needed to substantiate the claimed robustness. In the revised version we will add finite-size scaling plots for the phase boundaries extracted from ΔS^A, report disorder-averaged values with standard deviations for the quasiperiodic and binary-disorder ensembles, and state the criteria used to exclude rare configurations with anomalously large fluctuations. These additions will allow a direct, quantitative comparison of the stability of ΔS^A versus Q. revision: yes

Circularity Check

0 steps flagged

No circularity; phase boundaries anchored in independent transfer-matrix calculation

full rationale

The paper obtains exact phase boundaries from Lyapunov exponents via transfer matrices, an independent external benchmark unrelated to the entanglement entropy computations. Numerical ΔS^A results are compared against these boundaries and the topological invariant Q rather than being used to fit or define them. Subsystem tuning for distinguishing topological zero modes is presented via analysis of domain-wall configurations without reducing to a self-definitional loop, fitted input renamed as prediction, or load-bearing self-citation. The derivation chain remains self-contained against these external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard assumptions of gapped ground states in one-dimensional tight-binding models and the validity of transfer-matrix methods for Lyapunov exponents; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Ground states of the SSH variants are gapped and can be classified by edge-state localization properties.
    Invoked to interpret why ΔS^A vanishes only in the topological phase.
  • standard math Transfer-matrix Lyapunov exponents provide exact phase boundaries independent of the entanglement calculation.
    Used as the benchmark that numerical ΔS^A results are compared against.

pith-pipeline@v0.9.0 · 5752 in / 1458 out tokens · 39935 ms · 2026-05-18T21:26:57.323492+00:00 · methodology

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