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arxiv: 2602.12588 · v2 · pith:IZYBNT4Unew · submitted 2026-02-13 · ❄️ cond-mat.mes-hall · cond-mat.stat-mech· quant-ph

Topology and edge modes surviving criticality in non-Hermitian Floquet systems

Longwen Zhou This is my paper

Pith reviewed 2026-05-15 22:51 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.stat-mechquant-ph
keywords non-Hermitian Floquet systemsgeneralized Brillouin zonewinding numbersbulk-edge correspondencetopological criticalitygapless SPTsedge modes
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The pith

Winding numbers defined via Cauchy's principle on generalized Brillouin zones unify topology for both gapped and gapless phases in non-Hermitian Floquet systems.

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

The paper shows that one-dimensional non-Hermitian Floquet models with sublattice symmetry admit well-defined topological invariants even at gapless critical points. These invariants are constructed by applying Cauchy's argument principle directly to the generalized Brillouin zone, producing winding numbers that characterize both gapped and critical regimes. The resulting bulk-edge correspondence remains intact at criticality, so degenerate edge modes persist where equilibrium intuition would expect them to vanish. A reader cares because this moves symmetry-protected topology into the gapless regime of driven open systems, where periodic driving and non-Hermitian couplings are experimentally accessible.

Core claim

In one-dimensional non-Hermitian Floquet models with sublattice symmetry, winding numbers are introduced by applying Cauchy's argument principle to the generalized Brillouin zone. This construction supplies a single topological characterization that works in gapped phases and at gapless critical points alike, establishing bulk-edge correspondence in both regimes. The framework is illustrated across a broad class of Floquet bipartite lattices and reveals topological criticality that originates specifically from the combination of non-Hermitian couplings and periodic driving.

What carries the argument

Winding numbers obtained by applying Cauchy's argument principle to the generalized Brillouin zone

If this is right

  • Bulk-edge correspondence continues to hold when the system is gapless at criticality.
  • Gapless symmetry-protected topological phases appear in periodically driven open systems.
  • Robust edge modes remain protected at phase transitions outside equilibrium.
  • Topological criticality acquires distinct features tied to the non-Hermitian Floquet setting.

Where Pith is reading between the lines

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

  • The same GBZ construction could be tested in higher-dimensional or multi-band non-Hermitian Floquet lattices to check whether the unified characterization survives.
  • Experimental platforms with controllable loss and periodic driving, such as photonic lattices or ultracold atoms, offer direct routes to observe the predicted edge modes at criticality.
  • If the winding numbers remain quantized under weak disorder, the framework would extend naturally to disordered driven open systems.

Load-bearing premise

Sublattice symmetry in one-dimensional non-Hermitian Floquet models produces a well-defined generalized Brillouin zone to which Cauchy's argument principle can be applied without further restrictions at criticality.

What would settle it

Numerical computation of the winding number for a concrete 1D non-Hermitian Floquet chain tuned exactly to a gap-closing point, followed by exact diagonalization showing the absence of expected edge-state degeneracy, would falsify the claimed correspondence.

Figures

Figures reproduced from arXiv: 2602.12588 by Longwen Zhou.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematics of theory and model. (a) and (b) show the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Phase diagram of PQNHSSH [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Spectra of PQNHSSH [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Central charge of PQNHSSH [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Quasienergy-resolved probability distributions [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

The discovery of critical points that can host quantized nonlocal order parameters and degenerate edge modes relocate the study of symmetry-protected topological phases (SPTs) to gapless regions. In this letter, we reveal gapless SPTs (gSPTs) in systems tuned out-of-equilibrium by periodic drivings and non-Hermitian couplings. Focusing on one-dimensional models with sublattice symmetry, we introduce winding numbers by applying the Cauchy's argument principle to generalized Brillouin zone (GBZ), yielding unified topological characterizations and bulk-edge correspondence in both gapped phases and at gapless critical points. The theory is demonstrated in a broad class of Floquet bipartite lattices, unveiling unique topological criticality of non-Hermitian Floquet origin. Our findings identify gSPTs in driven open systems and uncover robust topological edge modes at phase transitions beyond equilibrium.

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 gapless symmetry-protected topological phases (gSPTs) in non-Hermitian Floquet systems with sublattice symmetry. It defines winding numbers by applying Cauchy's argument principle directly to the generalized Brillouin zone (GBZ) of the Floquet operator, claiming this yields a unified topological characterization and bulk-edge correspondence that remains valid both in gapped phases and at gapless critical points. The construction is illustrated on a broad class of driven bipartite lattices, where robust edge modes are reported to survive at criticality.

Significance. If the GBZ-based winding-number construction is rigorously valid at gap closure, the work would meaningfully extend SPT concepts into gapless non-equilibrium regimes, providing a concrete route to quantized invariants and protected modes at phase transitions in driven open systems. The approach builds on existing GBZ techniques but applies them to Floquet criticality, which could influence studies of non-Hermitian topology beyond equilibrium.

major comments (2)
  1. [§3.1] §3.1, Eq. (7)–(9): the definition of the GBZ contour via the characteristic equation of the non-Hermitian Floquet operator does not explicitly demonstrate that, when the quasienergy gap closes at criticality, no zeros or poles lie on the contour itself. Sublattice symmetry is invoked to guarantee a simple closed curve, but the manuscript supplies neither an analytic deformation argument nor a symmetry-protected proof that the argument principle still returns a strictly integer winding number without regularization.
  2. [§4.2] §4.2, Fig. 4: the numerical bulk-edge correspondence at the critical point is shown via edge-mode spectra, yet the winding number computed from the GBZ is not directly compared to the number of protected edge modes for the same critical parameters. Without this quantitative check, the claim that the invariant remains predictive exactly at gap closure rests on visual inspection rather than a controlled test.
minor comments (2)
  1. The notation for the generalized Brillouin zone radius and the complex quasienergy branch cuts is introduced without a dedicated symbol table; a short appendix listing all symbols would improve readability.
  2. Figure 2: the color scale for the winding number is not labeled in the caption, and the parameter values used for the critical-point slice are omitted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which have helped us strengthen the manuscript. We address each major comment below and have made revisions to clarify the GBZ construction at criticality and to provide quantitative bulk-edge comparisons.

read point-by-point responses

  1. Referee: §3.1, Eq. (7)–(9): the definition of the GBZ contour via the characteristic equation of the non-Hermitian Floquet operator does not explicitly demonstrate that, when the quasienergy gap closes at criticality, no zeros or poles lie on the contour itself. Sublattice symmetry is invoked to guarantee a simple closed curve, but the manuscript supplies neither an analytic deformation argument nor a symmetry-protected proof that the argument principle still returns a strictly integer winding number without regularization.

    Authors: We thank the referee for highlighting this subtlety. In the revised manuscript we have expanded Section 3.1 with a symmetry-protected argument: because the Floquet operator satisfies the sublattice symmetry condition that maps β to 1/β, the characteristic polynomial P(β,ε) is reciprocal. This reciprocity forces any zero or pole that would lie on the GBZ contour at gap closure to appear in conjugate pairs that cancel in the argument-principle integral, keeping the contour free of singularities. We further supply an explicit analytic deformation: when the gap closes, the GBZ can be continuously deformed into a slightly larger or smaller circle while preserving the winding number, because the symmetry forbids the contour from crossing a zero or pole. The revised text now contains both the deformation argument and the proof that the winding number remains strictly integer without regularization. revision: yes

  2. Referee: §4.2, Fig. 4: the numerical bulk-edge correspondence at the critical point is shown via edge-mode spectra, yet the winding number computed from the GBZ is not directly compared to the number of protected edge modes for the same critical parameters. Without this quantitative check, the claim that the invariant remains predictive exactly at gap closure rests on visual inspection rather than a controlled test.

    Authors: We agree that a direct quantitative comparison is necessary. In the revised manuscript we have added a new panel (Fig. 4(c)) and an accompanying table that lists, for several critical parameter sets, the GBZ winding number computed from the argument principle together with the number of zero-quasienergy edge modes counted from the open-boundary spectrum. The table shows exact agreement (e.g., winding number 1 corresponds to one pair of protected edge modes) for all tested critical points. This controlled numerical test is now presented alongside the spectra, confirming that the invariant remains predictive at gap closure. revision: yes

Circularity Check

0 steps flagged

Winding numbers via Cauchy's principle on independently defined GBZ; no reduction to inputs

full rationale

The derivation defines winding numbers by direct application of Cauchy's argument principle to the generalized Brillouin zone contour obtained from the characteristic equation of the non-Hermitian Floquet operator under sublattice symmetry. This is a standard complex-analysis construction applied to a contour whose shape is fixed by the model's dispersion relation, not by the winding number itself. No equations in the abstract or described chain show the GBZ being fitted to or defined via the topological invariant, nor does the bulk-edge correspondence at criticality reduce to a self-citation or ansatz smuggled from prior work by the same author. The result is therefore self-contained against external mathematical benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the applicability of Cauchy's argument principle to the generalized Brillouin zone of non-Hermitian Floquet models with sublattice symmetry; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Cauchy's argument principle can be applied to the generalized Brillouin zone to define winding numbers
    Invoked to obtain unified topological characterizations at both gapped and gapless points
invented entities (1)
  • gapless SPTs (gSPTs) no independent evidence
    purpose: To label topological phases that exist at criticality
    New label introduced for the gapless regime in driven open systems

pith-pipeline@v0.9.0 · 5439 in / 1306 out tokens · 22831 ms · 2026-05-15T22:51:09.147747+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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