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arxiv: 2604.07442 · v3 · submitted 2026-04-08 · 🪐 quant-ph · cond-mat.stat-mech

Locked Subharmonic Oscillations in the Entanglement Spectrum of a Periodically Driven Topological Chain

Rishabh Jha This is my paper

Pith reviewed 2026-05-10 17:52 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords entanglement spectrumFloquet topologysubharmonic responseperiod doublingSu-Schrieffer-Heeger chainedge modesdriven quantum systemsnonequilibrium coherence
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The pith

A coherent superposition of zero and pi edge modes in a driven SSH chain produces subharmonic period-doubling in the entanglement spectrum.

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

The paper studies a two-step periodically driven Su-Schrieffer-Heeger chain whose Floquet operator hosts symmetry-protected edge modes at quasienergies zero and pi. When the initial state is prepared as a coherent superposition of these two edge sectors, the subsystem correlation matrix alternates between two distinct stroboscopic structures. This alternation makes the full entanglement spectrum period-doubled as a set and causes a tracked entanglement level to oscillate robustly at half the drive frequency. The subharmonic signature vanishes if the initial state is instead a stationary Floquet eigenstate or if the chain is tuned into the topologically trivial phase. The results position entanglement spectroscopy as a sharp, subsystem-resolved witness of Floquet topological coherence in free-fermion systems.

Core claim

In the two-step driven SSH chain, zero and pi edge modes exist due to Floquet topology. Initializing the system in a coherent superposition of these modes makes the subsystem correlation matrix alternate between two structures at successive drive periods. Consequently the entanglement spectrum repeats every two periods as a set, and an overlap-tracked entanglement eigenvalue shows a clean Fourier peak at half the drive frequency. The same signature is absent both for any stroboscopically stationary combination of the same modes and for the entire trivial phase where edge modes are missing.

What carries the argument

The overlap-tracked entanglement level extracted from the stroboscopic subsystem correlation matrix, which alternates because of the coherent superposition of zero and pi edge sectors.

If this is right

  • Entanglement spectroscopy provides a subsystem-resolved probe of Floquet topological coherence even in non-interacting systems.
  • Diagonal edge densities stay constant by sublattice symmetry, while an off-diagonal edge-bond observable supplies the corresponding one-body linear response.
  • The period-doubling is locked to the drive and requires both zero-pi topology and nonequilibrium coherent preparation.
  • The effect is absent for stationary Floquet eigenstates, confirming that the subharmonic response is a nonequilibrium phenomenon.

Where Pith is reading between the lines

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

  • Similar locked subharmonic signatures could appear in other Floquet topological phases that host coexisting zero and pi modes.
  • Quantum simulators with controllable initial-state coherence could test whether the period-doubling survives weak interactions or disorder.
  • The result suggests entanglement measurements may detect hidden Floquet order that is invisible in local observables.

Load-bearing premise

The initial state must be prepared as a coherent superposition of the zero and pi edge sectors; the subharmonic response disappears for any stroboscopically stationary Floquet eigenstate built from the same modes.

What would settle it

Prepare the driven chain in a coherent superposition of its zero and pi edge modes and measure the entanglement spectrum at successive stroboscopic times; absence of period-doubling in the tracked level, or appearance of the same period-doubling in the trivial phase, would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.07442 by Rishabh Jha.

Figure 1
Figure 1. Figure 1: Subharmonic response in the driven SSH chain. (a) Tracked entanglement energy [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Phase diagram of the entanglement-spectrum subharmonic response in the ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Fourier power spectra of all three left-edge observables for all three initial conditions: topological super [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 1
Figure 1. Figure 1: Three-panel subharmonic diagnostic comparing the topological superposition state (blue/orange, even/odd [PITH_FULL_IMAGE:figures/full_fig_p028_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Fourier power spectra of all three left-edge observables for all three initial conditions: topological super [PITH_FULL_IMAGE:figures/full_fig_p028_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Phase diagrams recomputed with δtol = 0.025, in the same format as [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Phase diagrams recomputed with wthr = 0.10, in the same format as [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Robustness of the masked F1/2 phase diagram to the subsystem size LA at fixed total size L = 500. Each row corresponds to one value of LA (from top to bottom: 12, 100, 200); left column is the coherent zero–π superposition state and right column is the π-mode Floquet eigenstate control, in the same format as [PITH_FULL_IMAGE:figures/full_fig_p031_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Robustness of the masked F1/2 phase diagram to the total system size at fixed subsystem size LA = 22. Each row corresponds to one value of L (top: L = 400; bottom: L = 450); left column is the coherent zero–π superposition state and right column is the π-mode Floquet eigenstate control, in the same format as [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
read the original abstract

Periodically driven quantum systems can exhibit subharmonic response, usually characterized through physical observables and often discussed in interacting settings. Here we show that a sharp subharmonic signature already appears in the entanglement spectrum of a number-conserving free-fermion system. We study a two-step driven Su-Schrieffer-Heeger chain whose Floquet operator supports symmetry-protected edge modes at quasienergies $0$ and $\pi$. When the initial state is a coherent superposition of these two edge sectors, we show that the subsystem correlation matrix alternates between two stroboscopic structures, and the entanglement spectrum is period-doubled as a set, while an overlap-tracked entanglement level shows a robust period-doubling response with Fourier weight concentrated at half the drive frequency. By contrast, diagonal edge densities remain flat by sublattice symmetry, while an off-diagonal edge-bond observable provides the corresponding linear one-body comparator. The effect disappears both when the initial state is replaced by a stroboscopically stationary Floquet eigenstate built from the same topological mode content, and when the system is placed in the topologically trivial phase where no edge modes exist. Altogether, these establish zero-$\pi$ Floquet topology as a necessary condition and coherent nonequilibrium preparation as the additional sufficient ingredient. Our results identify entanglement spectroscopy as a sharp subsystem-resolved probe of Floquet topological coherence.

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

0 major / 3 minor

Summary. The manuscript studies subharmonic response in the entanglement spectrum of a free-fermion periodically driven topological chain, focusing on a two-step driven Su-Schrieffer-Heeger model whose Floquet operator hosts symmetry-protected zero and π edge modes. When the initial state is prepared as a coherent superposition of these edge sectors, the subsystem correlation matrix alternates between two stroboscopic structures, the entanglement spectrum is period-doubled as a set, and an overlap-tracked entanglement level exhibits robust period-doubling with Fourier weight concentrated at half the drive frequency. The effect is absent both for a stroboscopically stationary Floquet eigenstate constructed from the same modes and in the topologically trivial phase; diagonal edge densities remain flat by sublattice symmetry while an off-diagonal edge-bond observable serves as a linear comparator. The work concludes that zero-π Floquet topology is necessary and coherent nonequilibrium preparation is the additional sufficient ingredient, positioning entanglement spectroscopy as a sharp subsystem-resolved probe of Floquet topological coherence.

Significance. If the central claims hold, the result supplies a clean, subsystem-resolved signature of Floquet topological coherence that is sharper than conventional physical observables in this free-fermion setting. The manuscript's strengths include explicit two-step-drive SSH numerics, sublattice-symmetry arguments establishing flat diagonal densities, and direct comparisons that close the loop on necessity and sufficiency; these controls are reproducible and falsifiable within the model. The identification of both necessary and sufficient conditions for the locked subharmonic response adds clarity to the literature on nonequilibrium topological phases.

minor comments (3)
  1. [Abstract] The abstract states that 'the entanglement spectrum is period-doubled as a set'; a brief parenthetical clarification of what 'as a set' means (e.g., the multiset of eigenvalues alternates between two distinct collections) would remove potential ambiguity for readers.
  2. [Results] The overlap-tracked entanglement level is central to the subharmonic claim; its precise definition (which single-particle orbital or combination is tracked) should be stated explicitly in the main text rather than deferred to supplementary material.
  3. [Figures] Figure captions for the Fourier spectra should explicitly label the drive frequency and the half-frequency peak to facilitate immediate visual comparison with the text claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and detailed summary of our manuscript, as well as the recommendation for minor revision. We are pleased that the central claims, numerical controls, symmetry arguments, and identification of necessary and sufficient conditions are viewed favorably. As no specific major comments were raised in the report, we provide no point-by-point rebuttals below but remain ready to incorporate any minor editorial or technical adjustments requested by the editor.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit numerics and symmetry

full rationale

The paper's central results follow from direct computation on the driven SSH model: the Floquet operator is constructed explicitly, edge modes at 0 and π are identified, and the correlation matrix for a coherent superposition is shown to alternate between two stroboscopic forms whose eigenvalues period-double. Sublattice symmetry is invoked to prove flat diagonal densities independently of the dynamics. Controls (stationary Floquet eigenstate and trivial phase) are computed in the same framework and shown to lack the effect. No parameter is fitted to data and then re-predicted; no uniqueness theorem is imported from self-citation; the period-doubling is not renamed from a known result but extracted from the overlap-tracked level and Fourier analysis of the explicit spectrum. All load-bearing steps remain within the free-fermion calculation and are falsifiable by the supplied numerics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of symmetry-protected zero and pi edge modes in the Floquet operator of the two-step driven SSH chain and on the ability to prepare a coherent superposition of those sectors. No free parameters are explicitly fitted in the abstract; the result is presented as a qualitative consequence of the topological phase and initial-state choice.

axioms (2)
  • domain assumption The two-step drive produces a Floquet operator with symmetry-protected edge modes at quasienergies 0 and pi in the topological phase.
    Invoked throughout the abstract as the necessary condition for the effect.
  • standard math The system is a number-conserving free-fermion chain whose entanglement spectrum is obtained from the subsystem correlation matrix.
    Standard for free-fermion entanglement calculations.

pith-pipeline@v0.9.0 · 5541 in / 1453 out tokens · 33244 ms · 2026-05-10T17:52:27.252227+00:00 · methodology

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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/Breath1024.lean neutral8 / flipAt512 echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the entanglement spectrum is period-doubled as a set, while an overlap-tracked entanglement level shows a robust period-doubling response with Fourier weight concentrated at half the drive frequency

  • IndisputableMonolith/Foundation/ArrowOfTime.lean arrow_from_z unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    zero–π Floquet topology as a necessary condition and coherent nonequilibrium preparation as the additional sufficient ingredient

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

  1. [1]

    Peschel, Calculation of reduced density matrices from correlation functions, J

    I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A: Math. Gen.36, L205 (2003)

    work page 2003

  2. [2]

    Peschel and V

    I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, Journal of Physics A: Mathematical and Theoretical42, 504003 (2009). 26 TABLE III: Overlap-ranked diagnostic forL= 1000,L A = 31 =⌊ √ 1000⌋,δ 0 =−0.30,δ K = 0.80,N= 800 periods. Columns as in Table II. The subharmonic fractionF 1/2 ≈0.960 is identical to th...

    work page 2009