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arxiv: 2512.02470 · v2 · pith:XWZE4FEEnew · submitted 2025-12-02 · ❄️ cond-mat.mes-hall · cond-mat.stat-mech· quant-ph

Su-Schrieffer-Heeger model driven by sequences of two unitaries: periodic, quasiperiodic, aperiodic, and random protocols

Maitri Ganguli , Diptiman Sen This is my paper

Pith reviewed 2026-05-22 11:43 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.stat-mechquant-ph
keywords Su-Schrieffer-Heeger modelFloquet drivingLoschmidt echotopological edge modesquasiperiodic sequenceswinding numberunitary evolutionone-dimensional chains
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The pith

Driving the Su-Schrieffer-Heeger model alternately with two unitaries can produce end modes whose number does not match the winding number topological invariant.

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

The paper investigates the Su-Schrieffer-Heeger chain under driving by two unitaries that differ in inter-cell hopping strengths, applied through periodic, quasiperiodic, aperiodic, and random sequences. For periodic alternation of the two unitaries, localized end modes emerge at the boundaries, yet their count sometimes differs from the winding number computed for the combined evolution operator. When the unitaries act in quasiperiodic or aperiodic patterns such as Fibonacci or Thue-Morse sequences, the Loschmidt echo initiated from an end mode of one Hamiltonian oscillates around a constant value for a prolonged duration before eventually falling to zero, with the initial deviation from perfect return scaling quadratically with the small hopping difference. Random ordering causes rapid decay of the echo. These findings illustrate how the regularity of the driving sequence controls both the appearance of boundary states and the persistence of dynamical correlations in a driven topological wire.

Core claim

When two unitaries constructed from Hamiltonians with slightly different inter-cell hoppings are applied in alternation to the Su-Schrieffer-Heeger model, end modes appear whose multiplicity fails to equal the winding number of the effective operator, while quasiperiodic application of the same unitaries produces a Loschmidt echo that lingers near its starting value for times dependent on the small parameter before decaying.

What carries the argument

The Loschmidt echo and Loschmidt amplitude computed from an initial end mode state under different driving sequences, together with direct counting of zero-quasienergy end modes versus the winding number for periodic protocols.

If this is right

  • End modes can exist in periodically driven systems without being fully accounted for by the Z-valued winding number.
  • The Loschmidt echo remains close to one for long times in quasiperiodic driving, with deviation proportional to the square of the hopping difference at fixed period.
  • Random sequences of the two unitaries lead to fast loss of the initial state memory in the Loschmidt echo.
  • The time scale for decay of the echo in quasiperiodic cases shows a nontrivial dependence on both the small difference and the driving period.

Where Pith is reading between the lines

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

  • This mismatch suggests that Floquet topological invariants may require additional structure when the driving consists of two distinct unitaries rather than a single periodic Hamiltonian.
  • Similar long-lived echoes might serve as a signature to experimentally distinguish quasiperiodic from random driving in cold-atom or photonic realizations of the SSH chain.
  • The quadratic scaling of the deviation offers a way to estimate the hopping mismatch from measured return probabilities without needing full state tomography.

Load-bearing premise

The difference between the inter-cell hoppings of the two Hamiltonians is small and the driving period is short enough that the effective distance between the unitaries scales linearly with that small difference times the period.

What would settle it

Numerical simulation of the periodic protocol showing that the number of end modes always exactly equals the computed winding number for all parameter choices would contradict the reported disagreement.

Figures

Figures reproduced from arXiv: 2512.02470 by Diptiman Sen, Maitri Ganguli.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic picture of the hopping amplitudes of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plots (a) and (b) show the imaginary part vs real part [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plot of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Plot (a) shows the imaginary part versus real part of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots of the real versus imaginary parts of the Floquet [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Plot of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: (b) shows the modulus squared of the Fourier transform, |g(Ω)| 2 of the LE shown in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Plots of the [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Plot of the deviation of [PITH_FULL_IMAGE:figures/full_fig_p010_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Plots of the [PITH_FULL_IMAGE:figures/full_fig_p011_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Plot of [PITH_FULL_IMAGE:figures/full_fig_p011_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Plot of the [PITH_FULL_IMAGE:figures/full_fig_p012_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Plots of the [PITH_FULL_IMAGE:figures/full_fig_p013_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Plot of the time [PITH_FULL_IMAGE:figures/full_fig_p014_18.png] view at source ↗
read the original abstract

We study the effect of driving the Su-Schrieffer-Heeger model using two unitary operators $U_1$ and $U_2$ in different combinations; the unitaries differ in the values of the inter-cell hopping amplitudes. Specifically, we study the cases where the unitaries are applied periodically, quasiperiodically, aperiodically and randomly. For a periodic protocol, when $U_1 = e^{-i H_1 T/2}$ and $U_2 = e^{-i H_2 T/2}$ are applied alternately, we find that end modes may appear, but the number of end modes does not always agree with the winding number which is a $Z$-valued topological invariant. We then study the Loschmidt amplitude ($LA$) starting with a initial state which is an end mode of $H_1$. We find that the $LA$ exhibits pronounced oscillations whose Fourier transform has a peak at a frequency which is equal to the quasienergy of an end mode of $U$. Next, when $U_1$ and $U_2$ are applied in a quasiperiodic or aperiodic way (we consider the Fibonacci and Thue-Morse protocols as examples), we study the Loschmidt echo ($LE$) starting with an initial state which is an end mode of the Hamiltonian $H_1$. When the inter-cell hoppings differ by a small amount denoted by $\epsilon$, and the time period $T$ of each unitary is also small, the distance between the unitaries is found to be proportional to $\epsilon T$. We then find that the $LE$ oscillates around a particular value for a very long time before decaying to zero. The deviation of the value of the $LE$ from 1 scales as $\epsilon^2$ for a fixed value of $T$, while the time after which the $LE$ starts decaying to zero has an interesting dependence on $\epsilon$ and $T$. Finally, when $U_1$ and $U_2$ are applied in a random order, the $LE$ rapidly decays to zero with increasing time. We have presented a qualitative understanding of the above results.

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 studies the Su-Schrieffer-Heeger chain driven by two unitaries U1 and U2 that differ only in inter-cell hopping strength. It considers four classes of driving sequences: periodic alternation, quasiperiodic (Fibonacci and Thue-Morse), aperiodic, and random. For periodic driving the authors report that end modes can appear whose number does not always coincide with a Z-valued winding number; they also compute the Loschmidt amplitude starting from an H1 end mode and relate its Fourier peak to the quasienergy of a Floquet end mode. For small epsilon (hopping difference) and small T they find that the Loschmidt echo in quasiperiodic/aperiodic protocols oscillates about a non-unity value for a long time before decaying, with the deviation scaling as epsilon squared; random protocols produce rapid decay. Qualitative interpretations are supplied for all regimes.

Significance. If the central observations survive clarification of the topological invariant, the work would add concrete numerical evidence that static winding numbers can fail to predict the number of Floquet end modes under two-step periodic driving, and would supply quantitative scaling relations for Loschmidt-echo dynamics in weakly perturbed quasiperiodic drives. The systematic comparison across periodic, quasiperiodic, aperiodic and random protocols is a useful addition to the Floquet-topology literature.

major comments (2)
  1. [Abstract; periodic protocol paragraph] Abstract and periodic-protocol section: the claim that end-mode count 'does not always agree with the winding number' is load-bearing for the paper's main result on periodic driving. The text does not state whether the winding number is evaluated on the static Hamiltonians H1 or H2 or on the Floquet operator U = U2 U1 (or its effective Hamiltonian). Without an explicit definition or formula (e.g., the integral of the Berry connection over the Brillouin zone for the appropriate operator), it is impossible to judge whether the reported mismatch is a non-trivial feature of the driven system or an expected consequence of applying a static invariant to a Floquet problem.
  2. [Quasiperiodic section] Quasiperiodic/aperiodic section: the statement that 'the distance between the unitaries is proportional to epsilon T' for small epsilon and T is used to motivate the subsequent epsilon-squared scaling of the Loschmidt-echo deviation. The manuscript should specify the distance measure (operator norm, Hilbert-Schmidt, etc.) and provide either an analytic derivation or a direct numerical check of this proportionality, as it is the only quantitative link between the microscopic parameters and the reported scaling.
minor comments (2)
  1. [Abstract] The abstract states that 'we have presented a qualitative understanding of the above results' but does not indicate which observations rest on analytics versus which rest solely on numerics; a short clarifying sentence would help readers assess the strength of the claims.
  2. [Introduction / model definition] Notation for the two unitaries is introduced as U1 = exp(-i H1 T/2) and U2 = exp(-i H2 T/2); it would be helpful to state explicitly whether H1 and H2 are time-independent or already contain any time dependence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each major comment in detail below and will make the necessary revisions to improve the clarity and rigor of the paper.

read point-by-point responses

  1. Referee: Abstract and periodic-protocol section: the claim that end-mode count 'does not always agree with the winding number' is load-bearing for the paper's main result on periodic driving. The text does not state whether the winding number is evaluated on the static Hamiltonians H1 or H2 or on the Floquet operator U = U2 U1 (or its effective Hamiltonian). Without an explicit definition or formula (e.g., the integral of the Berry connection over the Brillouin zone for the appropriate operator), it is impossible to judge whether the reported mismatch is a non-trivial feature of the driven system or an expected consequence of applying a static invariant to a Floquet problem.

    Authors: We appreciate the referee's observation. The winding number in question is that of the static Hamiltonians H1 and H2, calculated as the integral of the Berry connection over the Brillouin zone for the SSH model. The end modes we report are those of the Floquet operator U = U2 U1. The mismatch arises because the topological properties of the driven system are governed by the Floquet operator rather than the static ones. This is a non-trivial aspect we aim to highlight. In the revision, we will include the explicit formula for the winding number and clarify the distinction between static and Floquet invariants. We will also add a brief discussion on the conditions under which the mismatch occurs. revision: yes

  2. Referee: Quasiperiodic/aperiodic section: the statement that 'the distance between the unitaries is proportional to epsilon T' for small epsilon and T is used to motivate the subsequent epsilon-squared scaling of the Loschmidt-echo deviation. The manuscript should specify the distance measure (operator norm, Hilbert-Schmidt, etc.) and provide either an analytic derivation or a direct numerical check of this proportionality, as it is the only quantitative link between the microscopic parameters and the reported scaling.

    Authors: We thank the referee for this suggestion. The distance measure we refer to is the operator norm ||U1 − U2||. For small ε and T, with U1 = exp(−i H1 T/2) and U2 = exp(−i H2 T/2) where H2 differs from H1 by a term proportional to ε, a perturbative expansion shows that ||U1 − U2|| ≈ (ε T / 2) * ||perturbation operator|| to first order. We will provide this analytic derivation in the revised text. Furthermore, we will include a numerical verification plot showing the proportionality for small values of ε and T. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on direct computation of driven SSH dynamics

full rationale

The paper computes end-mode spectra, winding numbers, Loschmidt amplitudes, and echoes explicitly from the time-evolution operators U1 and U2 applied in periodic, quasiperiodic, aperiodic, and random sequences. The reported disagreement between end-mode count and winding number for the periodic case follows from separate evaluation of the Floquet operator spectrum and the topological invariant; neither quantity is defined in terms of the other. Scaling relations for the Loschmidt echo deviation (~ε²) and decay time arise from perturbative expansion in the stated small-ε, small-T regime rather than from any fitted parameter or self-referential definition. No load-bearing self-citations, ansatzes smuggled via prior work, or renaming of known results are present. The derivation chain remains independent of its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard SSH Hamiltonian, unitary time evolution, and the assumption that epsilon and T are small enough for the stated proportionality and scaling to hold; no new entities are introduced.

free parameters (2)
  • epsilon
    Small difference between inter-cell hoppings of the two unitaries, used for scaling analysis.
  • T
    Time period of each unitary application, assumed small.
axioms (2)
  • domain assumption The two unitaries are applied exactly in the stated sequences (periodic alternation, Fibonacci, Thue-Morse, or random).
    Protocol definitions in the abstract.
  • standard math Standard quantum mechanics unitary evolution under time-independent Hamiltonians for each segment.
    Implicit in definition of U1 and U2.

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