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arxiv: 2510.04326 · v2 · submitted 2025-10-05 · ❄️ cond-mat.str-el · cond-mat.stat-mech· quant-ph

Integrable Floquet Time Crystals in One Dimension

Rahul Chandra , Mahbub Rahaman , Soumyabroto Majumder , Analabha Roy , Sujit Sarkar This is my paper

Pith reviewed 2026-05-18 09:51 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechquant-ph
keywords discrete time crystalFloquet systemsintegrable spin chainsnext-nearest-neighbor couplingsubharmonic responsequantum phase transitionsone-dimensional lattices
0
0 comments X

The pith

Integrable one-dimensional spin chains realize a rigid discrete time crystal phase whose lifetime grows exponentially with system size.

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

The paper sets out to establish that periodically driven quadratic lattice models derived from spin chains can host a stable discrete time crystal phase without any disorder. These models keep integrability intact while using next-nearest-neighbor couplings to create gaps at resonant quasienergies and lock in the modes that generate persistent subharmonic responses. The resulting time crystal remains stable across ranges of transverse field and coupling strength when the drive frequency is chosen to maximize the subharmonic signal. A phase diagram shows a Floquet paramagnet separated from the time crystal by sharp transitions, and scaling analysis indicates the lifetime lengthens exponentially as the chain grows longer. This route matters because it replaces fragile many-body-localized constructions that eventually thermalize with a clean, integrable alternative that restricts thermalization through conserved quantities.

Core claim

In a family of periodically driven one-dimensional quadratic lattice Hamiltonians obtained from spin chains, the discrete time crystal phase emerges when next-nearest-neighbor interactions open controllable gaps at resonant quasienergies and pin the emergent quasienergy modes responsible for subharmonics. The phase is rigid in the joint parameter space of transverse field strength and next-nearest-neighbor coupling, with drive frequency tuned for strongest subharmonic response. The phase diagram contains a Floquet paramagnet phase separated from the discrete time crystal by sharp quantum phase transitions. Finite-size scaling of the quasienergy splitting between the subharmonic mode and its

What carries the argument

Next-nearest-neighbor couplings in integrable quadratic Hamiltonians that open gaps at resonant quasienergies and pin the subharmonic quasienergy modes.

If this is right

  • The discrete time crystal phase remains stable over intervals of transverse field and next-nearest-neighbor coupling strength.
  • Sharp quantum phase transitions separate the Floquet paramagnet from the discrete time crystal.
  • The lifetime of the discrete time crystal increases exponentially with system size.
  • An optimized drive frequency produces the strongest subharmonic response within these models.

Where Pith is reading between the lines

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

  • This integrability-based construction could be implemented in clean quantum simulators that allow tunable longer-range couplings without introducing disorder.
  • The same pinning mechanism might extend to other integrable Floquet systems where conserved quantities are engineered to protect time-translation symmetry breaking.
  • Because the approach avoids many-body localization, it may allow longer observation windows for studying the interplay between time crystals and other dynamical phases.

Load-bearing premise

The added interactions must preserve integrability while opening controllable gaps at resonant quasienergies and pinning the modes that produce subharmonics.

What would settle it

Numerical or experimental data showing that the Floquet quasienergy splitting between the subharmonic mode and its conjugate remains finite or decreases only polynomially rather than exponentially as system size increases.

Figures

Figures reproduced from arXiv: 2510.04326 by Analabha Roy, Mahbub Rahaman, Rahul Chandra, Soumyabroto Majumder, Sujit Sarkar.

Figure 1
Figure 1. Figure 1: FIG. 1. Panels depicting the Bogolon energies [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Panels depicting the Floquet quasienergies [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Density plot of the long-time average (strobed at even multiples of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Density plot of the long-time average of the fidelity [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Center: Floquet phase diagram in the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Stroboscopic time series of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

We demonstrate the realization of a Discrete Time-Crystal (DTC) phase in a family of periodically driven, one-dimensional quadratic lattice Hamiltonians that can be obtained using spin chains. These interactions preserve integrability while opening controllable gaps at resonant quasienergies and pinning the emergent quasienergy modes that are responsible for subharmonics. We demonstrate that the DTC phase is rigid in the parameter space of transverse field and an additional interaction like Next-Nearest-Neighbor (NNN) coupling strength, with the drive frequency optimized to produce the strongest subharmonic response. We also provide a detailed phase diagram of the model, exhibiting a Floquet Paramagnet (FPM) phase, as well as sharp quantum phase transitions between the FPM and the DTC. Finite-size scaling of the Floquet quasienergy splitting between the emergent subharmonic mode and its conjugate shows that the DTC lifetime diverges exponentially with system size. Thus, our work establishes a novel mechanism for achieving robust long-lived DTCs in one dimension. Motivation for this work stems from the limitations of disorder-based stabilization schemes that rely on many-body localization and exhibit only prethermal or finite-lived plateaus, eventually restoring ergodicity. Disorder-free routes are therefore highly desirable. Integrable (or Floquet-integrable) systems provide an attractive alternative because their extensive set of conserved quantities and constrained scattering strongly restrict thermalization channels. Our construction exploits these integrable restrictions together with longer-range NNN engineering to produce a clean, robust DTC that avoids the prethermal fragility of disordered realizations.

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 constructs a family of periodically driven one-dimensional quadratic (free-fermion) lattice Hamiltonians, obtained from spin chains with next-nearest-neighbor (NNN) couplings, that realize a discrete time-crystal (DTC) phase. The central claims are that these interactions preserve integrability while opening controllable gaps at resonant quasienergies, pinning emergent subharmonic modes; that the DTC phase is rigid in the transverse-field and NNN-coupling parameter space (with drive frequency optimized for strongest subharmonic response); that a detailed phase diagram exhibits a Floquet paramagnet (FPM) phase separated from the DTC by sharp quantum phase transitions; and that finite-size scaling of the Floquet quasienergy splitting between the subharmonic mode and its conjugate demonstrates exponential divergence of the DTC lifetime with system size.

Significance. If the construction is shown to produce an L-independent gap at resonance while remaining integrable, the work would supply a clean, disorder-free mechanism for long-lived DTCs in one dimension, bypassing the prethermal fragility of many-body-localization routes. The reported phase diagram and explicit finite-size scaling constitute concrete, falsifiable evidence that strengthens the proposal.

major comments (2)
  1. [§3] §3 (Hamiltonian construction and integrability): The claim that NNN couplings open controllable gaps precisely at the resonant quasienergies while preserving integrability is load-bearing for the exponential lifetime result, yet the manuscript provides no explicit single-particle diagonalization or Floquet-mode analysis demonstrating that the gap remains finite and independent of L in the thermodynamic limit. Without this, the observed exponential scaling of the quasienergy splitting could reflect transient finite-size behavior rather than a true DTC.
  2. [§5] §5 (Finite-size scaling): The reported exponential divergence of the subharmonic-mode splitting with system size is central to the rigidity and lifetime claims, but the scaling is presented only numerically; an analytical argument (e.g., via the effective two-level splitting induced by the NNN term) is required to rule out power-law corrections or residual degeneracies inherent to free-fermion Floquet spectra.
minor comments (2)
  1. [§4] The drive frequency ω is optimized for strongest subharmonic response, but the precise optimization criterion (e.g., maximization of the Fourier component at ω/2) should be stated explicitly in the text and figure captions.
  2. [Introduction] Notation for the emergent subharmonic quasienergy modes and their conjugates is introduced in the abstract but should be defined with a consistent symbol (e.g., ε_sub) when first appearing in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the work's significance, and constructive suggestions. We address each major comment below and have revised the manuscript to incorporate additional analysis where appropriate.

read point-by-point responses

  1. Referee: [§3] §3 (Hamiltonian construction and integrability): The claim that NNN couplings open controllable gaps precisely at the resonant quasienergies while preserving integrability is load-bearing for the exponential lifetime result, yet the manuscript provides no explicit single-particle diagonalization or Floquet-mode analysis demonstrating that the gap remains finite and independent of L in the thermodynamic limit. Without this, the observed exponential scaling of the quasienergy splitting could reflect transient finite-size behavior rather than a true DTC.

    Authors: We agree that an explicit demonstration strengthens the central claim. The model remains quadratic after Jordan-Wigner transformation, so the Floquet operator is a single-particle Bogoliubov-de Gennes matrix whose spectrum can be obtained by exact diagonalization for any L. In the revised manuscript we add this analysis (new subsection in §3 together with a supplementary figure): the NNN term opens a finite gap at the resonant quasienergy whose magnitude saturates to a nonzero, L-independent value already for moderate system sizes, consistent with the local nature of the perturbation and the preservation of integrability. The gap does not close in the thermodynamic limit because the resonant condition is satisfied uniformly across the Brillouin zone for the chosen drive frequency. revision: yes

  2. Referee: [§5] §5 (Finite-size scaling): The reported exponential divergence of the subharmonic-mode splitting with system size is central to the rigidity and lifetime claims, but the scaling is presented only numerically; an analytical argument (e.g., via the effective two-level splitting induced by the NNN term) is required to rule out power-law corrections or residual degeneracies inherent to free-fermion Floquet spectra.

    Authors: We acknowledge the value of an analytical complement to the numerics. In the revision we include a perturbative effective two-level description (added to §5): the subharmonic mode and its conjugate form an isolated pair whose splitting is generated by the matrix element of the NNN term between the two Floquet eigenstates. Because the underlying single-particle states are delocalized plane-wave-like modes in the integrable quadratic chain, the overlap integral yields an exponentially small splitting ∼ exp(−cL) with c set by the imaginary part of the complex quasi-momentum at resonance. This form excludes power-law corrections or accidental degeneracies, which would require fine-tuned band crossings absent in the gapped DTC regime. Additional finite-size data up to larger L are also provided to confirm the exponential fit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a family of integrable quadratic (free-fermion) Hamiltonians with added NNN couplings, then analyzes the resulting Floquet spectrum, phase diagram, and finite-size scaling of quasienergy splitting. The DTC rigidity, subharmonic response optimization, and exponential lifetime divergence are derived from properties of this engineered model rather than from any self-referential definition, fitted input renamed as prediction, or load-bearing self-citation. No uniqueness theorem or ansatz is imported from prior work by the same authors in a way that reduces the central claims to inputs by construction. The derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the model preserving integrability under periodic driving and NNN terms, plus the existence of pinned quasienergy modes at resonant frequencies. No explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Interactions preserve integrability while opening controllable gaps at resonant quasienergies
    Stated directly in abstract as the basis for pinning subharmonic modes.

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Reference graph

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