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arxiv: 2512.24600 · v2 · pith:3MC2T6RPnew · submitted 2025-12-31 · ❄️ cond-mat.other · quant-ph

Dynamical Phase Transitions in Periodically Driving 1D Ising Model

Yuanyuan Cheng , Yuxia Zhang , Tianhui Qiu , Peipei Xin , Bao-Ming Xu This is my paper

Pith reviewed 2026-05-25 07:54 UTC · model grok-4.3

classification ❄️ cond-mat.other quant-ph
keywords dynamical quantum phase transitionsperiodically driven Ising modelFloquet topological phasestransverse field Ising modelnonequilibrium dynamicsresonant driveenergy level degeneracyquantum spin chains
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The pith

Periodic driving induces dynamical quantum phase transitions in the 1D Ising model through resonance within phases or low-frequency crossing of the critical point.

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

The paper establishes that dynamical quantum phase transitions can occur in a one-dimensional Ising model with a periodically modulated transverse field in two distinct scenarios. First, a resonant drive within either the ferromagnetic or paramagnetic phase triggers the transition when the drive frequency matches an energy-level transition, linking it to Floquet topological phases. Second, low-frequency drives that cross the critical point between phases induce the transition due to energy-level degeneracy at criticality, while high frequencies suppress it. A reader would care because these mechanisms show how periodic drives, unlike sudden quenches, can control nonequilibrium dynamics in quantum spin systems through resonance and degeneracy effects.

Core claim

In the periodically driven transverse-field Ising model, dynamical quantum phase transitions arise in two ways. Resonant periodic drives within a given phase (ferromagnetic or paramagnetic) induce DQPTs connected to Floquet topological phases, with the transition timescale following τ proportional to the energy gap at the critical mode times the inverse perturbation strength times csc of the critical wavevector. Drives across the critical point at low frequencies always induce DQPTs because of energy-level degeneracy ensuring excitation below the intrinsic frequency, whereas high frequencies inhibit the transitions.

What carries the argument

Resonance matching to energy transitions within phases and energy degeneracy at the critical point under periodic transverse-field modulation

If this is right

  • Resonant drives within a single phase produce DQPTs tied to Floquet topology.
  • The DQPT timescale scales as τ ∝ Δ_{k_c} λ'^{-1} csc k_c.
  • Low-frequency drives across the FM-PM boundary always excite the system and induce DQPTs.
  • High-frequency drives across the boundary suppress excitations and prevent DQPTs.

Where Pith is reading between the lines

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

  • This mechanism suggests periodic driving can be used to access Floquet topological phases in spin chains without crossing phase boundaries.
  • Similar resonance and degeneracy effects might appear in other periodically driven quantum many-body systems.
  • The scaling relation provides a testable prediction for the onset time of DQPTs in experiments.

Load-bearing premise

The results depend on the specific form of the periodically modulated transverse-field Ising Hamiltonian and the accuracy of the perturbative or analytic treatment used for the driven spectrum.

What would settle it

Numerical simulation or experiment showing whether DQPTs appear under resonant driving inside one phase only at matching frequencies, and whether crossing the critical point at low but not high frequencies consistently produces or suppresses DQPTs.

Figures

Figures reproduced from arXiv: 2512.24600 by Bao-Ming Xu, PeiPei Xin, Tianhui Qiu, Yuanyuan Cheng, Yuxia Zhang.

Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) Time evolution of the rate function [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) The time evolution of the rate function [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) The time evolution of the rate function [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
read the original abstract

This work investigates dynamical quantum phase transitions (DQPTs) in a one-dimensional Ising model subjected to a periodically modulated transverse field. In contrast to sudden quenches, we demonstrate that a DQPT can be induced in two distinct ways. First, when the system remains within a given phase--ferromagnetic (FM) or paramagnetic (PM), a resonant periodic drive can trigger a DQPTs when its frequency matches the energy-level transition of the system. This DQPT is intimately connected to the emergence of Floquet topological phases. The timescale for the transition is governed by the perturbation strength $\lambda'$, the critical mode $k_c$, and its energy gap $\Delta_{k_c}$, following the scaling relation $\tau\propto\Delta_{k_c}\lambda'^{-1}\csc k_c$. Second, for drives across the critical point between the FM and PM phases, low frequencies can always induce DQPT, regardless of resonance. This behavior stems from the degeneracy of the energy-level at the critical point, which ensures that any drive with a frequency lower than the system's intrinsic transition frequency will inevitably excite the system. However, in the high-frequency regime, such excitation will be strongly suppressed, thereby inhibiting the occurrence of DQPTs. This study provides deeper insight into the nonequilibrium dynamics of quantum spin chains.

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 examines dynamical quantum phase transitions (DQPTs) in the one-dimensional transverse-field Ising model under periodic driving of the transverse field. It claims two distinct induction mechanisms: resonant driving within a single phase (FM or PM) that triggers DQPTs via connection to Floquet topological phases, with timescale scaling τ ∝ Δ_{k_c} λ'^{-1} csc k_c; and low-frequency driving across the critical point, where energy-level degeneracy at h=1 ensures excitation (and thus DQPT) for ω below the intrinsic frequency, while high-frequency drives suppress it.

Significance. If the perturbative derivations and degeneracy arguments hold under scrutiny, the work clarifies how resonance and critical-point degeneracy control DQPTs in driven spin chains and links them to Floquet topology, providing concrete scaling predictions that could guide experiments on nonequilibrium quantum dynamics.

major comments (2)
  1. [Resonant periodic drive section] The resonant-drive scaling relation and its link to Floquet topology rest on a perturbative expansion around the undriven spectrum (implicit in the abstract's derivation of τ ∝ Δ_{k_c} λ'^{-1} csc k_c). This assumes weak λ' with negligible higher-order corrections and that resonance couples exclusively to the critical mode; the manuscript must supply explicit checks against exact Floquet quasienergy spectra or non-perturbative numerics to confirm the regime of validity.
  2. [Low-frequency drive across critical point section] The low-frequency across-critical-point claim asserts that degeneracy at h=1 guarantees level mixing and Loschmidt-echo non-analyticity for any ω below the intrinsic frequency. This depends on the specific modulation form always producing the required matrix elements; the derivation should be shown explicitly from the time-periodic Hamiltonian, with verification that the non-analyticity follows directly rather than from additional assumptions.
minor comments (2)
  1. The abstract states the scaling and mechanisms but does not indicate the method (perturbative analytic, exact diagonalization, etc.); a brief methods sentence would improve clarity.
  2. Notation for the perturbation amplitude (λ') and critical quantities (k_c, Δ_{k_c}) should be defined at first use with explicit reference to the Hamiltonian.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and will incorporate clarifications and additional verifications in a revised manuscript.

read point-by-point responses

  1. Referee: [Resonant periodic drive section] The resonant-drive scaling relation and its link to Floquet topology rest on a perturbative expansion around the undriven spectrum (implicit in the abstract's derivation of τ ∝ Δ_{k_c} λ'^{-1} csc k_c). This assumes weak λ' with negligible higher-order corrections and that resonance couples exclusively to the critical mode; the manuscript must supply explicit checks against exact Floquet quasienergy spectra or non-perturbative numerics to confirm the regime of validity.

    Authors: We agree that the scaling relation is obtained via a perturbative treatment. In the revision we will add direct comparisons of the perturbative prediction against exact Floquet quasienergy spectra (computed via time-evolution operator diagonalization) for representative values of λ' and system sizes, explicitly delineating the weak-driving regime where higher-order corrections remain small and confirming that resonance is dominated by the critical mode. revision: yes

  2. Referee: [Low-frequency drive across critical point section] The low-frequency across-critical-point claim asserts that degeneracy at h=1 guarantees level mixing and Loschmidt-echo non-analyticity for any ω below the intrinsic frequency. This depends on the specific modulation form always producing the required matrix elements; the derivation should be shown explicitly from the time-periodic Hamiltonian, with verification that the non-analyticity follows directly rather than from additional assumptions.

    Authors: We will expand the revised manuscript to derive the relevant matrix elements explicitly from the time-periodic Hamiltonian and demonstrate that the degeneracy at the critical point produces the necessary coupling for low-frequency drives. We will also include numerical verification that the Loschmidt-echo non-analyticity arises directly from this degeneracy without further assumptions. revision: yes

Circularity Check

0 steps flagged

Derivation chain self-contained; no reductions to inputs or self-citations

full rationale

The abstract and described derivation present the scaling τ∝Δ_kc λ'^{-1} csc k_c and the two DQPT induction mechanisms as following directly from perturbative treatment of the driven Ising Hamiltonian and energy degeneracy at h=1. No quoted step equates a prediction to a fitted parameter by construction, invokes a load-bearing self-citation, or renames a known result. The central claims retain independent content from the model spectrum and modulation assumptions, consistent with the default non-circular finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes the standard 1D transverse-field Ising Hamiltonian under periodic modulation and the existence of Floquet topological phases; no new entities are postulated. Free parameters are implicit in the drive amplitude and frequency but not fitted in the abstract. Axioms are standard quantum mechanics and the definition of DQPT via Loschmidt echo or return probability.

axioms (2)
  • domain assumption The driven system is described by the time-periodic transverse-field Ising Hamiltonian whose spectrum determines resonance conditions.
    Invoked when stating that resonance matches energy-level transitions and that degeneracy occurs at the critical point.
  • domain assumption Dynamical quantum phase transitions are diagnosed by non-analyticities in the Loschmidt echo or return probability under periodic driving.
    Standard definition used to claim that the two drive protocols induce DQPTs.

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