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arxiv: 2604.07714 · v1 · submitted 2026-04-09 · 🪐 quant-ph

Critical Entanglement Dynamics at Dynamical Quantum Phase Transitions

Kaiyuan Cao , Mingzhi Li , Xiang-Ping Jiang , Shu Chen , Jian Wang This is my paper

Pith reviewed 2026-05-10 18:17 UTC · model grok-4.3

classification 🪐 quant-ph
keywords dynamical quantum phase transitionsentanglement entropymomentum-space entanglementpost-quench eigenbasisSu-Schrieffer-Heeger modelHaldane modelgeometric conditiontwo-band models
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The pith

The geometric DQPT condition produces exact degeneracy of 1/2 in the post-quench entanglement spectrum, yielding maximal entropy ln 2.

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

This paper investigates critical behavior of momentum-space entanglement entropy at dynamical quantum phase transitions in two-band models. It establishes that the condition where the initial and final pseudospin vectors are orthogonal leads to exact half-probabilities in the entanglement spectrum when the bipartition uses the post-quench eigenstates. This results in a maximal and time-independent entropy of ln 2 at the critical momenta. The critical points are isolated in one dimension but form continuous curves in two dimensions. Other bipartitions like the sublattice one instead make the entropy time-dependent and minimal at the transitions, highlighting the importance of the basis choice.

Core claim

We establish that the geometric DQPT condition d̂_k^i · d̂_k^f = 0 manifests as exact degeneracy p_k*=1/2 in the entanglement spectrum defined with respect to the post-quench eigenbasis, yielding a maximal momentum-space entropy of ln 2. In one dimension, critical momenta appear as isolated points, whereas in two dimensions they form continuous one-dimensional manifolds, reflecting the dimensional dependence of the underlying critical structure. Alternative bipartitions such as the sublattice basis produce qualitatively different behavior where the entropy is time-dependent and attains a minimum at DQPT critical times.

What carries the argument

The entanglement spectrum defined in the post-quench eigenbasis, which develops exact degeneracy p=1/2 precisely at momenta satisfying the geometric condition d̂^i · d̂^f =0.

If this is right

  • Critical momenta appear as isolated points in one dimension but form continuous one-dimensional manifolds in two dimensions.
  • Momentum-space entanglement entropy provides a robust time-independent diagnostic of DQPTs when evaluated in the post-quench basis.
  • The geometric condition links directly to entanglement degeneracy, connecting entanglement measures to non-equilibrium criticality.
  • Alternative bipartitions produce time-dependent entropy that reaches a minimum at the DQPT times instead.

Where Pith is reading between the lines

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

  • Experimental measurements of entanglement after a quench could test this by confirming the basis dependence in cold-atom or superconducting systems.
  • The result suggests basis choice is essential for using entanglement as a probe in other non-equilibrium settings.
  • The dimensional structure of critical manifolds may appear in real-space entanglement observables as well.

Load-bearing premise

The bipartition for the entanglement entropy must be defined in the post-quench eigenbasis for the entropy to be time-independent and reach its maximum at the critical points.

What would settle it

Compute the eigenvalues of the reduced density matrix in the post-quench eigenbasis for a model such as the SSH chain after a quench that satisfies d_i · d_f =0 at some momentum k*, and check whether they equal exactly 1/2 at the critical time with no time dependence afterward.

Figures

Figures reproduced from arXiv: 2604.07714 by Jian Wang, Kaiyuan Cao, Mingzhi Li, Shu Chen, Xiang-Ping Jiang.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Momentum-space entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Contour plot of the momentum-space entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Momentum-space entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
read the original abstract

We investigate the critical behavior of momentum-space entanglement entropy at dynamical quantum phase transitions (DQPTs) in translationally invariant two-band insulators and superconductors. By analyzing the Su-Schrieffer-Heeger model, the quantum XY chain, and the Haldane model, we establish that the geometric DQPT condition $\hat{\textbf{d}}_{\textbf{k}}^{i} \cdot \hat{\textbf{d}}_{\textbf{k}}^{f} = 0$ manifests as exact degeneracy $p_{\textbf{k}^{*}}=1/2$ in the entanglement spectrum defined with respect to the post-quench eigenbasis, yielding a maximal momentum-space entropy of $\ln 2$. In one dimension, critical momenta appear as isolated points, whereas in two dimensions they form continuous one-dimensional manifolds, reflecting the dimensional dependence of the underlying critical structure. Importantly, alternative bipartitions such as the sublattice basis produce qualitatively different behavior: the entropy becomes explicitly time-dependent and attains a minimum at DQPT critical times, underscoring the essential role of basis selection. Our results establish that momentum-space entanglement entropy, when evaluated in the appropriate eigenbasis, provides a robust, time-independent diagnostic of DQPTs and offers a unified geometric perspective linking entanglement, topology, and non-equilibrium criticality.

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 / 4 minor

Summary. The manuscript investigates critical behavior of momentum-space entanglement entropy at dynamical quantum phase transitions (DQPTs) in translationally invariant two-band insulators and superconductors. It establishes that the geometric DQPT condition d̂_k^i · d̂_k^f = 0 produces exact degeneracy p_k*=1/2 in the entanglement spectrum when bipartitioned in the post-quench eigenbasis, yielding a time-independent maximal entropy contribution of ln 2 at critical momenta. This is demonstrated analytically and numerically for the Su-Schrieffer-Heeger model, quantum XY chain, and Haldane model. Critical momenta appear as isolated points in 1D and continuous manifolds in 2D. The paper explicitly contrasts this with the time-dependent entropy (attaining a minimum at DQPT times) obtained under sublattice bipartition, underscoring basis dependence.

Significance. If the central link holds, the work supplies a parameter-free geometric diagnostic for DQPTs via momentum-space entanglement entropy evaluated in the post-quench eigenbasis. The exact degeneracy, time-independence due to conserved occupations, and dimensional structure of critical manifolds provide a unified perspective connecting entanglement, topology, and non-equilibrium criticality. Explicit model calculations and the basis-contrast analysis are strengths; the result is falsifiable through direct computation of the entanglement spectrum at the stated geometric condition.

minor comments (4)
  1. [§2] §2 (or wherever the general derivation appears): the step connecting d̂_k^i · d̂_k^f = 0 to p_k* = 1/2 should be written out explicitly with the definition of the reduced density matrix in the post-quench basis, even if it is standard; this would allow readers to verify the degeneracy without model-specific algebra.
  2. [Table 1] Table 1 or equivalent summary of models: the reported entropy values at DQPT times should include the precise numerical tolerance (e.g., |p_k* - 1/2| < 10^{-10}) to confirm the claimed exactness beyond floating-point agreement.
  3. [Figure 3] Figure 3 (Haldane model): the 2D manifold of critical momenta is shown as a curve; clarify whether the plotted entropy is integrated over the manifold or evaluated at representative points, and state the Brillouin-zone discretization used.
  4. [Introduction] Notation: the vector d̂_k is used both for initial and final Hamiltonians; a brief reminder of the normalization ||d̂|| = 1 when first introduced would prevent any momentary ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation for minor revision. The referee's summary accurately captures the central results of our manuscript, including the link between the geometric DQPT condition and the exact degeneracy p_k^*=1/2 in the post-quench eigenbasis, the resulting time-independent ln 2 entropy contribution, the dimensional structure of critical manifolds, and the basis dependence highlighted by the sublattice contrast. We are encouraged by the recognition of the unified perspective connecting entanglement, topology, and non-equilibrium criticality, as well as the note that the result is falsifiable. Since no specific major comments were listed, we provide a response to the summary below.

read point-by-point responses

  1. Referee: REFEREE SUMMARY: The manuscript investigates critical behavior of momentum-space entanglement entropy at dynamical quantum phase transitions (DQPTs) in translationally invariant two-band insulators and superconductors. It establishes that the geometric DQPT condition d̂_k^i · d̂_k^f = 0 produces exact degeneracy p_k*=1/2 in the entanglement spectrum when bipartitioned in the post-quench eigenbasis, yielding a time-independent maximal entropy contribution of ln 2 at critical momenta. This is demonstrated analytically and numerically for the Su-Schrieffer-Heeger model, quantum XY chain, and Haldane model. Critical momenta appear as isolated points in 1D and continuous manifolds in 2D. The paper explicitly contrasts this with the time-dependent entropy (attaining a minimum at DQPT times) obtained under sublattice bipartition, underscoring basis dependence.

    Authors: We thank the referee for this accurate and detailed summary, which faithfully reflects the content and emphasis of our manuscript. The description correctly highlights the exact degeneracy arising from the geometric condition, the time-independence due to conserved occupations in the post-quench basis, the explicit demonstrations in the SSH, XY, and Haldane models, the dimensional dependence (isolated points in 1D vs. manifolds in 2D), and the contrasting time-dependent minimum under sublattice bipartition. We agree that these elements provide a robust, parameter-free diagnostic and a falsifiable connection between entanglement and DQPTs. No revisions are needed in response to this summary, as it requires no corrections or additions. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's core step is the direct algebraic consequence that the geometric condition d̂_k^i · d̂_k^f = 0 produces exact degeneracy p_k^*=1/2 in the entanglement spectrum when the bipartition is taken in the post-quench eigenbasis. This follows from the two-band structure and conserved occupations in that basis, without any fitting, parameter tuning, or redefinition that forces the outcome. The abstract and described claims explicitly contrast this with sublattice bipartition (which yields time-dependent entropy), showing the basis choice is justified by observable difference rather than smuggled in. No load-bearing self-citations, uniqueness theorems from prior author work, or renaming of known results appear in the provided text; the momentum-space entropy maximum of ln 2 is a straightforward per-mode sum once degeneracy is established. The derivation chain therefore remains independent of its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of the geometric DQPT condition in two-band translationally invariant systems and on the choice of post-quench eigenbasis for the entanglement spectrum; no free parameters, ad-hoc axioms, or new postulated entities are introduced.

axioms (2)
  • domain assumption The systems under study are translationally invariant two-band insulators and superconductors
    Explicitly stated as the setting for the analysis of SSH, XY, and Haldane models.
  • standard math The geometric DQPT condition is d̂_k^i · d̂_k^f = 0
    Invoked as the defining condition whose consequences for the entanglement spectrum are derived.

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