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REVIEW 3 major objections 3 minor 74 references

This paper claims that the Kramers-Wannier symmetry of the Ising chain is dynamically restored after quenches to criticality, and that this restoration can exhibit a quantum Mpemba effect.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 08:05 UTC pith:K3YJN5ZY

load-bearing objection A real lattice result for a non-invertible symmetry; the quench predictions are new and numerically well backed, but their foundation is an openly labeled conjecture that deserves a sharp referee. the 3 major comments →

arxiv 2607.21226 v1 pith:K3YJN5ZY submitted 2026-07-23 cond-mat.stat-mech hep-thquant-ph

Entanglement asymmetry and quantum Mpemba effect for Kramers-Wannier duality

Milo Vescovo , Pasquale Calabrese , Filiberto Ares This is my paper
classification cond-mat.stat-mech hep-thquant-ph
keywords Kramers-Wannier dualitynon-invertible symmetryentanglement asymmetryquantum Mpemba effecttransverse-field Ising chainquantum quenchquasiparticle picture
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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 give the first quantitative, subsystem-level account of what happens to the Kramers-Wannier duality—the archetypal non-invertible symmetry—when the Ising chain is taken away from criticality and then driven back to it. It defines a Kramers-Wannier entanglement asymmetry, shows that it is nonzero in both gapped phases and exactly zero at the self-dual point, and derives an analytical description, with numerical support, of its relaxation to zero after quenches to criticality. Along the way it finds a quantum Mpemba effect: for quenches starting in the ordered phase, a state that initially breaks the duality more strongly can restore it faster. If right, it establishes entanglement asymmetry as a practical probe of non-invertible symmetries and gives a concrete prediction—a t^{-3} tail with a specific prefactor—that quantum simulators could look for.

Core claim

The paper introduces a Kramers-Wannier entanglement asymmetry for a subsystem of the transverse-field Ising chain, defined by symmetrizing the reduced density matrix with its dual, ΔS_A^(2)=2 log 2 + log Tr ρ_A^2 − log Tr(ρ_A^2 + ρ̂_A^2 + 2 ρ_A ρ̂_A). Its equilibrium result is that in each gapped phase the asymmetry saturates to a finite, field-dependent value as the subsystem grows, while at the self-dual point h=1 the overlap term Tr(ρ_A ρ̂_A) is no longer negligible and forces the asymmetry exactly to zero, producing a critical dip that sharpens with subsystem size. Its dynamical result is that after quenches from h0≠1 to h=1 the asymmetry tends to zero at long times, with the asymptotic

What carries the argument

The central object is the mixed overlap Tr(ρ_A ρ̂_A) between the reduced density matrix and its Kramers-Wannier dual—the only quantity in the second-Rényi asymmetry that is not fixed by the purities. In equilibrium the paper computes its exponential decay rate with subsystem size from a block-Toeplitz determinant asymptotics. After a quench it conjectures a quasiparticle formula: each momentum mode relaxes independently, weighted by the same min(1,2|v_k|t/ℓ) counting factor used for entanglement growth, interpolating between the known initial and stationary values. This conjecture is what produces the intermediate plateau at log 2, the universal t^{-3} tail, and the Mpemba ordering.

Load-bearing premise

Everything rests on the conjecture that the overlap between the reduced state and its dual relaxes mode by mode, independently, with the same min(1,2|v|t/ℓ) factor used for entanglement growth; if modes instead relax collectively, the long-time decay law and the Mpemba conditions change.

What would settle it

Measure the mixed overlap Tr(ρ_A ρ̂_A)(t) or the second-Rényi asymmetry directly for a quench from a known h0, e.g. h0=2, over long times. Eq. (60) fixes the curve and Eq. (64) fixes the tail; failure to observe the t^{-3} decay with prefactor ℓ^4/(192π)[(h0−1)/(h0+1)]^2, or the absence of the intermediate plateau, settles the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • In equilibrium, the KW entanglement asymmetry saturates to a finite value in each gapped phase, depends nontrivially on the transverse field, and vanishes exactly at the self-dual point with a critical dip whose width shrinks as the subsystem grows.
  • After a quench from either gapped phase to h=1, the asymmetry decays to zero: the non-invertible KW symmetry is dynamically restored, not just in the full state but for the subsystem.
  • For quenches from the ordered phase there is an intermediate plateau at ΔS=log 2, signalling a regime where the reduced state and its dual are effectively orthogonal.
  • The universal long-time decay is t^{-3}, with initial-state dependence only in the prefactor ℓ^4/(192π)[(h0−1)/(h0+1)]^2.
  • The quantum Mpemba effect occurs for pairs of quenches both starting in the ordered phase, never for pairs both in the disordered phase, and case-by-case for opposite phases.

Where Pith is reading between the lines

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

  • Inference: the same mode-by-mode relaxation of the overlap should be testable in other free-fermion or free-boson critical quenches; a failure there would pinpoint exactly which part of the conjecture is model-specific.
  • Inference: the plateau at log 2 may act as an experimentally accessible order parameter for non-invertible symmetry breaking in the ordered phase, distinguishable from the ordinary Z2 asymmetry's plateau.
  • Inference: if restoration is generic, it would contrast with known failures of restoration for ordinary invertible symmetries and suggest non-invertible symmetries are more strongly constrained by dynamics; the paper itself leaves this as an open question.
  • Inference: the prefactor formula is sharp enough that a quantum-simulator measurement of the asymmetry at two or three initial fields could confirm or rule out the t^{-3} law and the Mpemba ordering.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper introduces a Kramers-Wannier (KW) entanglement asymmetry for the transverse-field Ising chain, defined by symmetrizing the full density matrix with its KW dual and then tracing to a subsystem. In equilibrium, it is shown that gapped ground states have a finite large-subsystem asymmetry, with exponentially small overlap Tr(ρ_A ρ̂_A) and a pronounced dip at the critical self-dual point that sharpens with subsystem size. For quenches to the critical point, the paper proposes a quasiparticle picture for log Tr(ρ_A ρ̂_A), leading to a plateau at log 2, a universal long-time decay ΔS_A^(2)(t) ~ ℓ^4/(192π t^3) [(h0-1)/(h0+1)]^2, and precise conditions for the quantum Mpemba effect. Higher-Rényi and von Neumann extensions are also given using corner-transfer-matrix results.

Significance. If the central dynamical claims hold, this is a valuable first quantitative probe of non-invertible symmetry breaking and dynamical restoration in a lattice model. The equilibrium results are checked against exact numerics, and the long-time scaling law is concrete and falsifiable. The manuscript is unusually honest about its conjectural ingredients, which is a strength; however, the main nonequilibrium predictions rest on a conjecture that has so far only been tested at moderate subsystem sizes. The paper will be influential if that conjecture is confirmed by a systematic scaling-limit check or a derivation within the Gaussian framework.

major comments (3)
  1. [Sec. 5.1, Eq. (60)] The central dynamical prediction is built on the conjecture in Eq. (60). The mixed overlap log Tr(ρ_A ρ̂_A) is a determinant of I + T[G(t)] T[Ĝ(t)] with two different, non-commuting block-Toeplitz symbols; there is no factorization theorem that guarantees independent relaxation of each momentum mode. Equation (60) interpolates between the known initial value (57) and the assumed stationary value (59), and Eq. (59) already assumes symmetry restoration. Because Eq. (64) and the Mpemba conditions in Sec. 5.2 inherit Eq. (60), a single ℓ=50 comparison in Fig. 5 is not sufficient support. A scaling collapse of the exact ratio for several ℓ and t/ℓ, and ideally a controlled derivation of the min(1,2|v_k|t/ℓ) envelope, would make the central claim load-bearing rather than conditional.
  2. [Sec. 5 and Eq. (64)] The illustrative Mpemba example given near Fig. 4 appears inconsistent with the long-time formula. Equation (64) has prefactor C = ((h0-1)/(h0+1))^2, so C(0.7) ≈ 0.031 and C(2) ≈ 0.111. Since h0 = 2 has the larger initial asymmetry, this pair cannot satisfy the definition (63) for all t > t_M: at asymptotically long times the h0 = 2 curve is larger. If the observation is only a finite-time crossing before the two curves eventually re-cross, it should not be described as faster relaxation. Otherwise the example should be replaced by a pair consistent with Eq. (64), e.g. h0 = 0.1 or 0.2 versus h0 = 2.
  3. [Appendix A, Eq. (67)] Equation (67) is a conjecture for the product of block-Toeplitz matrices, not a proven theorem, as the text itself acknowledges. The main text in Sec. 4, however, states that the 'Widom-Szegő theorem' provides the full derivation and reports Eq. (42) as the result. This is an overstatement. The numerical verification in Fig. 1 is reassuring, and this ingredient does not affect the quench conclusions, but the manuscript should clearly distinguish the theorem (for a single Toeplitz symbol) from the numerically verified conjecture (for a product of symbols). This is important because the supposed derivation in Appendix A is currently not a derivation.
minor comments (3)
  1. [Appendix B, Eq. (80)] There is a factor-of-two inconsistency between Eq. (80) and the final result Eq. (81). As written, Eq. (80) evaluates to ℓ^4/(96π t^3) [(h0-1)/(h0+1)]^2, missing the 1/2 from Eq. (76). The final prefactor in Eq. (81) appears to be the correct one, but the intermediate displayed equation should be corrected.
  2. [Fig. 5 caption] Typo: 'susbystem size' should be 'subsystem size'.
  3. [Sec. 2] The definition of the subsystem KW asymmetry via symmetrization of the full state and then partial trace is a convention choice and not intrinsic to the subsystem algebra. The paper states this clearly, but it may deserve a more prominent caveat, since other definitions could give different finite-ℓ results.

Circularity Check

1 steps flagged

The analytic derivation of dynamical restoration assumes the restored stationary value as the endpoint of the conjectured overlap formula, then re-derives that endpoint; the independent numerics mitigate but do not remove this circular step.

specific steps
  1. self definitional [Sec. 5.1, Eqs. (58)-(60); used in Appendix B]
    "The asymptotic value for t→∞ is also known. Indeed, assuming that the Kramers-Wannier symmetry is dynamically restored at long times, one has TrρA ˆρA(t→∞) = Trρ2A(t→∞). ... By construction, Eq. (60) reproduces exactly both the initial condition, Eq. (57), and the stationary limit, Eq. (59)."

    Equation (58) sets the long-time mixed overlap equal to the purity, which is exactly the condition ΔS_A^{(2)}(∞)=0 in Eq. (41) when TrρA^2=Trρ̂A^2. Equation (60) is then built to interpolate to that endpoint, so the claimed analytic proof of dynamical restoration—and the t^{-3} asymptotics in Eq. (64)/Appendix B, which start from Eq. (60)—inherit the assumed conclusion. The exact numerics in Fig. 4 provide independent evidence for restoration, so the circularity is confined to the analytic derivation, not the physical result.

full rationale

The main equilibrium results are self-contained: the overlap asymptotics are derived (with a stated unproved Widom-Szegő-type conjecture, Eq. (67)) and the purities come from independent corner-transfer-matrix results [46-48], not from a self-citation chain. The definition of the KW asymmetry is introduced explicitly, with [9] used as a cross-check for boundary states rather than as the load-bearing input. No uniqueness theorem or ansatz is smuggled in via citation. The only genuine circular step is in the quench section: Eq. (60) is a conjecture for log Tr ρA ρ̂A(t) whose stationary limit is fixed by assuming the symmetry is restored, and the subsequent analytic derivation of the long-time decay and Mpemba ordering starts from this assumed endpoint. Because this step is transparent and backed by exact numerical data, the paper is not wholly circular; however, the analytic claim 'the KW symmetry is dynamically restored' is, within the derivation, an input rather than an output.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters are fitted anywhere; the asymptotic formulas contain no hand-tuned constants. The load-bearing inputs are standard free-fermion tools, prior CTM results, and two explicitly conjectured analytic steps (product Widom-Szegő and quasiparticle interpolation), both verified numerically in the paper. The invented-entity list is empty: no new particles, forces, or conserved quantities are introduced.

axioms (6)
  • ad hoc to paper Product Widom-Szegő conjecture: log det[I + ∏ T_ℓ[g_m]] = Aℓ + o(ℓ) for products of block-Toeplitz matrices (Eq. (67)).
    Appendix A invokes this unproved extension for Γ Γ̂ to get the exponential decay of Tr ρ_A ρ̂_A. The paper states it is not rigorous and verifies it numerically.
  • ad hoc to paper Quasiparticle independence of momentum modes for Tr ρ_A ρ̂_A (Eq. (60)): each mode relaxes independently with factor min(1, 2|v_k|t/ℓ).
    Sec. 5.1 states this as an assumption ('we assume that each momentum mode relaxes independently'), validated by comparison with exact numerics.
  • standard math Gaussian-state reduction: all traces of products of reduced density matrices are computed from correlation matrices via Eq. (29) (Wick's theorem).
    Sec. 3: ground and quenched states are Gaussian; the trace formula is from Refs. [36,37]. Standard free-fermion technology.
  • domain assumption Corner-transfer-matrix expressions for Rényi traces (Eqs. (44), (52)) taken from Refs. [46-48].
    Used as exact inputs for the equilibrium asymptotic asymmetry. These are prior literature results, not re-derived here.
  • domain assumption Replica-trick analytic continuation n→1 for the von Neumann asymmetry (Eq. (53)).
    Sec. 4.1: standard practice in the entanglement-asymmetry literature, not rigorously justified in the paper.
  • ad hoc to paper KW asymmetry is defined by symmetrizing the full state and then tracing (Sec. 2), rather than by an intrinsic subsystem-level duality.
    Because KW duality is non-local, no canonical subsystem-level dual exists. The paper adopts the full-then-trace procedure following Refs. [30,31]; this is a modeling choice that defines the measure.

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read the original abstract

The Kramers-Wannier duality is the prototypical example of a non-invertible symmetry, yet little is known about its fate away from criticality and out of equilibrium. We introduce the Kramers-Wannier entanglement asymmetry, a quantum-information measure that quantifies the breaking of this non-invertible symmetry in the transverse-field Ising chain. We first investigate its equilibrium properties, showing that it exhibits a striking crossover between the ordered and disordered phases together with a pronounced dip at the critical point that becomes increasingly sharp with subsystem size. We then study quantum quenches from both gapped phases to criticality and show that the Kramers-Wannier entanglement asymmetry decays to zero, signaling the dynamical restoration of the non-invertible symmetry. Remarkably, we uncover the emergence of a quantum Mpemba effect: under suitable conditions, states initially farther from equilibrium restore the Kramers-Wannier symmetry faster than states prepared closer to it. We provide both analytical and numerical evidence for this phenomenon and identify the mechanism responsible for its occurrence. Our work establishes entanglement asymmetry as a powerful probe of non-invertible symmetries beyond equilibrium and opens new perspectives on the dynamics of dualities in quantum many-body systems.

Figures

Figures reproduced from arXiv: 2607.21226 by Filiberto Ares, Milo Vescovo, Pasquale Calabrese.

Figure 1
Figure 1. Figure 1: (a) Tr(ρAρˆA) as function of the subsystem size ℓ for the ground state of the quantum Ising chain for different magnetic fields h, showing the exponential decay. The symbols are the exact values, computed using the corresponding formula in Eq. (31). The solid lines represent the asymptotic behavior in Eqs. (42)-(43), neglecting the subleading terms. (b) The symbols are obtained from linear fits to the exac… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Rényi-2 KW entanglement asymmetry ∆S (2) A as a function of the subsys￾tem size ℓ in the ground state of the quantum Ising chain for different magnetic fields h. (b) ∆S (2) A as a function of h for ℓ = 25, 45, 100. In both panels, the symbols show the exact numerical values obtained from the correlation matrices at finite ℓ using Eq. (31), while the solid curves correspond to the asymptotic expression … view at source ↗
Figure 3
Figure 3. Figure 3: Rényi KW entanglement asymmetry for several Rényi indices n as function of the magnetic field h in the ground state of the quantum Ising chain. (a) Asymptotic behavior in Eqs. (51)-(52), including the analytic continuation for n = 1. (b) Comparison with exact numerics at finite ℓ for n = 3, showing the typical dip at criticality, signaling the restoration of symmetry not captured by Eq. (51) as in [PITH_F… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Time evolution of Rényi-2 KW entanglement asymmetry ∆S (2) A (t) for a quench to the critical point of the quantum Ising chain for different values of the initial magnetic field h. The symbols are the exact numerical results for ℓ = 50, computed from the correlation matrix using Eq. (31). The full lines represent the asymptotic results from the quasiparticle picture. (b) Time evolution of ∆S (2) A (t) … view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of the traces entering the Rényi-2 Kramers–Wannier entan￾glement asymmetry (41) after a quench from h0 = 2 to the critical point. Here the susbystem size is ℓ = 50. The symbols are the exact numerical values, computed us￾ing the formulas in Eq. (31), while the curves represent the quasiparticle predictions in Eqs. (55) and (60). Observe that the quasiparticle prediction for Trρ 2 A and its K… view at source ↗

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