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arxiv: 2604.28117 · v1 · submitted 2026-04-30 · ❄️ cond-mat.stat-mech

Strong Mpemba Effect Through a Reentrant Phase Transition

Kristian Blom , Doron Benyamin , Uwe Thiele , Oren Raz , Aljaz Godec This is my paper

Pith reviewed 2026-05-07 05:34 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords antiferromagnetic Ising modelMpemba effectreentrant phase transitionparamagnetic phaserelaxation modesstaggered modetemperature quenchespair approximation
0
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The pith

Quenches to the paramagnetic phase in the antiferromagnetic Ising model produce a strong Mpemba effect via a staggered slow mode.

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

The paper examines how temperature quenches across a reentrant phase transition in the antiferromagnetic Ising model lead to anomalous relaxation known as the strong Mpemba effect. It finds that both direct and inverse strong Mpemba effects appear when the final state is in the paramagnetic phase because the slowest relaxation mode there is a purely staggered one that gets selectively excited or avoided depending on the initial phase. A sympathetic reader would care since this connects the structure of equilibrium phases directly to how quickly a system can relax after a sudden change in temperature. The work uses the pair approximation to calculate the modes and shows that without the reentrant behavior the effect vanishes. This suggests that phase transitions can be harnessed to control relaxation dynamics in spin systems.

Core claim

We show that the strong direct and inverse Mpemba effects arise when quenches terminate in the paramagnetic phase of the antiferromagnetic Ising model in a magnetic field. These anomalous relaxation phenomena originate from the selective excitation of the slowest relaxation mode, which in the paramagnetic phase is purely staggered. Consequently, quenches starting from the paramagnetic phase have zero overlap with the slow mode and exhibit a strong (inverse) Mpemba effect. Quenches from the antiferromagnetic phase excite the staggered mode and display a slow-relaxation tail. By varying the lattice coordination number we show that the strong Mpemba effect disappears in the absence of reentranc

What carries the argument

The staggered relaxation mode that becomes slowest in the paramagnetic phase after the reentrant transition, with its overlaps computed under the pair approximation.

If this is right

  • Quenches terminating in the paramagnetic phase have zero overlap with the slowest mode and therefore relax without the slow tail.
  • Quenches that begin in the antiferromagnetic phase excite the staggered mode and therefore exhibit a pronounced slow-relaxation tail.
  • The strong Mpemba effect vanishes when the coordination number is changed so that reentrance disappears.
  • The pair approximation is sufficient to establish the direct link between the reentrant transition and the anomalous relaxation behavior.

Where Pith is reading between the lines

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

  • The same selective excitation of a staggered slow mode could appear in other lattice models or real materials that possess reentrant transitions, offering a route to phase-controlled relaxation speeds.
  • Experiments on antiferromagnetic samples in applied fields could directly test whether starting temperature determines the presence of the slow tail after a quench.
  • Mapping equilibrium phase diagrams may become a practical tool for predicting and designing fast-relaxation protocols in systems far from equilibrium.

Load-bearing premise

The pair approximation accurately captures the relaxation modes and their overlaps for the antiferromagnetic Ising model across the reentrant transition.

What would settle it

Numerical simulations of the full dynamics of the antiferromagnetic Ising model that find nonzero overlap between paramagnetic initial conditions and the slowest mode, or that retain the strong Mpemba effect even when reentrance is removed, would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.28117 by Aljaz Godec, Doron Benyamin, Kristian Blom, Oren Raz, Uwe Thiele.

Figure 1
Figure 1. Figure 1: b–d. This is consistent with the exact result for the square lattice [45]. In contrast, for z > z∗ the low￾temperature slope becomes positive and the critical line turns non-monotonic as shown in Fig. 1f–h [73]. Such view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 view at source ↗
read the original abstract

We investigate temperature quenches across the reentrant phase transition of the antiferromagnetic Ising model in a magnetic field and show that the strong direct and inverse Mpemba effects arise when quenches terminate in the paramagnetic phase. These anomalous relaxation phenomena originate from the selective excitation of the slowest relaxation mode, which in the paramagnetic phase is purely staggered. Consequently, quenches starting from the paramagnetic phase have zero overlap with the slow mode and exhibit a strong (inverse) Mpemba effect. Quenches from the antiferromagnetic phase excite the staggered mode and display a slow-relaxation tail. By varying the lattice coordination number we show that the strong Mpemba effect disappears in the absence of reentrance. Our results provide the first demonstration of the strong (inverse) Mpemba effect in the antiferromagnetic Ising model based on the pair-approximation, and establish a link between anomalous relaxation and the equilibrium phase behavior.

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

3 major / 3 minor

Summary. The manuscript investigates temperature quenches across the reentrant phase transition in the antiferromagnetic Ising model in a magnetic field within the pair approximation. It claims that strong direct and inverse Mpemba effects occur precisely when quenches terminate in the paramagnetic phase, because the slowest relaxation mode there is purely staggered (zero overlap with uniform magnetization). Quenches from the antiferromagnetic phase excite this mode and show slow tails, while the effect vanishes when reentrance is suppressed by reducing coordination number. The results are derived from linearizing the closed dynamical equations for one- and two-point functions and analyzing eigenvalue overlaps.

Significance. If the pair-approximation spectrum is faithful, the work supplies a concrete mechanism linking the strong Mpemba effect to equilibrium phase structure via selective mode excitation, and it is the first such demonstration for the antiferromagnetic Ising model. The coordination-number control provides a falsifiable link between reentrance and anomalous relaxation that could be tested in other mean-field or lattice models.

major comments (3)
  1. [§3.2] §3.2 (pair-approximation closure and mode analysis): The central claim that the slowest eigenvalue in the paramagnetic phase has strictly zero overlap with the uniform magnetization (hence exact strong inverse Mpemba for paramagnetic initial conditions) rests on the pair closure. Because the closure truncates the BBGKY hierarchy at the two-spin level, it can artificially decouple staggered and uniform sectors; the reentrant transition itself is also a mean-field artifact of the same closure. No comparison to the exact master equation on small clusters (e.g., 2×2 or 4-site lattices) or to Monte Carlo trajectories is reported, leaving the zero-overlap result unverified.
  2. [§4.3] §4.3 (coordination-number scan): The disappearance of the strong Mpemba effect for coordination numbers that eliminate reentrance is presented as evidence that reentrance is necessary. However, the same scan simultaneously changes the accuracy of the pair approximation (higher z improves mean-field character), so the observed correlation could be an artifact of the closure rather than a generic feature of the model. A direct comparison at fixed z between pair-approximation and Monte Carlo relaxation spectra would be required to separate these effects.
  3. [§5] §5 (quantitative Mpemba measures): The reported crossing times and relaxation-rate ratios that quantify the “strong” Mpemba effect are obtained from the linearized pair dynamics without error bars or sensitivity analysis to the closure parameters. Because the pair approximation is uncontrolled for dynamics, it is unclear whether the reported effect sizes survive restoration of higher-order correlations.
minor comments (3)
  1. [Figure 2] Figure 2: the eigenvector components plotted for the slowest mode should include an explicit statement that the uniform-magnetization projection is numerically zero to machine precision, together with the tolerance used.
  2. [§2 and §3] Notation: the symbols for the staggered and uniform magnetizations are introduced inconsistently between the equilibrium phase diagram (§2) and the dynamical equations (§3); a single consistent notation table would help.
  3. [Introduction] The abstract states “first demonstration … based on the pair-approximation”; a brief sentence in the introduction citing prior Mpemba studies on Ising models (even if they used different approximations) would clarify the novelty claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments. We will revise the manuscript to clarify the limitations of the pair approximation and to include additional analysis addressing the concerns raised. Our responses to each major comment are provided below.

read point-by-point responses

  1. Referee: §3.2 (pair-approximation closure and mode analysis): The central claim that the slowest eigenvalue in the paramagnetic phase has strictly zero overlap with the uniform magnetization (hence exact strong inverse Mpemba for paramagnetic initial conditions) rests on the pair closure. Because the closure truncates the BBGKY hierarchy at the two-spin level, it can artificially decouple staggered and uniform sectors; the reentrant transition itself is also a mean-field artifact of the same closure. No comparison to the exact master equation on small clusters (e.g., 2×2 or 4-site lattices) or to Monte Carlo trajectories is reported, leaving the zero-overlap result unverified.

    Authors: We agree that the pair approximation truncates the BBGKY hierarchy and that the reentrant transition is a mean-field artifact. Within this approximation, the linearized dynamics in the paramagnetic phase decouple the uniform and staggered sectors due to symmetry, resulting in zero overlap of the slowest (staggered) mode with the uniform magnetization. We will add to the revised manuscript a comparison of the pair-approximation spectrum with the exact master equation on a 4-site cluster to verify the decoupling at least qualitatively for small systems. A full Monte Carlo study on larger lattices is beyond the present scope. revision: partial

  2. Referee: §4.3 (coordination-number scan): The disappearance of the strong Mpemba effect for coordination numbers that eliminate reentrance is presented as evidence that reentrance is necessary. However, the same scan simultaneously changes the accuracy of the pair approximation (higher z improves mean-field character), so the observed correlation could be an artifact of the closure rather than a generic feature of the model. A direct comparison at fixed z between pair-approximation and Monte Carlo relaxation spectra would be required to separate these effects.

    Authors: We acknowledge that the coordination-number scan affects both reentrance and the accuracy of the pair approximation. The results show that within the pair approximation the strong Mpemba effect vanishes when reentrance is eliminated. In the revision we will add explicit discussion of this limitation and note that a fixed-z comparison with Monte Carlo would be needed to confirm the effect is not an artifact of the closure. The mechanism we identify remains valid within the approximation employed. revision: partial

  3. Referee: §5 (quantitative Mpemba measures): The reported crossing times and relaxation-rate ratios that quantify the “strong” Mpemba effect are obtained from the linearized pair dynamics without error bars or sensitivity analysis to the closure parameters. Because the pair approximation is uncontrolled for dynamics, it is unclear whether the reported effect sizes survive restoration of higher-order correlations.

    Authors: Since the measures are computed from deterministic linearized equations, no error bars apply. The pair closure has no free parameters. We will include a sensitivity analysis with respect to quench parameters and initial conditions in the revised §5. We will also add a statement noting that quantitative values may change with higher-order correlations, while the qualitative link to the reentrant transition persists. revision: yes

Circularity Check

0 steps flagged

Derivation follows directly from pair-approximated dynamical equations; no circularity

full rationale

The paper closes the master equation at the pair level, linearizes the resulting ODEs for the two-point correlations, and solves the eigenvalue problem for the relaxation spectrum. The claim that the slowest mode in the paramagnetic phase is purely staggered (zero overlap with uniform magnetization) is obtained by explicit diagonalization of that matrix; the strong Mpemba effect is then a direct algebraic consequence of the overlap integral being zero for paramagnetic initial conditions. No parameter is fitted to Mpemba data and then re-used as a prediction, no self-citation supplies a uniqueness theorem that forces the result, and the reentrant transition itself is an independent output of the same closure. The logical chain is therefore self-contained within the stated approximation and does not reduce to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the pair approximation for both equilibrium phases and non-equilibrium relaxation, plus the assumption that the slowest mode is purely staggered in the paramagnetic phase.

axioms (2)
  • domain assumption The pair approximation provides a quantitatively reliable description of both statics and dynamics for the antiferromagnetic Ising model in a field.
    Invoked to obtain the relaxation spectrum and mode overlaps.
  • domain assumption The slowest relaxation mode in the paramagnetic phase is purely staggered.
    Central to the selective-excitation argument.

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discussion (0)

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