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Prethermal inverse Mpemba effect

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper demonstrates that a colder initial state can heat up faster and reach a higher-energy plateau than a warmer initial state during prethermalization in a periodically driven fermion chain.

desk verdict Genuinely new prethermal IME with a clean but narrow numerical demonstration; the window-dependence of the prethermal energy needs testing before I'd trust the headline claim. read the letter →

arxiv 2507.04669 v1 pith:TEGJGCAR submitted 2025-07-07 cond-mat.stat-mech quant-ph

Koudai Sugimoto , Tomotaka Kuwahara , Keiji Saito This is my paper · ORCID
classification cond-mat.stat-mechquant-ph
keywords prethermalizationinverseMpembaeffectperiodicallydrivenquantumsystemsFloquetdynamicsthermalpurestatesweaklycoupledsubsystemsthermalizationentanglemententropy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces the prethermal inverse Mpemba effect: during relaxation toward a long-lived prethermal state, a system initialized at lower temperature can heat up faster and to a higher energy than one initialized at higher temperature. It proves the concept by simulating a one-dimensional spinless fermion chain weakly coupled to an ancilla chain and driven by a periodic gauge field on one leg. The cold initial state ($\beta^{-1}=0.1$) reaches its prethermal plateau sooner and at higher energy than the warm state ($\beta^{-1}=0.5$), satisfying both criteria of their Eq. (2). The authors also sketch a general mechanism---conserved particle number dividing Hilbert space into weakly coupled sectors, with one subsystem heating rapidly and the other slowly---so the effect should transfer to other driven platforms. If correct, the Mpemba phenomenon is not limited to final thermal equilibrium but appears throughout multistage relaxation, with implications for energy control in driven quantum systems.

What carries the argument

The core object is the two-leg fermion lattice with Hamiltonian $\hat{H}_0 = -J_A\sum_j(\hat{c}^\dagger_{j,A}\hat{c}_{j+1,A}+\mathrm{h.c.}) + V_A\sum_j \hat{n}_{j,A}\hat{n}_{j+1,A} - J_\perp\sum_j(\hat{c}^\dagger_{j,A}\hat{c}_{j,B}+\mathrm{h.c.})$, driven through a Peierls phase on chain A only. The central mechanism is the combination of weak interchain coupling $J_\perp$ and particle-number conservation, which partitions the Hilbert space into sectors $(N_A,N_B)$ that communicate only on long timescales. The Floquet drive heats chain A quickly while chain B remains cold; the observable $\mathrm{NIPR}_A$, built from the Schmidt decomposition and normalized so that a completely random state gives $1$, diagnoses when chain A has prethermalized. The energy-spectrum inclusion $[E_B^{\min},E_B^{\max}]\subset[E_A^{\min},E_A^{\max}]$ ensures the cold initial state, with particles concentrated in A, can absorb energy to the top of A's band, producing both a faster and a higher prethermal plateau.

What would settle it

Repeat the simulation with larger system sizes or smaller $J_\perp$ and check whether the $\beta^{-1}=0.1$ trajectory still crosses the $\beta^{-1}=0.5$ trajectory and whether the prethermal energy (the average over $t/T\in[40,50]$) remains stable when the averaging window is shifted to $[60,70]$ or $[80,90]$; if the crossing disappears or the plateau drifts toward infinite-temperature energy on the same timescale as the crossing, the prethermal IME claim would be falsified.

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Extended reading notes

Core claim

The central claim is that the inverse Mpemba effect occurs not only for relaxation to true thermal equilibrium, but also for relaxation to a prethermal state in an isolated, periodically driven quantum system. In the model studied---a chain of interacting spinless fermions (chain A) coupled by weak hopping $J_\perp=0.04$ to an undriven ancilla chain (chain B), with total particle number conserved---the state prepared at inverse temperature $\beta^{-1}=0.1$ has a shorter prethermal relaxation time and a larger energy gap between its initial and prethermal energies than the state at $\beta^{-1}=0.5$. The crossing of the energy trajectories and the crossing of the subsystem inverse-participation-ratio curves $\mathrm{NIPR}_A$ indicate that chain A has fully randomized while global thermalization is still suppressed, giving a two-stage relaxation. The authors argue the mechanism is generic: a conserved quantity splits the Hilbert space into weakly coupled sectors, the fast-heating subsystem has a broader energy spectrum that contains the slow subsystem's spectrum, and at low initial temperatures more particles sit in the fast-heating subsystem.

Load-bearing premise

The demonstration rests on an assumed timescale separation: with $J_\perp=0.04$ the weak interchain hopping keeps the two chains nearly decoupled long enough for chain A to prethermalize within the observation window, and this separation is read off the numerics rather than derived from a controlled approximation.

Editorial extensions

If this is right

  • If the mechanism is generic, prethermal IME should appear in any driven system with a conserved quantity and weakly coupled subsystems whose spectra overlap unevenly.
  • Cold-atom ladders, superconducting qubit arrays with conserved particle number or parity, and Rydberg chains are concrete platforms where the crossing could be searched for.
  • The effect turns the Mpemba phenomenon into a multistage phenomenon, so relaxation protocols in quantum control may be able to exploit faster and higher energy uptake before full thermalization.
  • Because the entanglement entropy also shows a crossing, the prethermal IME has a quantum-information signature, not just an energy signature.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the authors leave implicit: the speedup may be tunable by shaping the drive spectrum, for instance by driving chain A harder or chain B at a higher frequency, rather than by changing only the initial temperature.
  • One testable prediction is that the effect should vanish or reverse when $J_\perp$ becomes comparable to $J_A$, since the sector bottleneck disappears and the two chains thermalize together.
  • The criterion in Eq. (2) could be recast in terms of thermomajorization or phase-space distances, potentially linking prethermal IME to non-monotonic relaxation rates in Markovian settings.
  • A possible concern: the observed crossing may depend on the finite particle density $N/L$; checking whether the effect survives at higher densities would clarify whether it is a generic feature or specific to dilute fillings.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes a prethermal inverse Mpemba effect (IME): in an isolated, periodically driven quantum system, a state initialized at a lower temperature can heat up faster and absorb more energy en route to a long-lived prethermal state than a state initialized at a higher temperature. The authors define quantitative criteria in Eq. (2) in terms of a relaxation time t* to a prethermal energy Epre_β, and demonstrate the effect numerically in a one-dimensional spinless fermion chain (chain A) weakly coupled to an ancilla chain (chain B), driven by a periodic gauge field. Using canonical thermal pure quantum states, they show energy trajectories crossing, a rapid saturation of the chain-A normalized inverse participation ratio (NIPR_A ≈ 1), and a two-stage relaxation in which global thermalization occurs on a much longer timescale. A heuristic mechanism is then proposed based on particle-number conservation, weakly coupled sectors, differing energy bandwidths, and asymmetric heating rates of the two chains.

Significance. If the result is robust, it extends the Mpemba/IME phenomenon from relaxation to thermal equilibrium to relaxation toward prethermal states in closed quantum systems, which is a conceptually valuable and potentially experimentally accessible generalization. The numerical strategy—using canonical thermal pure quantum states, NIPR diagnostics, and entanglement-entropy crossing—is well suited to the claim and the two-stage relaxation evidenced in Figs. 2 and 3 is compelling. However, the central demonstration depends on operational definitions (the averaging window for Epre_β and the threshold δ) and on a criterion in Eq. (2) that is weaker than the title's literal claim of heating to a higher prethermal energy. These points need to be tightened before the paper's main assertion is fully supported.

major comments (3)
  1. [Demonstration with a simple model, Fig. 3(a)] The prethermal energy Epre_β is defined as the average of the energy over t/T ∈ [40,50], but Fig. 3(a) and the accompanying text state that for β^{-1}=0.1 the energy is still drifting downward in this window, decreasing toward the infinite-temperature value. The sentence 'Estimating the prethermal energy over different time intervals does not significantly affect the behavior of the prethermal relaxation time' is not backed by a systematic test. Since the inequalities in Eq. (2) and the extracted t*_β values both depend on Epre_β, the demonstrated IME could in principle be an artifact of the chosen window. Please provide a sensitivity analysis over several averaging windows (e.g., [30,40], [50,60], [70,80]) and over several values of δ, showing that the ordering t*(β=0.1) < t*(β=0.5) and the second criterion in Eq. (2) are stable.
  2. [Eq. (2) and abstract/title] The second line of Eq. (2) requires Epre_βℓ − Eβℓ(0) > Epre_βh − Eβh(0), which is a statement about energy uptake, not about the ordering of the prethermal energies themselves. Because Eβℓ(0) < Eβh(0), this inequality can hold even when Epre_βℓ < Epre_βh, i.e., even when the colder state does not heat to a higher prethermal energy. The abstract and title claim that the colder state 'heats up ... to a higher prethermal energy,' so the criterion in Eq. (2) is weaker than the stated phenomenon. Please either add the condition Epre_βℓ > Epre_βh or revise the terminology to describe a larger energy-gap condition rather than a higher prethermal energy.
  3. [Mechanism and generalization] The proposed general mechanism (Sec. 'Mechanism and generalization') is presented as universal and controllable, but the timescale separation between fast intra-chain heating and slow inter-chain particle transfer is only observed for one parameter set (JA=VA=1, J⊥=0.04, Ω=1, A0=1) and is not derived from a controlled approximation. The paper would be substantially strengthened by a parameter sweep showing that the prethermal IME disappears as J⊥/JA grows or as the drive frequency changes in the directions predicted by the mechanism, thereby testing the claimed universality.
minor comments (6)
  1. [Fig. 2(b) inset] The inset shows relaxation times t*/T as a function of β^{-1} without markers or error bars; please clarify whether these are single estimates or averaged over the ten samples, and include the bootstrap standard errors shown elsewhere.
  2. [Fig. 3 caption] The caption of Fig. 3 contains a duplicated '(b)(b)' typographical error in the second panel label.
  3. [Eq. (4)] The ensemble average over random vectors is indicated by an overline in Eq. (4), but the notation is not defined in the equation itself; please define it in the text immediately preceding the equation.
  4. [High-temperature energy estimate] When stating that the high-temperature random state gives energy VA/2 * N(N−1)/(L(L−1)), the text should specify that this is the per-site interaction energy and that the kinetic energy averages to zero under the random-state assumption.
  5. [Fig. 4 caption] The caption refers to panels (a) and (b), but the figure panels are not labeled in the displayed schematic; please add the labels.
  6. [End Matter, Fig. 6] The temperature dependence of NIPR_B is described as nonmonotonic, but the physical interpretation would be clearer if the text explained why NIPR_B returns to ≈1 at low temperatures (vacuum state) and how this relates to the prethermal IME mechanism.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructional circularity: the prethermal IME is demonstrated by direct numerical energy trajectories; only minor non-load-bearing self-citations and an unquantified averaging-window choice appear.

full rationale

The central claim is a direct numerical observation: with fixed parameters JA=VA=1, J⊥=0.04, Ω=1, and A0=1, the energy trajectories for different initial temperatures cross and the extracted relaxation times satisfy both criteria in Eq. (2). No parameter is fitted to enforce the inequalities; Epre_β is an independently measured average of the same energy observable, and the conditions t*(β_l)<t*(β_h) and Epre_β_l−Eβ_l(0)>Epre_β_h−Eβ_h(0) are not identities that follow from the definition of Epre_β. The self-citations [43,44] are used only to justify suppression of heating in the weakly driven chain B and are standard published Floquet heating bounds, not load-bearing uniqueness claims. The only mild self-referential element is that Epre_β is defined by averaging the energy over t/T∈[40,50] in the same simulation used to extract t*; the paper asserts 'Estimating the prethermal energy over different time intervals does not significantly affect the behavior of the prethermal relaxation time' without showing a systematic scan, which is a robustness caveat rather than a constructional circularity. Similarly, Fig. 3(a) indicates that for β^-1=0.1 the energy is still drifting downward after the averaging window, but this affects confidence in the quantitative t* values, not whether the result reduces to its inputs by definition. No renaming, ansatz-smuggling, or self-citation chain forces the central result, so the appropriate score is low.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The central demonstration depends on the choice of J⊥, A0, Ω and on the operational definitions of the prethermal energy and relaxation time. It also relies on standard results in Floquet prethermalization and on the typicality approximation for thermal pure quantum states. No new physical entities are introduced.

free parameters (5)
  • Interchain hopping J⊥ = 0.04
    Chosen much smaller than JA=VA=1 to create the timescale separation between fast thermalization of chain A and slow interchain relaxation; the prethermal plateau and the observed crossing depend on this separation.
  • Drive amplitude A0 = 1
    Chosen to make chain A heat up rapidly under the periodic field; not scanned, so the sensitivity of the effect to this amplitude is unknown.
  • Drive frequency Ω = 1
    Chosen together with A0 to place chain A in the fast-heating regime while chain B remains essentially undriven; no frequency scan is reported.
  • Prethermal energy averaging window = t/T ∈ [40,50]
    The prethermal energy E_pre^β is defined as the time average of the energy over this window. The paper claims other intervals do not change the behavior qualitatively but does not provide a quantitative convergence analysis.
  • Relaxation threshold δ = 1/e
    Used to define the relaxation time t* via |E_pre - E(t*)| = δ |E_pre - E(0)|. The classification of the IME depends on this metric choice.
assumptions (4)
  • standard math Floquet prethermalization: high-frequency driving suppresses energy absorption for long times (rigorous bounds exist)
    Invoked via refs [43,44] to justify the existence of long-lived prethermal states in driven systems.
  • domain assumption Eigenstate thermalization hypothesis holds for the driven nonintegrable chain A, so it rapidly heats to an infinite-temperature-like state within each particle-number sector
    Used in the 'Mechanism and generalization' section to argue chain A reaches NIPR_A ≈ 1 while chain B remains cold.
  • standard math Canonical thermal pure quantum states provide accurate finite-temperature expectation values (typicality)
    Used to compute E_β(t) from random states with only 10 samples.
  • domain assumption Particle number conservation holds throughout the dynamics, dividing the Hilbert space into (N_A, N_B) sectors with suppressed inter-sector transitions when J⊥ is small
    Used to define the mechanism: weak coupling keeps N_A and N_B approximately constant on the prethermal timescale.

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Cite this review

Pith. "Pith review of Prethermal inverse Mpemba effect." pith.science (2026). https://pith.science/paper/TEGJGCAR

@misc{pith2026250704669,
  author       = {Pith},
  title        = {Pith review of: Prethermal inverse Mpemba effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TEGJGCAR}},
  note         = {Machine review of arXiv:2507.04669}
}
read the original abstract

The inverse Mpemba effect is a counterintuitive phenomenon in which a system, initially in thermal equilibrium and prepared at different temperatures below that of the final equilibrium state, relaxes to the final state more rapidly when starting from a lower initial temperature. We extend this concept to the relaxation toward a prethermal state in isolated quantum systems. By examining a simple model that exhibits prethermalization, we demonstrate that this effect indeed manifests under periodic driving. We further discuss the realization of this phenomenon in a variety of systems within a unified theoretical framework.

Figures

Figures reproduced from arXiv: 2507.04669 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic pictures of conventional IME (left panel) [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (a) Schematic picture of the model employed in this [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Time evolution of (a) the energy per site and (b) [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic picture of (a) the energy spectrum [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Temperature dependence of (a) the particle density [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Temperature dependence of the NIPR. Chains A [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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