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REVIEW 3 major objections 5 minor 54 references

In a holographic superfluid, states with stronger initial symmetry breaking relax faster toward the symmetry-restored equilibrium — the quantum Mpemba effect.

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2026-08-01 09:05 UTC pith:7VN3U3O3

load-bearing objection The direct QME observation at ρ_f=3.0 looks real, but the 'wide parameter range' claim rests on a single measurement time t_f=10 that is likely too short near criticality. the 3 major comments →

arxiv 2607.20899 v1 pith:7VN3U3O3 submitted 2026-07-23 hep-th cond-mat.stat-mech

Quantum Mpemba effect in holography

Xian-Hui Ge , Shuta Ishigaki , Yu-Qi Lei , Yu Tian This is my paper
classification hep-th cond-mat.stat-mech
keywords quantum Mpemba effectholographic superfluidquasinormal modessymmetry restorationAdS/CFTnonequilibrium dynamicsblack holeshifted free energy
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.

This paper claims that the quantum Mpemba effect—where a state starting farther from equilibrium relaxes faster than a closer one—occurs in a holographic superfluid. In a quench from a symmetry-broken phase to a symmetry-restored equilibrium, more strongly broken initial states settle into equilibrium sooner than weakly broken ones. The authors use the shifted free energy, obtained from energy flux through the black hole horizon, as a monotonic distance measure that cleanly shows the effect. They explain the anomaly through quasinormal-mode competition: the slowest decaying mode is suppressed and the second mode amplified, steering relaxation through a faster channel. This provides a holographic example of the quantum Mpemba effect in a strongly coupled, non-Lindblad system.

Core claim

The paper's central discovery: after a quench in a holographic superfluid from ρ_i > ρ_c to ρ_f = 3.0 (< ρ_c), relaxation is non-monotonic in the initial condensate—initial condensates near |⟨O2(0)⟩|≈3.4–10 relax faster than smaller ones, even though they start farther from the symmetric equilibrium. This is the quantum Mpemba effect. The authors verify it through the late-time condensate at t_f=10 and the shifted free energy Δε(t), which is monotonic in time. The mechanism is uncovered by fitting the nonlinear bulk scalar field to a quasinormal-mode expansion: the slowest mode's weight decreases while the second mode's weight grows in the QME parameter window. This mode competition mirrors

What carries the argument

The key machinery is the quasinormal-mode decomposition of the bulk scalar field, Φ(t,z) ≃ z Σ_{n=1}^{4} b_n(t) δψ_n(z), where δψ_n are the radial QNM profiles of the final state and b_n(t) are time-dependent amplitudes capturing nonlinear interactions. The QNM weights w_n, defined by time-integrated |b_n(t)|², quantify how much each channel contributes to relaxation. The shifted free energy Δε(t) = ∫_{t}^{∞} √-g T^z_t|_{z_h} dt'—the energy flux into the black hole horizon—is used as a monotonic distance measure. These two tools together turn a nonlinear far-from-equilibrium process into a competition among discrete decay modes.

Load-bearing premise

The mechanistic explanation rests on fitting the nonlinear bulk scalar field with a truncated sum of four quasinormal modes—an expansion the paper itself notes is not generally justified because quasinormal modes do not form a complete basis.

What would settle it

Extend the numerical integration well beyond t=10 and check whether the ordering of relaxation (the non-monotonic late-time condensate) persists or whether the curves recross as the slowest mode eventually dominates. A recrossing would show the effect is a transient overshoot rather than a genuine quantum Mpemba effect.

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

If this is right

  • In strongly coupled superfluids described by holography, a state with a larger initial order parameter can equilibrate faster than one with a smaller order parameter, even when both are quenched to the same final parameters.
  • The shifted free energy computed from horizon flux is a valid monotonic distance measure for quench dynamics, offering a way to define the approach to equilibrium in holography.
  • The QME in this model shares its underlying mechanism with open quantum systems: suppression of the slowest-decaying mode. Here, however, it arises from the nonlinear bulk dynamics, not from a Lindblad generator.
  • The relaxation timescale τ₁ ≈ 1.34/T is of the same order as the Planckian dissipation time, suggesting the model's fast thermalization is consistent with known bounds.

Where Pith is reading between the lines

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

  • If the effect is generic across symmetry-breaking patterns and quench protocols, the quantum Mpemba effect may be a universal feature of holographic phase transitions, not an accident of the U(1) model.
  • A careful test would be to repeat the quench with a fixed chemical potential rather than fixed density; if the ordering of relaxation times is inverted, the effect is protocol-dependent and not a property of the model alone.
  • The QNM-fit mechanism could be checked against a complete-basis or exact nonlinear spectral decomposition, which would confirm whether the suppression/amplification pattern is genuine or an artifact of the incomplete QNM basis.
  • For platforms like ultracold atomic gases across a superfluid transition, a sudden quench of the interaction strength might show the same non-monotonic relaxation; if seen, it would provide experimental weight to the holographic prediction.

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 / 5 minor

Summary. The paper studies relaxation dynamics in a holographic superfluid in the probe limit. Initial states are stationary U(1)-broken solutions at charge densities ρ_i > ρ_c, and at t=0 the charge density is quenched to ρ_f < ρ_c so that the system relaxes toward the symmetry-restored normal phase. The authors report a quantum Mpemba effect: for example, at ρ_f=3.0 the state with ρ_i=12.0 decays faster than states with smaller initial condensates, as seen in the time evolution of the condensate (Fig. 2A). They also construct a 'shifted free energy' from the energy flux into the horizon, Eq. (8), and show that it decreases monotonically (Fig. 2B). A QNM-based fit, Eq. (10), is used to extract time-dependent mode weights w_n, Eq. (11), and the authors argue that the QME arises from suppression of the slowest QNM and amplification of the second mode. Fig. 3 gives a phase diagram of the late-time condensate at t_f=10.0 and claims a wide QME parameter range, with the w1=w2 crossing falling inside the QME region.

Significance. If the central claim is correct, this would be one of the first demonstrations of the quantum Mpemba effect in a strongly coupled holographic system, and the proposed horizon-flux distance measure is a physically motivated addition to the QME toolkit. The QNM mechanism is consistent with the well-known mode-suppression picture in open quantum systems and provides a concrete interpretation in terms of nonlinear holographic dynamics. However, the direct observation and the phase diagram rely on a fixed finite measurement time, and the mechanistic conclusions rest on an admittedly incomplete QNM projection. With the requested convergence checks, the paper would be a solid contribution; as it stands, the 'wide parameter range' and the mode-competition mechanism are not yet fully established.

major comments (3)
  1. [Fig. 3 and the 'wide parameter range' claim] The QME region in Fig. 3 is defined using t_f=10.0, and the authors explicitly justify a late-time reading only for ρ_f=3.0 ('From t≈2, the curves exponentially decay with this decay rate'). For ρ_f approaching ρ_c=4.06, the slowest QNM lifetime τ_1=1/|Im ω_1| diverges, so at t_f=10 the condensate still contains substantial higher-QNM contamination. The boundary ∂|⟨O2(t_f)⟩|/∂|⟨O2(0)⟩|=0 and the associated 'wide parameter range' may therefore be finite-time artifacts. The authors should show t_f=15,20 (or extract the asymptotic single-mode amplitude) for representative ρ_f values, and either strengthen or appropriately restrict the parameter-range claim.
  2. [Eq. (9) and Fig. 2(B)] Eq. (9) defines Δε(t) = lim_{tf→∞} ∫_t^{tf} dt' √−g T^z_t|_{z_h}. The text then says 'Here we set t_f=10.0 to perform the time integration in Eq. (9) numerically.' With this replacement, Δε(10)=0 by construction, not because equilibrium has been reached. The omitted tail ∫_10^∞ dt' (−√−g T^z_t) is positive and depends on the initial state through the amplitude of the slowest QNM. The comparisons of Δε among different ρ_i in Fig. 2(B) are therefore biased by a ρ_i-dependent term. The authors should quantify this tail, e.g. by comparing with the extrapolated e^{−2 Im ω_1 t} decay, or use a controlled upper cutoff in the definition.
  3. [Eq. (10) and the QNM mechanism] All mechanistic conclusions rest on the fit Φ(t,z) ≃ z Σ_{n=1}^{4} b_n(t) δψ_n(z), but the paper itself states that QNMs do not form a complete basis even at N_max=∞. No quantitative fit residual is reported; the statement that the model 'fits the nonlinear result ... almost perfectly' is based on visual inspection. Since the coefficients b_n(t) are extracted from the same data used to identify the QME, the w1=w2 crossing in Fig. 3 is not an independent check of the mechanism. The authors should provide L2 residuals as a function of t and repeat the extraction with N_max=5,6 to demonstrate that the mode weights and the w1=w2 line are stable.
minor comments (5)
  1. [Eq. (9)] The symbol t_f is overloaded: it is the upper integration limit in Eq. (9) and also a numerical cutoff. Use a separate symbol such as T for the cutoff and clearly distinguish the limit from the chosen numerical value.
  2. [Appendix, Eq. (18)] The notation A_v appears without definition; from context it should be A_t. Please correct.
  3. [Numerical methods] The appendix states N_z=50 and Δt=0.01, but no convergence or error estimate is given for the order parameter, Δε, or the QNM weights. At least a statement of the observed discretization error would help the reader judge the significance of the small effects in Fig. 3.
  4. [Fig. 5(A)] The fitted curves and data points are described as 'mostly overlapping'; a separate residual plot as a function of z would be far more informative and would support the 'almost perfect' claim.
  5. [Introduction] The phrase 'fast scrambling system' in Ref. [6] is cited as a prior observation of symmetry restoration; please clarify its relation to the present quench protocol in the bibliography or text.

Circularity Check

0 steps flagged

No significant circularity: the QME observation comes from direct holographic time evolution, and the QNM fit is explicitly acknowledged as an approximate post hoc interpretation rather than a load-bearing prediction.

full rationale

The central QME claim is obtained from direct numerical evolution of the bulk equations, not from the fitted QNM expansion. The paper evolves initial stationary broken-phase solutions after a quench to a fixed final charge density and directly reads off the condensate and the energy flux: the non-monotonic late-time condensate in Fig. 3 and the faster decay of the shifted free energy for larger initial condensates are outputs of the simulation. The shifted free energy in Eq. (9) is defined as an integral of the dissipation rate and is monotonic because of Eq. (8), but that does not predetermine the ordering of different initial states; the QME is a nontrivial property of the computed relaxation rates. The QNM decomposition is introduced explicitly as a fit ('we fit the nonlinear bulk profile of the scalar field using the following model') and the paper concedes its limited status: 'the expansion (10) is not generally justified even for Nmax=∞ since QNMs do not form a complete basis.' Therefore the mode-competition discussion is an interpretive overlay, not a fitted parameter renamed as a prediction, and the QME evidence does not depend on it. The finite cutoff t_f=10, used both in Fig. 3 and in the numerical evaluation of Eq. (9), is a numerical accuracy concern flagged in the text ('Here we set t_f=10.0 to perform the time integration in Eq. (9) numerically'), but it does not make the endpoint value equivalent to equilibrium by construction. The self-citations ([6], [17], [47]) are methodological or contextual: the free-energy flux is computed directly from Eq. (8), and the QNM fitting scheme is described in the Appendix, so no load-bearing claim reduces to an unverified prior-result chain. No circular step is present.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central QME detection uses only standard holographic superfluid ingredients. The free parameters are the quench protocol choices and the QNM truncation. No new entities are introduced. The main model-dependent assumptions are the probe limit and the incomplete QNM basis used for the mechanism.

free parameters (4)
  • post-quench charge density ρ_f = 3.0
    Chosen for the main demonstration; Fig. 3 also scans ρ_f from ~1 to 4.
  • QNM truncation N_max = 4
    Number of modes in the fitting ansatz Eq. (10); higher modes assumed negligible.
  • measurement time t_f = 10.0
    Window for late-time condensate and QNM weights; slowest mode decayed by e^{-1.77}.
  • bulk mass m^2 and charge q = -2, 1
    Hand-chosen standard holographic superfluid parameters; not fitted to data.
axioms (4)
  • domain assumption Gauge/gravity duality maps the boundary superfluid to bulk Maxwell-scalar theory on AdS4-Schwarzschild.
    Everything in the paper uses the holographic dictionary; introduced in 'Holographic setup'.
  • domain assumption Probe limit: matter fields do not backreact on the metric.
    The background is fixed Schwarzschild-AdS4; backreaction is not checked.
  • ad hoc to paper QNMs of the final normal phase can be used as a basis to expand the nonlinear bulk profile with time-dependent coefficients.
    Eq. (10); the paper itself notes QNMs are not a complete basis, so this is an unproven modeling assumption for the mechanism claim.
  • domain assumption Horizon energy flux equals boundary free energy dissipation rate (Eq. 8), following Refs. [16,17].
    Used to define the shifted free energy distance measure.

pith-pipeline@v1.3.0-alltime-deepseek · 11011 in / 19749 out tokens · 176451 ms · 2026-08-01T09:05:35.438658+00:00 · methodology

0 comments
read the original abstract

We investigate the quantum Mpemba effect in a holographic superfluid, in which states with stronger initial symmetry breaking relax faster toward the symmetry-restored equilibrium. We demonstrate its emergence by identifying the shifted free energy computed from the energy flux into the black hole horizon as monotonic distance measure. By decomposing the nonlinear bulk dynamics based on quasinormal modes, we reveal that the anomalous relaxation is governed by a dynamical competition in which the slowest-decaying mode is suppressed while the second mode is amplified. These findings provide a holographic perspective on the quantum Mpemba effect in nonequilibrium relaxation involving strongly coupled degrees of freedom.

Figures

Figures reproduced from arXiv: 2607.20899 by Shuta Ishigaki, Xian-Hui Ge, Yu-Qi Lei, Yu Tian.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of our setup. The system is initially pre [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (A) Time evolution of the condensate for various [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Late-time condensate [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Bulk profiles of the first four QNMs for [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (A) Fitted curves of the bulk scalar field for [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Condensate as a function of the charge density for [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

discussion (0)

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Reference graph

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