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REVIEW 3 major objections 3 minor 52 references

Signature of nonequilibrium quantum phase transition in the long time average of Loschmidt echo

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The long-time average of the Loschmidt echo is a detector of nonequilibrium quantum phase transitions.

desk verdict Solid numerical diagnostics for NQPT via long-time averaged LE, but the fidelity-susceptibility connection is derived for the wrong object (χδ vs χλf) and should be reframed. read the letter →

arxiv 1908.11572 v3 pith:4QVHXVMK submitted 2019-08-30 quant-ph cond-mat.quant-gas

Bozhen Zhou , Chao Yang , Shu Chen This is my paper · ORCID
classification quant-phcond-mat.quant-gas
keywords Loschmidtecholong-timeaveragenonequilibriumquantumphasetransitionquenchfidelitysusceptibilityratefunctiondynamicalnonanalyticity
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

This paper establishes that the long-time average of the Loschmidt echo—the time-averaged return probability of a state after a sudden quantum quench—carries a sharp signature of quantum phase transitions in the post-quench Hamiltonian. Across three representative models (Aubry-André, transverse-field Ising, and Haldane), the echo average or its rate function develops a nonanalytic feature when the final quench parameter crosses a critical point, independent of which phase the system starts in. For quenches of small amplitude, the paper derives a proportionality between the second derivative of the echo average and the fidelity susceptibility, tying the dynamical signal to a well-established ground-state probe. The payoff is a time-independent, computable observable that can reveal nonequilibrium quantum criticality even when the usual dynamical quantum phase transitions—zeros of the echo at critical times—are absent or masked.

What carries the argument

The load-bearing object is the long-time average of the Loschmidt amplitude, $\overline{\mathcal{L}}(\lambda_f)=\sum_n |\langle \psi_n(\lambda_f)|\Psi(0)\rangle|^4$, which has the form of an inverse participation ratio of the initial state in the post-quench eigenbasis; its intensive logarithm is the rate function $\eta$. The bridge to known quantum-critical physics is the small-quench perturbation theory: expanding the eigenstates of $H(\lambda+\delta)$ to first order in $\delta$ gives $\overline{\mathcal{L}}_\delta = 1 - 2\delta^2 \sum_{m\neq 0} |H_{m0}|^2/(E_0-E_m)^2$, while the ground-state fidelity susceptibility is $\chi_F = \sum_{m\neq 0}|H_{m0}|^2/(E_0-E_m)^2$, yielding $\chi_\delta=4\chi_F$. This identity is what converts a time-averaged dynamical quantity into a ground-state criticality detector.

What would settle it

Take the transverse-field Ising model with initial field $h_i=0$, evaluate the analytic rate function $\eta(h_f)$ on progressively larger chains, and compute the second derivative $\chi_{h_f}$ near $h_f/J=1$; if the peak height does not grow without bound as $L\to\infty$, the claimed critical divergence is absent. A complementary check is to compute $\chi_\delta$ and $4\chi_F$ exactly for small but finite $\delta$ at the critical point and see whether the proportionality $4$ survives when the perturbative expansion fails.

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Extended reading notes

Core claim

The central claim is that the long-time average $\overline{\mathcal{L}}(\lambda_f)$ of the Loschmidt echo, or its rate function $\eta(\lambda_f)=-(1/L)\log \overline{\mathcal{L}}(\lambda_f)$, is a nonequilibrium order parameter in quench-parameter space: it is nonanalytic exactly when $\lambda_f$ crosses a quantum phase-transition point. The nonanalyticity appears as a sharp change in $\overline{\mathcal{L}}$ or $\eta$ in the Aubry-André and Ising models, and, in the Haldane model where $\eta$ looks smooth, as a diverging peak in the second derivative $\chi_{\lambda_f}=-\partial^2\eta/\partial\lambda_f^2$. In the small-quench limit $\lambda_f=\lambda_i+\delta$, the paper derives $\chi_\delta=-\partial^2\overline{\mathcal{L}}_\delta/\partial\delta^2 = 4\chi_F$, with $\chi_F$ the fidelity susceptibility, so the divergence at criticality follows from the known divergence of $\chi_F$. The signature is shown to be independent of the initial phase, and finite-size data show the peak height growing with system size, consistent with a true thermodynamic-limit divergence.

Load-bearing premise

The small-quench identity $\chi_\delta=4\chi_F$ rests on first-order perturbation theory in $\delta$, which requires nonzero energy gaps between the ground state and all excited states; at the critical point the gap closes, so the expansion is not controlled for the modes that would produce the divergence.

Editorial extensions

If this is right

  • The long-time averaged Loschmidt echo gives a time-independent probe of nonequilibrium quantum phase transitions, complementing dynamical quantum phase transitions that require zeros of the echo at special times.
  • The second derivative $\chi_{\lambda_f}$ can expose topological phase transitions in models such as the Haldane model, where the rate function itself appears analytic.
  • Because the signature is independent of the initial phase, it works for quenches starting on either side of the critical point.
  • The proportionality $\chi_\delta=4\chi_F$ means the divergence of the fidelity susceptibility at criticality is inherited by the quench-averaged return probability.

Reading between the lines

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

  • A natural extension is to use the measured long-time averaged return probability for small quenches as an experimental estimator of fidelity susceptibility, which is normally inferred from ground-state wavefunction overlaps.
  • The same logic may apply to other time-averaged overlaps, such as survival probabilities of excited states or spin autocorrelators, giving a family of nonequilibrium criticality detectors.
  • For finite quench amplitude $\delta$, corrections to $\chi_\delta=4\chi_F$ should appear; tracking how the peak position and scaling exponent change with $\delta$ would test how robust the fidelity-susceptibility link is away from the perturbative regime.
  • Because the long-time average removes time dependence, the signature may be measurable in platforms with limited coherence times, as long as the time average converges before decoherence sets in; this is an experimentally testable prediction.
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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 / 3 minor

Summary. This paper studies the infinite-time average of the Loschmidt echo after a sudden quench, L = Σ_n |⟨ψ_n(λ_f)|Ψ(0)⟩|^4, and its rate function η(λ_f) = -(1/L) log L. The authors present numerical evidence that L or η changes sharply and that χ_{λ_f} = -∂²η/∂λ_f² develops peaks when λ_f crosses a quantum phase boundary in the Aubry-André model, the transverse-field Ising chain, and the Haldane model. They also derive a small-quench identity, χ_δ = 4χ_F, connecting the second derivative of L to the fidelity susceptibility, and use this relation to argue that the rate-function second derivative diverges at criticality.

Significance. If the central claim holds, the long-time-averaged Loschmidt echo would provide a simple, model-independent diagnostic for nonequilibrium quantum phase transitions, and the connection to fidelity susceptibility would give an analytic explanation of the divergence. The paper tests the idea on three different models with known transition points, uses no fitted parameters, and provides an exact integral expression for the Ising rate function in Eq. (15), which are clear strengths. However, the fidelity-susceptibility identity is derived for L, not for the rate function η, so the analytic explanation for the divergences in χ_{λ_f} is currently unsupported.

major comments (3)
  1. [Sec. II.E, Eq. (31) vs Eq. (23)] The identity χ_δ = -∂²L_δ/∂δ² = 4χ_F concerns the long-time-averaged echo L, whereas the quantity plotted in Figs. 4-6 and highlighted in the abstract is χ_{λ_f} = -∂²η/∂λ_f² for the rate function η. Because η = -(1/S) log L, the small-δ expansion L_δ ≈ 1 - 2Aδ² with A = Σ_{m≠0} |H_{m0}|²/(E_0-E_m)² gives η ≈ 2Aδ²/S and hence χ_{λ_f} ≈ -4χ_F/S. This differs from Eq. (31) by an overall sign and a 1/S prefactor, so Eq. (31) does not by itself explain the positive peaks in Figs. 4-6. The authors need to state explicitly how χ_{λ_f} is related to χ_F, or remove the claim that the fidelity-susceptibility connection explains the rate-function divergence.
  2. [Sec. II.E, Eqs. (25)-(27)] The perturbation expansion in δ assumes non-degenerate eigenstates with finite energy denominators. At a quantum critical point the gap closes, so the first-order expansion is not controlled for the modes that dominate the fidelity susceptibility. Using Eq. (31) to argue for a divergence of χ at criticality therefore interchanges the limits δ→0 and the approach to the critical point. The authors should provide a non-perturbative argument or explicitly restrict the identity to quenches away from criticality.
  3. [Sec. III, Fig. 6] The claim that χ_{λ_f} diverges in the thermodynamic limit rests on Fig. 6, which shows peak heights increasing with lattice size. The paper does not provide a scaling collapse, an extrapolation to L→∞, or a comparison with the 1/S prefactor that arises from the corrected rate-function relation. The finite-size trends are suggestive but do not by themselves establish a true divergence, especially because the small-quench analysis indicates that χ_{λ_f} and χ_δ scale differently with system size.
minor comments (3)
  1. [Eq. (27) and Eq. (30)] Equations (27) and (30) omit the summation over m≠0 that appears in Eq. (26); as written, they read as if only a single matrix element contributes. The sums should be restored.
  2. [Eq. (5) and throughout] The symbol L is used both for the lattice size and for the long-time-averaged Loschmidt echo, which makes expressions such as η = -(1/L) log L confusing. A different symbol, such as S or N, for the system size would improve readability.
  3. [Eq. (23), Figs. 4-6] The sign convention for χ_{λ_f} should be checked against the plotted quantity. The text defines χ_{λ_f} = -∂²η/∂λ_f², but the figures appear to show positive peaks; if the plotted quantity is actually ∂²η/∂λ_f² or |χ_{λ_f}|, this should be stated in the captions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central derivations are self-contained and use external benchmarks.

full rationale

The paper's central claim is that the long-time average of the Loschmidt echo and its rate function become nonanalytic when the post-quench parameter crosses a phase transition. This is supported by direct numerical evaluation in the AA and Haldane models and by an analytic expression for the Ising model (Eq. 15), with the transition locations taken from independent band-topology criteria (winding number, Chern number). No parameter is fitted to the claimed signature and then renamed as a prediction. The small-quench relation in Sec. II.E is a genuine derivation: starting from Lδ = Σ_n |⟨ψ_n(λ+δ)|ψ_0(λ)⟩|^4, first-order perturbation theory yields Lδ = 1 − 2δ² Σ_{m≠0} |H_m0|²/(E0−Em)², and the same expansion gives the fidelity susceptibility χF = Σ |H_m0|²/(E0−Em)², hence χδ = 4χF (Eq. 31). This is an identity derived from the same perturbation expansion, not an input assumed to prove the conclusion. The cited results on fidelity-susceptibility divergence (Refs. 8-10, including some by the authors) are external benchmarks and not the load-bearing justification for the numerical peaks. The only caveat is that Eq. (31) concerns χδ = −∂²Lδ/∂δ², whereas the figures plot χλf = −∂²η/∂λf² for the rate function; that is a possible technical gap in how the connection is applied, but it is not a circularity because neither quantity is defined in terms of the other or fitted to the target data.

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

The central claim rests on standard quantum mechanics and the diagonal-ensemble identity for the long-time average of the Loschmidt echo (Eq. 4). No free parameters are fitted; all model parameters are inputs. The most fragile premise is the use of first-order perturbation theory near the phase transition, where the gap closes.

assumptions (4)
  • domain assumption The long-time average of the Loschmidt echo equals Σ_n |⟨ψ_n(λf)|Ψ(0)⟩|^4 (Eq. 4), valid when the post-quench Hamiltonian has a non-degenerate spectrum.
    This identity is the basis of the method. If degenerate eigenstates exist, the time average includes cross terms; the paper assumes generic parameters away from exact degeneracies.
  • standard math In the thermodynamic limit, discrete momentum sums can be replaced by integrals over the Brillouin zone (Eqs. 14, 22).
    Standard Riemann-sum replacement for translation-invariant models.
  • domain assumption First-order perturbation theory in δ describes the eigenstates of H(λ+δ) (Eq. 25).
    Used to derive χδ = 4χF. The expansion is not controlled when the gap closes at the phase transition, a limitation not addressed in the paper.
  • domain assumption The initial state is the ground state of H(λi).
    Central to the quench setup; the results are shown for this specific choice of initial state.

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Pith. "Pith review of Signature of nonequilibrium quantum phase transition in the long time average of Loschmidt echo." pith.science (2026). https://pith.science/paper/4QVHXVMK

@misc{pith2026190811572,
  author       = {Pith},
  title        = {Pith review of: Signature of nonequilibrium quantum phase transition in the long time average of Loschmidt echo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QVHXVMK}},
  note         = {Machine review of arXiv:1908.11572}
}
read the original abstract

We unveil the role of the long time average of Loschmidt echo in the characterization of nonequilibrium quantum phase transitions by studying sudden quench processes across quantum phase transitions in various quantum systems. While the dynamical quantum phase transitions are characterized by the emergence of a series of zero points at critical times during time evolution, we demonstrate that nonequilibrium quantum phase transitions can be identified by nonanalyticities in the long time average of Loschmidt echo. The nonanalytic behaviours are illustrated by a sharp change in the long time average of Loschmidt echo or the corresponding rate function or the emergence of divergence in the second derivative of rate function when the driving quench parameter crosses the phase transition points. The connection between the second derivative of rate function and fidelity susceptibility is also discussed.

Figures

Figures reproduced from arXiv: 1908.11572 by the authors.

Figure 1
Figure 1. FIG. 1. (Color online) The behavior of the long time average [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. (Color online) The behavior of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (Color online) (a) Illustration of the honeycomb lat [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. (Color online) (a) The behavior of [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 4
Figure 4. Figure 4: FIG. 4. (Color online) (a) The behavior of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. (Color online) [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (Color online) (a) [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

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