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Quantum criticality and universality in stationary state of long-range Kitaev model

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Pith's one-line read The paper claims that after a critical-to-critical quench in the long-range Kitaev chain, the stationary state's mutual information and logarithmic negativity scale with the same effective central charge as the ground state, identifying…

desk verdict Plausible qualitative extension of hidden quantum criticality to the long-range Kitaev chain, but the central charge match is not yet documented. read the letter →

arxiv 2412.01076 v2 pith:VQHSPKNL submitted 2024-12-02 quant-ph cond-mat.stat-mechcond-mat.str-el

Akash Mitra , Sanku Paul , Shashi C. L. Srivastava This is my paper · ORCID
classification quant-phcond-mat.stat-mechcond-mat.str-el
keywords quantumcriticalityquenchdynamicslong-rangeKitaevmodelstationarystatecentralchargemutualinformationlogarithmicnegativityuniversalityclass
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 investigates whether quantum criticality and its universality class survive in the long-time stationary state of an integrable long-range Kitaev chain after a sudden quench. The model is a one-dimensional fermionic chain with pairing interactions decaying with distance as $1/l^\alpha$, and the authors study quenches of the chemical potential and pairing strength. Their central claim is that when both the pre- and post-quench Hamiltonians are at the critical point, the stationary state shows the same effective central charge as the critical ground state: $c=1$ for the all-to-all pairing limit $\alpha=0$ and $c=1/2$ for $\alpha=2$. This matters because it would mean the universality class of a quantum critical point can be read off from a nonequilibrium steady state, even in systems where fermionic correlators decay algebraically away from criticality.

What carries the argument

The exactly solvable free-fermion structure of the long-range Kitaev chain is the central object: after a Fourier transform and Bogoliubov rotation, the Hamiltonian becomes a sum of independent modes with dispersion $\lambda_\alpha(k)=\sqrt{(\mu+\cos k)^2+(\Delta g_\alpha(k))^2}$, where $g_\alpha(k)$ is expressed through polylogarithms. Time-dependent correlations after a quench are computed from the difference of pre- and post-quench Bogoliubov angles, and the long-time stationary state is obtained by dropping the oscillatory terms. Entanglement is extracted from the correlation matrix; mutual information cancels the volume-law part of the entropy and isolates the logarithmic scaling $I \sim (c^I_{\rm eff}/3)\ln L$, while logarithmic negativity is estimated through the upper bound of Eq. (18) for fermionic Gaussian states, with scaling $\xi \sim (c^N_{\rm eff}/4)\ln L$. The soft-mode occupation probability $n_k$ at $k=\pi$, given in Eq. (23), is the mechanism that produces the characteristic peak for critical-to-critical quenches and dip for noncritical-to-critical quenches.

What would settle it

Take the stationary state after a critical-to-critical quench for a moderate system size, construct the fermionic reduced density matrix, perform the partial transpose numerically, and compute the exact logarithmic negativity; if the exact $c^N_{\rm eff}$ deviates from the upper-bound estimate or from the ground-state central charge, the quantitative match claimed here would not survive. A second check is to vary the pairing quench amplitude $\Delta_f$ at fixed $\Delta_i=-1$ and verify that the fitted $c^I_{\rm eff}$ remains constant; any drift with $\Delta_f$ or with subsystem size $L$ would show that the logarithmic scaling is not the CFT form.

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Extended reading notes

Core claim

The central discovery claimed is that, in the long-range Kitaev chain, a critical-to-critical quench produces a stationary state whose entanglement scaling identifies the same universality class as the ground state. Extracting effective central charges from the logarithmic growth of mutual information, $I \sim (c^I_{\rm eff}/3)\ln L$, and from the upper bound on logarithmic negativity, $\xi \sim (c^N_{\rm eff}/4)\ln L$, the authors find $c^I_{\rm eff}=c^N_{\rm eff}=1$ at $\alpha=0$ and $1/2$ at $\alpha=2$, matching the ground-state values. The same measures show a nonanalytic peak at the post-quench critical point $\mu=1$ for all studied $\alpha$, and this is explained through the occupation probability of the soft mode at $k=\pi$, which controls the entanglement of the union of the two subsystems. For noncritical-to-critical and critical-to-noncritical quenches, the effective central charge takes the noncritical ground-state value, so the authors do not attribute those nonanalyticities to criticality. For $\alpha=1$ the stationary state, like the ground state, has no universal central charge.

Load-bearing premise

The argument assumes that the stationary state's mutual information and the upper bound on logarithmic negativity follow the same logarithmic scaling laws with the same prefactors as the critical ground state, so that the fitted slope can be identified with the ground-state central charge; for log-negativity, only a bound is computed, so the match also depends on that bound being tight.

Editorial extensions

If this is right

  • For $\alpha=0$ and $\alpha=2$, the effective central charge extracted from the stationary state after a critical-to-critical quench equals the ground-state central charge, independent of the quench amplitude in the pairing strength $\Delta$.
  • For noncritical-to-critical and critical-to-noncritical quenches, the effective central charge takes the noncritical ground-state value ($1/2$ for $\alpha=0$, $0$ for $\alpha=2$), so the nonanalytic behavior of entanglement measures at $\mu=1$ is not by itself a signature of universality.
  • Logarithmic negativity confirms that the long-range correlations detected in the stationary state are genuinely quantum, since $c^N_{\rm eff}$ matches $c^I_{\rm eff}$ for both $\alpha=0$ and $\alpha=2$.
  • Tripartite mutual information also peaks at the critical point for a critical-to-critical quench, and its sign distinguishes delocalized information ($\mu<1$) from information redundancy ($\mu>1$) for $\alpha=1$.
  • For $\alpha=1$, the stationary state, like the ground state, cannot be assigned a universal central charge, but long-range correlations still develop for the critical-to-critical quench protocol.

Reading between the lines

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

  • A testable extension would be to measure mutual information after a critical-to-critical quench in a trapped-ion simulator of the long-range Ising chain, whose Jordan-Wigner form is this model; observing the predicted logarithmic slope would provide an experimental route to the central charge from a nonequilibrium steady state.
  • Because the paper uses only an upper bound for logarithmic negativity, an exact partial-transpose calculation for small system sizes would show whether $c^N_{\rm eff}$ genuinely equals $c^I_{\rm eff}$ or whether the match is an artifact of the bound.
  • The paper's distinction between nonanalyticity and universality suggests a general methodological lesson: in long-range interacting systems, where correlation decay is algebraic even off criticality, scaling slopes rather than peaks or dips should be used to identify criticality in quench experiments.
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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 / 5 minor

Summary. The manuscript studies quench dynamics in the one-dimensional long-range Kitaev chain, focusing on the stationary state after sudden quenches of the chemical potential and pairing strength. Using free-fermion correlation-matrix techniques, the authors compute mutual information, tripartite mutual information, and an upper bound on logarithmic negativity in the long-time limit. They report that a critical-to-critical quench produces stationary-state signatures of criticality for all pairing exponents α considered, and they extract effective central charges cI_eff and cN_eff from the logarithmic scaling of mutual information and log-negativity. The central quantitative claim is that, for α=0 and α=2, these effective central charges agree with the ground-state central charges compiled in Table I, so that the universality class of the ground state can be inferred from the stationary state. The paper also explains the nonanalytic features at μ_f=1 through the soft-mode occupation probability n_k in Eq. (23).

Significance. If the claimed agreement is correct, the result is significant: it would show that for an integrable long-range fermionic model, the stationary state after a critical-to-critical quench retains enough information about the underlying conformal critical point to reproduce the ground-state central charge, even though fermionic correlators decay algebraically at noncritical points for α≤1. The manuscript has real strengths: the model is exactly solvable, the stationary-state correlation functions in Eq. (17) are analytic and clearly presented, the soft-mode occupation mechanism in Eqs. (23)-(25) gives a concrete and testable explanation of the peak/dip structure, and the comparison against the XY-model precedent in Ref. [40] is a natural methodological link. However, the central quantitative claim is currently supported only by fitted-coefficient phase maps and a table of final values; no scaling curves, system sizes, fit details, or uncertainties are shown, and the log-negativity analysis is based on an upper bound whose asymptotic prefactor is not established.

major comments (3)
  1. [§IV B, Fig. 3] The central claim that cI_eff equals the ground-state central charge for α=0 and α=2 is not supported by the displayed evidence. Figures 3 and 4 present only color maps of the fitted coefficient over the (μ,Δ) plane, with no scaling curves of I_A1:A2 or ξ^u versus ln L, no statement of total system sizes N, no subsystem length range, no fit function, and no error bars. Without such information, the reader cannot check whether the quoted values are robust or whether the agreement with Table I is genuine.
  2. [§III, Eqs. (18)-(20), Appendix A] The log-negativity claim is extracted from the upper bound ξ^u defined in Eq. (18), not from the true log-negativity of Eq. (11). The comparison with the ground-state central charge assumes that ξ^u scales with the same asymptotic coefficient c/4 as the true negativity in Eq. (13). No argument or numerical check is given that the bound is tight for the stationary state of the long-range Kitaev model; if the upper bound has a different coefficient, the reported cN_eff values would not measure the central charge. Appendix A also does not provide the promised finite-size analysis but merely restates the final values.
  3. [§III, Eq. (10)] The effective central charge cI_eff is defined through the assumed CFT scaling form I_A1:A2 ∼ (cI_eff/3) ln L for the stationary state, and no derivation or numerical verification of this form is provided for the post-quench generalized Gibbs ensemble of this model. Since the prefactor is fixed by assumption, the subsequent agreement with the ground-state central charges in Table I could be a consequence of the assumed scaling form rather than an independent test of universality. The authors should either derive this prefactor from the Fisher-Hartwig structure of the stationary-state mode occupation or demonstrate it with explicit finite-size scaling data.
minor comments (5)
  1. [Fig. 1 caption] The quantity plotted in the middle row is labeled ξ_A1:A2, but the method computes only the upper bound ξ^u via Eqs. (18)-(20); the caption should distinguish the bound from the true log-negativity.
  2. [Eq. (18)] The operators O_+ and O_- are used in Eq. (18) but are not explicitly defined in the text; a brief definition or a pointer to the decomposition used in Refs. [84-89] would improve clarity.
  3. [Appendix A] The main text says the details of the log-negativity finite-size scaling are in Appendix A, but the appendix reports only the extracted cN_eff values; the promised scaling analysis should either be included or the reference to it should be removed.
  4. [§IV B] In the sentence discussing α=1, the text contains the garbled expression 'ΔΔ 0'; this should be corrected to a clear statement about the sign or magnitude of Δ_f Δ_i.
  5. [Eq. (16)] The notation λ^f_α(k_n) in Eq. (16) and the surrounding text is not fully consistent with the earlier notation λ_α(k_n); unify the superscripts and subscripts used for the post-quench dispersion.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the effective central charges are fit outputs compared against external ground-state benchmarks, and the one self-citation (Ref. [40]) motivates an explicitly stated scaling assumption rather than supplying the result.

full rationale

The derivation chain is self-contained against external benchmarks. The Hamiltonian is exactly diagonalized; stationary-state correlators are computed from the quench protocol via Eqs. (16) and (17); mutual information and the log-negativity upper bound are evaluated from the correlation matrix. The effective central charges cI_eff and cN_eff are defined in Eqs. (10) and (13) as the coefficients of ln L in those measures, not as the ground-state central charges. The agreement with Table I is therefore a fitted output compared with literature values, not an input. The only author-overlapping citation is Ref. [40] (co-author Sanku Paul), used to motivate the log-negativity scaling form in Eq. (13) and the upper-bound method; the paper explicitly says 'Based on this, we assume,' so the scaling form is an announced assumption rather than a smuggled ansatz. The cI_eff comparison is independently supported by Refs. [39, 73-76]. No uniqueness theorem is imported, and no fitted parameter is renamed as a prediction. The main weakness is evidentiary, not circular: Appendix A summarizes cN_eff values without showing the promised finite-size scaling fits or error bars, and Eq. (18) computes only an upper bound whose logarithmic coefficient need not equal that of the true log-negativity. These are validation concerns, not definitional reductions, so the circularity score is low.

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

The central claim rests on fitting the scaling slopes of mutual information and an upper bound on log-negativity, so the main assumptions are the scaling forms and the identification of the slope with central charge; no new entities are introduced.

assumptions (4)
  • domain assumption The equal-time correlation functions in the long-time limit are obtained by dropping all oscillatory terms in Eq. (16), resulting in Eq. (17).
    Standard for integrable quenches; relies on infinite time limit and absence of degeneracies.
  • domain assumption The mutual information of the stationary state scales as I ~ (c_eff/3) ln L + const.
    Adopted from CFT and previous quench studies (Refs. [39,40,73-76]); no derivation for the long-range stationary state is provided.
  • domain assumption The log-negativity of the stationary state scales as ξ ~ (c_eff/4) ln L + const, and the upper bound in Eq. (18) yields the same coefficient as the true log-negativity.
    The upper bound is used in place of the exact log-negativity; its scaling coefficient is assumed to match the CFT prediction.
  • standard math The ground-state effective central charges in Table I, taken from Refs. [42,44,63,64], are correct.
    External literature results used as benchmarks.

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Pith. "Pith review of Quantum criticality and universality in stationary state of long-range Kitaev model." pith.science (2026). https://pith.science/paper/VQHSPKNL

@misc{pith2026241201076,
  author       = {Pith},
  title        = {Pith review of: Quantum criticality and universality in stationary state of long-range Kitaev model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQHSPKNL}},
  note         = {Machine review of arXiv:2412.01076}
}
abstract

We investigate the signature of quantum criticality in the long-time stationary state of the long-range Kitaev chain by performing various quench protocols. In this model, the pairing interaction decays with distance according to a power law with exponent $\alpha$. Using quantum information-theoretic measures, such as mutual information and logarithmic negativity, we show that, irrespective of the values of $\alpha$, critical-to-critical quench displays quantum criticality even in the stationary state. Remarkably, in the presence of long-range pairing interactions, where fermionic correlators decay algebraically even at non-critical points, signature of quantum criticality persists in the stationary state. Furthermore, the effective central charge, calculated from both mutual information and logarithmic negativity of stationary state following a critical-to-critical quench, agrees with the central charge of the corresponding ground states for both $\alpha = 0$ and $\alpha = 2$. Therefore, information of the universality class can be inferred from the stationary state.

Figures

Figures reproduced from arXiv: 2412.01076 by the authors.

Figure 1
Figure 1. FIG. 1. Top row: The mutual information between two sub [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. We plot the tripartite mutual information for different [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Phase plot of effective central charge [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Phase plot of the effective central charge from log-negativity [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

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