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REVIEW 3 major objections 4 minor 69 references

A Hydrodynamic Theory for Non-Equilibrium Full Counting Statistics in One-Dimensional Quantum Systems

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Pith's one-line read After a quantum quench, charge fluctuations are governed by the current fluctuations of a biased partitioning protocol, reducible to steady-state hydrodynamic data.

desk verdict A transparent, carefully argued paper that makes the assumptions behind a known ballistic hydrodynamic formula for quench FCS explicit and checks them in free fermions; the load-bearing Asymptotic Commutativity step is honestly flagged as unproven. read the letter →

arxiv 2507.05954 v2 pith:DH2XAQAY submitted 2025-07-08 cond-mat.stat-mech cond-mat.quant-gascond-mat.str-elquant-ph

David X. Horvath , Benjamin Doyon , Paola Ruggiero This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.quant-gascond-mat.str-elquant-ph MSC 82C1082C70
keywords fullcountingstatisticsquantumquenchballistictransportnon-equilibriumsteadystatefluctuationtheoryfreefermionsintegrablesystemshydrodynamics
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 tries to establish that, in one-dimensional systems with ballistic transport, the dynamical part of the full counting statistics of a conserved charge after a homogeneous quench is equal to the current full counting statistics in a specially constructed non-equilibrium steady state. The construction is a charge-biased, spatially inhomogeneous version of the original quench, the standard partitioning protocol. If the argument is right, the non-equilibrium fluctuation problem reduces to thermodynamic and hydrodynamic data of the steady state, computable in integrable systems through the Ballistic Fluctuation Theory flow equations. The authors identify three physical conditions under which this reduction holds and verify them in detail for non-relativistic free fermions, where exact results provide cross-checks. A sympathetic reader would take the central claim to be that the quench FCS encodes no additional dynamical information beyond the initial-state fluctuations and the steady-state current fluctuations.

What carries the argument

The argument is carried by a contour deformation: using the continuity equation, the charge $\\int_0^X dx\\,q(x,T)$ is rewritten as the initial charge plus the integrated currents through the two boundaries, $\\int_0^X dx\\,q(x,0)+\\int_0^T dt\\,j(0,t)-\\int_0^T dt\\,j(X,t)$. Two factorisation properties then do the heavy lifting. Factorisation Property 1 separates left and right contributions at large $X$, relying on spatial clustering and, where available, a Lieb-Robinson bound. Factorisation Property 2 is the 'Asymptotic Commutativity' step, replacing $e^{\\lambda(\\tilde J+\\tilde Q)}$ by $e^{\\lambda\\tilde J}e^{\\lambda\\tilde Q}$ up to $o(T)$ corrections for large $T$; this is the mathematically least secure step and is checked only perturbatively in $\\lambda$ for free fermions. The final ingredient is the replacement of the biased initial state by its NESS, justified by local relaxation and temporal clustering, at which point the Ballistic Fluctuation Theory flow equations evaluate the current FCS from thermodynamic and hydrodynamic data.

What would settle it

A concrete falsifier is to compute, in free non-relativistic fermions, the fourth-order coefficient in $\\lambda$ of the difference $\\ln\\langle\\Psi|e^{\\lambda(\\tilde J+\\tilde Q)}|\\Psi\\rangle - \\ln\\langle\\Psi|e^{\\lambda\\tilde J}e^{\\lambda\\tilde Q}|\\Psi\\rangle$ and check whether it grows linearly in $T$. If it does, the Asymptotic Commutativity step fails at that order and the central identity (8) collapses; conversely, a $o(T)$ result would directly support the conjectured all-order validity.

Watch

Extended reading notes

Core claim

The central claim is the hydrodynamic identity for the dynamical scaled cumulant generating function: $f_{\\rm dyn}(\\lambda)=\\lim_{T\\to\\infty} \\frac{2}{T}\\ln\\mathrm{Tr}[e^{\\lambda J_0^{0|T}}\\rho_{\\rm NESS}(\\lambda)]$, where $\\rho_{\\rm NESS}(\\lambda)$ is the non-equilibrium steady state reached after a bipartite quench from the biased initial state $e^{\\lambda/2 Q|_0^\\infty}|\\Psi\\rangle$, and $J_0^{0|T}$ is the integrated current through the origin. This follows from $f_{\\rm dyn}(\\lambda)=\\lim_{T\\to\\infty}\\frac{2}{T}\\ln\\mathrm{Tr}[e^{\\lambda J_0^{0|T}}\\rho_{\\rm in}(\\lambda)]$ under a spatial clustering condition, and then from the additional conditions of local relaxation to a NESS and temporal clustering of current correlations. For free non-relativistic fermions with integrable quenches, the authors evaluate the formula explicitly through the BFT flow equations, obtaining $f_{\\rm dyn}(\\lambda)=2\\int\\frac{dk}{2\\pi}|E'(k)|\\ln[(1-\\theta_{\\rm bpp}(k,\\lambda))+\\theta_{\\rm bpp}(k,\\lambda)e^{\\lambda\\,\\mathrm{sgn}(k)q(k)}]$, with the NESS filling function determined by the two GGEs of the left and right reservoirs. Microscopic computations at low orders in the counting field demonstrate convergence to the NESS, absence of strong temporal correlations, and agreement with known exact results.

Load-bearing premise

The load-bearing premise is the Asymptotic Commutativity step, Eq. (33): for large $T$, the operator $e^{\\lambda(\\tilde J+\\tilde Q)}$ can be replaced by $e^{\\lambda\\tilde J}e^{\\lambda\\tilde Q}$ up to $o(T)$ corrections; this is verified only to third order in $\\lambda$ for free fermions, with higher orders supported by a scaling heuristic, and if it fails the hydrodynamic reduction breaks before the steady-state replacement.

Editorial extensions

If this is right

  • If the hydrodynamic identity holds, the full counting statistics after a quench in the regime $X\\gg T\\gg1$ is determined by the initial-state scaled cumulant generating function plus a term linear in $T$ coming from the current FCS in a NESS.
  • For integrable systems, the current FCS in the NESS is computable in closed form through the BFT flow equations, so the quench FCS becomes a calculable function of the GGE filling functions of the two reservoirs.
  • The three stated conditions provide a precise applicability criterion: spatial clustering suffices for the intermediate identity, but the full steady-state reduction requires local relaxation and absence of strong temporal current correlations.
  • The theory extends to charges beyond particle number, to non-fluctuating initial states such as the Néel state, and to generic local charges, provided the same conditions are met.
  • When temporal clustering is violated, as in the quenched Lieb-Liniger model with the hydrodynamic waft effect identified in the companion paper, the fully hydrodynamic formula is not expected to hold, although the first few cumulants may still be reproduced.

Reading between the lines

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

  • The Asymptotic Commutativity step is the most fragile link; a direct test would be to compute the fourth-order term in $\\lambda$ for free fermions and check that the difference between the two sides of Eq. (33) remains $o(T)$, which would strengthen the case beyond the third order treated here.
  • If the hydrodynamic reduction is correct, it suggests a general principle: for ballistically propagating conserved charges, the long-time dynamical fluctuations after a homogeneous quench contain no information about the quench beyond what is encoded in the initial state and in the steady-state current statistics, a sharpening of generalized thermalization for fluctuations.
  • The same contour-deformation logic may apply to symmetry-resolved Rényi entropies or to charges with sub-extensive initial fluctuations, where the biased state becomes trivial and the NESS is simply the homogeneous steady state of the original quench.
  • A testable extension is to compare the hydrodynamically predicted $f_{\\rm dyn}(\\lambda)$ with exact numerics for interacting integrable models beyond free fermions, in cases where condition iii) holds; agreement would support the conjectured generality of the reduction.
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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 / 4 minor

Summary. The paper proposes a hydrodynamic theory for the full counting statistics (FCS) of conserved charges after homogeneous quantum quenches in one-dimensional systems with ballistic transport. The central claim is that, under three conditions (spatial clustering of the initial state, local relaxation after a biased bipartite quench, and temporal clustering of current correlations), the dynamical part of the post-quench FCS equals the current FCS in the non-equilibrium steady state (NESS) of a partitioning protocol, Eq. (9). For free non-relativistic fermions with integrable quenches, the authors evaluate this via ballistic fluctuation theory, obtaining explicit flow equations and cross-checking against known exact results. The paper also provides microscopic checks of the three conditions in the free-fermion setting, and discusses cases where the temporal clustering condition is violated.

Significance. If the central identity (9) holds, the paper provides a valuable conceptual bridge between quench dynamics and NESS fluctuation theory, with concrete closed-form predictions for integrable models. The strengths are the clear identification of the three conditions, the explicit free-fermion computations demonstrating (parts of) them, and the transparent use of the companion paper for the BFT flow equations. The paper does not fit parameters and its predictions are falsifiable by comparison with the exact FCS results of Refs. [30,31]. However, the significance is tempered by the fact that the key step, Asymptotic Commutativity, is not proven to all orders, and the verification of the conditions is partial.

major comments (3)
  1. [Sec. 4.2, Eq. (33)] The identity (33) (Asymptotic Commutativity) is the step that converts the contour-deformed expression (28) into the biased-state formula (8). The paper verifies it only to third order in lambda for free fermions (Sec. 4.2.2-4.2.3), and the higher-order argument in Sec. 4.2.4 explicitly excludes multi-point functions containing nested commutators together with additional J or Q insertions. Such terms are exactly those that could produce O(T) contributions at higher orders, so the central identity (8) is not established beyond O(lambda^3). Since the paper itself states in Sec. 2.3 that no general statement is known, the claim as a general result requires either a rigorous all-order proof for the free-fermion case, a numerical check to higher orders, or a clear restatement of the result as holding only to third order.
  2. [Sec. 6, condition iii)] Condition iii) requires temporal clustering of current correlations at all orders and for all multi-point functions. The microscopic verification in Sec. 6 is carried out only for the two-point function, and only to first non-trivial order in lambda (Eq. (211) and following). The derivation of Eq. (9), however, uses condition iii) for the replacement of rho_in by rho_NESS in the full generating function, which involves all connected multi-point functions. The paper acknowledges this limitation in Sec. 7, but as the result (9) is the main claim, the incomplete verification should be stated more prominently, and the order in lambda to which (9) is demonstrated should be quantified.
  3. [Sec. 5, Eq. (21)] The identification rho_NESS = rho_bpp, used to obtain the filling function (69) and hence the explicit formula (77), is verified only for the current and for a restricted set of conserved densities, and only to low orders in lambda. While the paper is transparent about this, the explicit BFT prediction (77) inherits this limitation. The authors should either extend the verification or clearly state that (77) is conjectured beyond the orders checked.
minor comments (4)
  1. [Sec. 1, paragraph 2] There is a typo in the phrase 'an infnite LR velocity'; it should read 'an infinite Lieb-Robinson velocity'.
  2. [Eq. (4)] The notation o(X^0)_T is used without an explicit definition; please clarify the meaning of this term, particularly how the limit in X and T is ordered.
  3. [Sec. 4.2, Eq. (140)] The Zassenhaus expansion is written with an ellipsis 'x(...)' in the exponent; specify the series more rigorously or refer to a standard reference for the Zassenhaus formula.
  4. [Fig. 1 caption] The caption refers to 'Turquoise bars', but this color may not be visible in black-and-white print; consider using a pattern-based legend instead.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the central identity is derived from the continuity equation and clustering conditions, not assumed; the known formula (77) is independently benchmarked against exact results.

full rationale

The paper's central claim, Eq. (8), is not equivalent to its input by construction. It is obtained by contour deformation of the integrated charge using the continuity equation (27), followed by Factorisation Property 1 (Sec. 4.1) and Asymptotic Commutativity (Eq. (33), Sec. 4.2). The latter is explicitly checked order by order in lambda, up to third order, rather than being assumed as the result. The quoted limitation that no general rigorous statement of Asymptotic Commutativity is known, and that higher orders are supported by a scaling heuristic, is an acknowledged rigor gap and a correctness risk, not a circular step: the heuristic does not define f_dyn(λ) in terms of itself. Formula (77) was previously obtained in Refs. [30,31,34], but the paper uses it as a benchmark and independently reproduces the ingredients for free fermions via microscopic computations (Appendix C), so the reliance is transparent and not load-bearing in a circular sense. No parameters are fitted to the target quantity, and no prediction is a renamed input. The self-citations to the companion paper [34] are clearly flagged and are used for context or for results that are also derived here or independently checkable; they do not force the main conclusion. Overall the derivation chain has independent content, with the main open issue being the completeness of the Asymptotic Commutativity verification at higher orders, which belongs to rigor rather than circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The core claims rest on several explicitly stated physical conditions plus one unproved commutativity principle. No data fitting or invented entities are involved.

assumptions (6)
  • domain assumption Linear-in-T large-deviation principle for current fluctuations after the biased bipartite quench is assumed.
    Stated in Section 7 as an implicit assumption; needed for fdyn to be defined as a finite T slope. Explicitly acknowledged, not proven.
  • ad hoc to paper Asymptotic Commutativity: for large T, e^{λ(\tilde J + \tilde Q)} can be replaced by e^{λ \tilde J} e^{λ \tilde Q} up to o(T).
    Introduced in Section 2.3 after Eq. (33); no general proof, checked to O(λ^3) for free fermions in Section 4.2 and Appendix B.
  • domain assumption Condition i: exponential or strong power-law clustering of the initial state.
    Eq. (13) in Section 2.2; used for Factorization Property 1 and 2, and verified for integrable squeezed states in Appendix A.
  • domain assumption Condition ii: local relaxation of the biased bipartite quench to a steady state ρ_NESS(λ).
    Eq. (16); shown for conserved densities in free fermions to low orders in λ, expected generically but not proven in general.
  • domain assumption Condition iii: temporal clustering of current connected correlation functions after the biased bipartite quench.
    Eq. (17); nontrivial and can fail, for example in the quenched Lieb-Liniger model discussed in the companion paper. Checked for 2-point functions in free fermions at leading orders.
  • domain assumption Lieb-Robinson bound with finite velocity for lattice models, used as an auxiliary condition.
    Section 2.2 and Section 4.1.1; needed for the proof of Factorization Property 1 in bounded-operator lattice systems. The authors expect it can be relaxed.

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Cite this review

Pith. "Pith review of A Hydrodynamic Theory for Non-Equilibrium Full Counting Statistics in One-Dimensional Quantum Systems." pith.science (2026). https://pith.science/paper/DH2XAQAY

@misc{pith2026250705954,
  author       = {Pith},
  title        = {Pith review of: A Hydrodynamic Theory for Non-Equilibrium Full Counting Statistics in One-Dimensional Quantum Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DH2XAQAY}},
  note         = {Machine review of arXiv:2507.05954}
}
read the original abstract

We study the dynamics of charge fluctuations after homogeneous quantum quenches in one-dimensional systems with ballistic transport. For short but macroscopic times where the non-trivial dynamics is largely dominated by long-range correlations, a simple expression for the associated full counting statistics can be obtained by hydrodynamic arguments. This formula links the non-equilibrium charge fluctuation after the quench to the fluctuations of the associated current after a charge-biased inhomogeneous modification of the original quench which corresponds to the paradigmatic partitioning protocol. Under certain assumptions, the fluctuations in the latter case can be expressed by explicit closed form formulas in terms of thermodynamic and hydrodynamic quantities via the Ballistic Fluctuations Theory. In this work, we identify precise physical conditions for the applicability of a fully hydrodynamic theory, and provide a detailed analysis explicitly demonstrating how such conditions are met and how this leads to such hydrodynamic treatment. We discuss these conditions at length in non-relativistic free fermions, where calculations become feasible and allow for cross-checks against exact results. In physically relevant cases, strong long-range correlations can complicate the hydrodynamic picture, but our formula still correctly reproduces the first cumulants.

Figures

Figures reproduced from arXiv: 2507.05954 by the authors.

Figure 1
Figure 1. Illustration of the universal hydrodynamic principles for full counting statistics [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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