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arxiv: 2604.28151 · v1 · submitted 2026-04-30 · ❄️ cond-mat.stat-mech · quant-ph

Recognition: unknown

Domain-wall melting in all-to-all QSSEP from random-matrix theory

Denis Bernard , Lorenzo Piroli , Stefano Scopa
Authors on Pith no claims yet

Pith reviewed 2026-05-07 07:28 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords all-to-all QSSEPdomain wall meltingJacobi processrandom matrix theoryvon Neumann entanglement entropyfull counting statisticsthermodynamic limitquantum simple exclusion process
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The pith

The eigenvalues of the correlation matrix in the all-to-all quantum simple exclusion process evolve as a Jacobi process, giving exact dynamics for entanglement entropy and showing that quantum and classical charge full-counting statisticsco

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

The paper examines domain-wall melting in the quantum simple exclusion process with all-to-all hoppings. It applies random-matrix spectral results to show that the eigenvalues of the correlation matrix for the initially charged subsystem obey a closed set of stochastic differential equations known as the Jacobi process. This mapping produces the time evolution of all eigenvalue moments. Two applications follow: an explicit formula for the averaged von Neumann entanglement entropy in the thermodynamic limit, and an exact expression for the full-counting statistics of charge transfer. The latter expression reveals that the quantum and classical statistics become identical once the system size goes to infinity, with no finite-time corrections remaining.

Core claim

The authors establish that the eigenvalues of the correlation matrix corresponding to the initially charged subsystem evolve according to a Jacobi process defined by a closed system of stochastic differential equations. This observation yields the real-time dynamics of all eigenvalue moments. They then obtain a fully explicit expression for the averaged von Neumann entanglement entropy in the thermodynamic limit. For the full-counting statistics of the charge, they derive an analytical formula and demonstrate that it coincides exactly with the corresponding statistics of the classical simple exclusion process in the thermodynamic limit, with no finite-time corrections.

What carries the argument

The Jacobi process for the eigenvalues of the correlation matrix, obtained from random-matrix theory spectral results applied to the all-to-all hoppings.

If this is right

  • Explicit closed-form expressions become available for the time-dependent averaged von Neumann entanglement entropy once the system size is taken to infinity.
  • All moments of the correlation-matrix eigenvalues possess known real-time dynamics through the Jacobi process.
  • The full-counting statistics of transferred charge can be computed analytically and agree exactly with the classical simple-exclusion-process result at infinite size.
  • Quantum and classical descriptions of charge transport become indistinguishable in the thermodynamic limit for this model.

Where Pith is reading between the lines

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

  • The same random-matrix mapping may simplify entanglement and fluctuation calculations in other long-range or fully connected quantum spin models.
  • The lack of finite-time corrections implies that the thermodynamic limit is reached uniformly in time, a property that finite-size numerical checks could test directly.
  • The equivalence invites examination of whether other quantum signatures, such as off-diagonal coherences, also reduce to their classical counterparts in the same limit.

Load-bearing premise

The eigenvalues of the correlation matrix evolve exactly according to the Jacobi process given by the random-matrix spectral results for this all-to-all model.

What would settle it

A numerical computation of the charge full-counting statistics in a large but finite system at times comparable to the system size that deviates from the classical prediction would disprove the exact coincidence in the thermodynamic limit.

Figures

Figures reproduced from arXiv: 2604.28151 by Denis Bernard, Lorenzo Piroli, Stefano Scopa.

Figure 1
Figure 1. Figure 1: Illustration of the setup. We consider a one-dimensional quantum system of size L, where spinless fermions can hop between any pair of sites j → k, with amplitudes given by different realization of the complex Brownian motion. The hopping amplitudes are reset at each time step. The system is initialized in a domain-wall configuration, with the leftmost M sites occupied and the rest of the chain empty. Duri… view at source ↗
Figure 2
Figure 2. Figure 2: Entropy density dynamics from a domain-wall initial state with L = 32 and M = ℓ = L/2. The solid line corresponds to the analytic prediction in Eq. (46), while the symbols are obtained from numerical simulations of the quantum dynamics (see Appendix A), averaged over 200 realizations. The dashed horizontal line marks the steady-state value s(∞) = 2 log(2) − 1. where L (1) n (x) is the n-th Laguerre polynom… view at source ↗
Figure 3
Figure 3. Figure 3: Charge full-counting statistics dynamics from a domain-wall initial state with L = 32 and M = ℓ = L/2. The solid lines correspond to the analytic predictions in Eq. (52), while the symbols are obtained from numerical simulations of the quantum dynamics (see Appendix A), averaged over 200 realizations. Several values of α are shown, as indicated by the color legend. This expression is exact in the thermodyn… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between the classical and quantum dynamics of the charge full￾counting statistics. Symbols: numerical data for the time evolution of F class ℓ (α, t) for ℓ = M = L/2 = 16 as a function of time for different values of α (see color legend), obtained from Eq. (81) by direct matrix exponentiation. Solid lines: analytic solution for the quantum dynamics F(α, t), given in Eq. (52). Dashed horizontal l… view at source ↗
Figure 5
Figure 5. Figure 5: Numerical analysis of the finite-size deviation |∆L(α, t)| between the classical and quantum charge full-counting statistics as a function of L, for different values of α and t. Quantum data are obtained from simulations of the all-to-all QSSEP as described in Appendix A, while classical results are computed from the corresponding Markov process in Eq. (81) via direct matrix exponentiation. For each system… view at source ↗
read the original abstract

We study the melting of a domain wall in the quantum simple exclusion process with all-to-all hoppings (a.k.a. the charged SYK$_2$ model). We show that the real-time dynamics of physical quantities of interest can be obtained exploiting spectral results in random matrix theory. We first show that the eigenvalues of the correlation matrix corresponding to the initially charged subsystem evolve according to a Jacobi process, which is defined in terms of a closed system of stochastic differential equations. In turn, this observation allows us to obtain the real-time dynamics of all the eigenvalue moments. We present two physical applications. First, we study the dynamics of the averaged von Neumann entanglement entropy, arriving at a fully explicit expression in the thermodynamic limit. Second, we compute analytically the full-counting statistics of the charge. Our formula allows us to perform a thorough comparison with the full-counting statistics of the classical simple exclusion process. Notably, we show that, in the thermodynamic limit, the quantum and classical full-counting statistics coincide, with no finite-time corrections.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies domain-wall melting in the all-to-all quantum simple exclusion process (QSSEP, or charged SYK2 model). It claims that the eigenvalues of the correlation matrix associated with the initially charged subsystem evolve according to a Jacobi process governed by a closed system of stochastic differential equations, obtained via spectral results from random matrix theory. This mapping yields the real-time dynamics of all eigenvalue moments, an explicit expression for the averaged von Neumann entanglement entropy in the thermodynamic limit, and an analytic formula for the full-counting statistics (FCS) of charge. The central result is that, in the N→∞ limit at any fixed t, the quantum FCS coincides exactly with the classical SEP FCS, with no finite-time corrections.

Significance. If the eigenvalue-to-Jacobi-process mapping is exact, the work supplies rare closed-form results for real-time non-equilibrium dynamics in an interacting quantum system, including explicit entanglement entropy and a parameter-free demonstration of quantum-classical FCS equivalence. The approach leverages independently established RMT results to close the dynamics without ad-hoc fits, providing a benchmark for numerics and potential extensions to other long-range models.

major comments (2)
  1. [Abstract and §2–3] Abstract and the derivation of the eigenvalue dynamics (presumably §2–3): the claim that the correlation-matrix eigenvalues evolve exactly according to the closed Jacobi-process SDEs at finite N and finite t is load-bearing for the 'no finite-time corrections' result. The manuscript must explicitly show that the SDE closure holds without invoking the thermodynamic limit inside the dynamics (i.e., that the process remains Markovian in the eigenvalues alone for finite N), as any hidden N-dependent approximation would invalidate the exact coincidence of quantum and classical FCS at fixed t after N→∞.
  2. [§4 (FCS)] The FCS section (presumably §4): the generating function for the charge FCS is asserted to match the classical expression exactly in the thermodynamic limit. The paper should provide either an explicit error bound on the finite-N deviation or a rigorous interchange of limits (N→∞ before or after t fixed) to confirm the absence of corrections; otherwise the strongest claim rests on an unverified assumption about the spectral mapping.
minor comments (2)
  1. Ensure consistent use of 'all-to-all QSSEP' versus 'charged SYK2 model' throughout the text and figures.
  2. The thermodynamic-limit expressions for entanglement entropy and FCS moments should be accompanied by a brief statement of the order of limits (N→∞ at fixed t) to avoid ambiguity.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below, providing the strongest honest defense of the results while clarifying the derivations. We have revised the manuscript to incorporate additional explanations where appropriate.

read point-by-point responses

  1. Referee: [Abstract and §2–3] Abstract and the derivation of the eigenvalue dynamics (presumably §2–3): the claim that the correlation-matrix eigenvalues evolve exactly according to the closed Jacobi-process SDEs at finite N and finite t is load-bearing for the 'no finite-time corrections' result. The manuscript must explicitly show that the SDE closure holds without invoking the thermodynamic limit inside the dynamics (i.e., that the process remains Markovian in the eigenvalues alone for finite N), as any hidden N-dependent approximation would invalidate the exact coincidence of quantum and classical FCS at fixed t after N→∞.

    Authors: We appreciate the referee highlighting the importance of this foundational step. Sections 2 and 3 derive the stochastic differential equation for the correlation matrix C(t) directly from the Heisenberg equations of motion for the all-to-all QSSEP Hamiltonian; this step is exact for any finite N with no large-N approximation. Because the noise is all-to-all and the matrix remains Hermitian, the application of Itô's formula to the eigenvalues yields a closed system of SDEs (the Jacobi process) whose drift and diffusion terms depend only on the eigenvalues themselves. The Markov property in eigenvalue space therefore holds exactly at finite N and finite t; the explicit N-dependence appears in the coefficients but introduces no approximation. The thermodynamic limit is invoked only when computing averaged observables such as the entanglement entropy or FCS. We have added a dedicated clarifying paragraph at the close of Section 2 that states this exactness and references the relevant finite-N random-matrix results supporting the closure. revision: yes

  2. Referee: [§4 (FCS)] The FCS section (presumably §4): the generating function for the charge FCS is asserted to match the classical expression exactly in the thermodynamic limit. The paper should provide either an explicit error bound on the finite-N deviation or a rigorous interchange of limits (N→∞ before or after t fixed) to confirm the absence of corrections; otherwise the strongest claim rests on an unverified assumption about the spectral mapping.

    Authors: In Section 4 the FCS generating function is written exactly as a functional of the eigenvalues of C(t), whose moments are obtained from the closed ODEs that follow from the Jacobi process. Taking the thermodynamic limit N→∞ at fixed t, the empirical spectral measure converges (in probability) to a deterministic density whose evolution is identical to the hydrodynamic limit of the classical SEP; consequently the generating functions coincide exactly. We do not supply an explicit finite-N error bound, as obtaining quantitative concentration estimates for the fluctuations of the Jacobi process would require additional analytic tools beyond the scope of the present work. We have revised Section 4 to state the order of limits explicitly (N→∞ first, t fixed) and to clarify that the convergence follows from the law of large numbers for the eigenvalue measure. While this supports the exact coincidence in the limit, a fully rigorous error analysis remains an open question. revision: partial

standing simulated objections not resolved
  • Deriving explicit finite-N error bounds or a complete rigorous proof of the limit interchange for the FCS, which would require quantitative concentration results for the Jacobi process not developed in the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation uses external RMT spectral results

full rationale

The paper maps the eigenvalues of the correlation matrix for the initially charged subsystem to a Jacobi process via a closed system of SDEs, citing spectral results from random matrix theory applied to the all-to-all QSSEP dynamics. From this, it derives the time evolution of eigenvalue moments, the averaged von Neumann entanglement entropy in the thermodynamic limit, and the full-counting statistics of charge. The central result—that quantum and classical FCS coincide exactly in the N→∞ limit with no finite-time corrections—follows directly as a consequence of the eigenvalue process matching the classical case, without any reduction to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The mapping is presented as derived from the model structure and standard RMT, making the chain self-contained against external benchmarks rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that random-matrix spectral statistics apply directly to the correlation matrix of the all-to-all QSSEP and that the resulting eigenvalue evolution is exactly captured by the Jacobi process SDEs. No free parameters are introduced or fitted; the thermodynamic limit is invoked to obtain closed expressions. No new physical entities are postulated.

axioms (2)
  • domain assumption Spectral properties of appropriate random matrices govern the eigenvalues of the correlation matrix in the all-to-all QSSEP
    Invoked to justify the mapping to the Jacobi process
  • domain assumption The thermodynamic limit exists and eliminates finite-size corrections in the full-counting statistics
    Required for the claimed exact coincidence with classical SEP

pith-pipeline@v0.9.0 · 5482 in / 1630 out tokens · 84105 ms · 2026-05-07T07:28:34.456391+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

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