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Inhomogeneous quenches and GHD in the $\nu = 1$ QSSEP model

T0 review · 2 major / 3 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Averaging the per-realization QGHD entropy over Brownian Fermi contours gives the exact noise-averaged entanglement: growth is (1/12) log t for a melting domain wall and (1/8) log t for free expansion.

desk verdict A solid extension of QGHD to stochastic free fermions, with clean numerics and one central typo (Eq. 11 has a factor-2 Itô diffusion coefficient) that needs fixing before publication. read the letter →

arxiv 2602.15122 v2 pith:OCX4TXQO submitted 2026-02-16 cond-mat.stat-mech quant-ph

Angelo Russotto , Filiberto Ares , Pasquale Calabrese , Vincenzo Alba This is my paper · ORCID
classification cond-mat.stat-mechquant-ph
keywords quantumsymmetricsimpleexclusionprocessstochasticfreefermionsgeneralizedhydrodynamicsentanglemententropyFermicontourdomain-wallmeltingexpansiondiffusivetransport
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 claims that a one-dimensional chain of free fermions with spatially uniform but time-random hopping amplitudes—the ν=1 quantum symmetric simple exclusion process—can be described at long times by a stochastic version of generalized hydrodynamics. Starting from a domain wall or a trapped gas, the coarse-grained occupation function obeys a diffusion equation after noise averaging, while each individual noise realization is a ballistic front with a random displacement. The paper then argues that the full, noise-averaged entanglement statistics are obtained exactly by averaging the quantum generalized hydrodynamics (twist-field) prediction for each realization's Fermi contour. If correct, this gives an exact account of diffusive entanglement growth, with the half-system entropy growing as (1/12) log t for domain-wall melting and (1/8) log t for free expansion, and with relative fluctuations that vanish at late times. It is the first extension of quantum generalized hydrodynamics to stochastic dynamics.

What carries the argument

The Brownian Fermi-contour ensemble is the central object: each noise realization moves the initial contour x0(k) to x0(k) + 2ρ sin(k + φ), with ρ Rayleigh-distributed and φ uniform. The contour's intersections with the subsystem cut are the Fermi points at which chiral twist fields are inserted; the twist-field correlation, together with a Fisher–Hartwig non-universal constant, supplies the single-realization entropy, and the ensemble average over ρ and φ reproduces the noise-averaged Rényi and von Neumann entropies.

What would settle it

Extract the late-time slope of the half-system entanglement for the domain wall: if ⟨S₀(t)⟩ − C does not approach (1/12) log t as both L and t grow with t ≪ L², the central claim fails. Equivalently, a single fixed noise realization compared to the per-realization formula would expose any missing stochastic correction beyond the QGHD prediction.

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Extended reading notes

Core claim

In the hydrodynamic limit, the noise-averaged Rényi entropy equals the average of the single-realization conformal-field-theory entropy over the Brownian Fermi-contour ensemble. For the domain wall, each realization is labeled by a Rayleigh-distributed radius ρ, and its von Neumann entropy is S_ℓ(ρ) = (1/6) log[2ρ(1 − (ℓ/2ρ)^2)^{3/2}] + κ₁ for ℓ < 2ρ, the same formula as the deterministic model with the replacement 2ρ → t. Averaging gives ⟨S_ℓ(t)⟩ ≃ e^{−ℓ²/(4t)}/12 log t + S(ℓ/√t), which reduces to (1/12) log t + const at ℓ = 0. For free expansion, an additional uniform phase φ appears, split Fermi seas with four Fermi points can arise, and the hard-wall limit gives (1/8) log t + const. The

Load-bearing premise

The load-bearing premise is that averaging the deterministic twist-field formula over the Brownian Fermi contour reproduces the noise-averaged entanglement entropy, with no additional stochastic corrections; a secondary fragile step is the Itô differentiation of the step-like occupation function, where a factor-of-two inconsistency appears.

Editorial extensions

If this is right

  • Average density profiles are exactly diffusive: the domain-wall front becomes an error-function profile with width √(Dt), so transport is not ballistic on average.
  • Half-system entanglement grows logarithmically with half the deterministic slope: (1/12) log t for domain-wall melting and (1/8) log t for free expansion, instead of (1/6) and (1/4) in the clean model.
  • Entanglement is self-averaging in the hydrodynamic limit: the relative standard deviation of the half-system entropy vanishes as t → ∞, so the average equals the typical value.
  • The single-realization formulas also describe deterministic dynamics with a fixed complex hopping phase φ; the stochastic result arises from averaging those fixed-phase evolutions.
  • The same averaging procedure gives all higher moments and all Rényi indices, not only the von Neumann entropy.

Reading between the lines

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

  • Beyond the paper itself, because each per-realization entropy depends only on ρ and φ, the late-time distribution of entanglement is a one- or two-parameter distribution; measuring its skewness or third cumulant would test the mechanism more sharply than the mean does.
  • The boundary effect near x = −L/2, which the paper explicitly flags as outside its hydrodynamic description, could plausibly be incorporated by regularizing the Fermi contour at the hard wall and including elastic reflection—an extension the paper suggests but does not derive.
  • The same contour-averaging prescription should carry over to symmetry-resolved entropies and entanglement asymmetry, which are listed as future work: if the per-realization contour fixes the entanglement spectrum, those quantities follow from the identical ensemble.
  • For ν > 1, where the noise is spatially periodic rather than homogeneous, the contour will no longer be a rigid translation, so the exact factorization into (ρ, φ) will break; the present result therefore serves as a benchmark rather than a full solution of the general ν-QSSEP family.
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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

2 major / 3 minor

Summary. The paper studies the ν=1 QSSEP, a free-fermion chain with spatially uniform stochastic complex hopping, starting from either a domain-wall state or the ground state of a trapped gas. The central proposal is that, in the hydrodynamic limit, every noise realization is characterized by a stochastically translated Fermi contour, and that the per-realization Rényi/von Neumann entanglement entropy is exactly given by the deterministic QGHD/CFT twist-field formula evaluated on that contour. Averaging over the exactly known Brownian distribution of contours yields Eq. (25), from which the authors derive explicit asymptotic predictions: half-system entanglement grows as (1/12) log t for domain-wall melting and as (1/8) log t for free expansion, with relative fluctuations vanishing at late times. These predictions are compared with exact numerics for systems up to L=160–400 over about 10^3 noise realizations, with good agreement in the bulk/hydrodynamic regime. The paper is clearly written and the numerical checks are extensive.

Significance. If the central claim holds, this is the first extension of quantum generalized hydrodynamics to stochastic quantum dynamics, and it provides an exact, parameter-free description of diffusive entanglement growth and its fluctuations. The paper has notable strengths: the contour ensemble is derived from the Brownian statistics of the noise, the density propagator is checked against the exact lattice result Eq. (19), the single-realization QGHD formulas are independently benchmarked in Appendix A against deterministic complex-hopping evolution, and all final predictions are compared to exact numerics with no fitted parameters. The main formal problem is the factor-of-two inconsistency in the displayed stochastic evolution equation, Eq. (11); this is a concrete, correctable error but it sits at the base of the advertised derivation, so it must be fixed before the paper can be accepted in its present form.

major comments (2)
  1. [Sec. 3, Eq. (11)] The displayed Itô equation has the wrong diffusion coefficient. From the definition dξ_k = 2√D(cos k dB_1 + sin k dB_2) with dB_i^2 = dt/2, one obtains dξ_k^2 = 2D dt. Itô's lemma applied to n_k(x,t) = n_k(x−ξ_k(t),0) gives dn_k = −n'_k dξ_k + (1/2)n''_k dξ_k^2 = −n'_k dξ_k + D n''_k dt, not 2D n''_k dt. As written, averaging Eq. (11) would yield ∂_t⟨n_k⟩ = 2D ∂_x^2⟨n_k⟩, contradicting Eq. (12), the density profile Eq. (15), and the exact lattice propagator Eq. (19), whose hydrodynamic limit is the D-diffusion kernel. Since Eq. (11) is advertised as the derived stochastic evolution of the occupation function and Fig. 1 explicitly refers to it, the derivation as displayed cannot be reproduced. The error is local and correctable: replace 2D by D and re-check the Itô computation.
  2. [Sec. 3, Eqs. (9)–(11)] The Itô computation is also formal for the domain-wall initial condition, because n_k(x,0) = θ(−x) is not differentiable, so ∂_x and ∂_x^2 appearing in Eq. (11) act on a step function. This is separate from the factor-of-two issue. The authors should state the regularized version of the argument—for example, start from a smooth initial contour as in the free-expansion setup with finite β and then take the hydrodynamic/β→∞ limit, or formulate Eq. (11) weakly. This regularity point does not invalidate the numerics, but it is needed if Eq. (11) is to serve as the rigorous basis for the stochastic GHD description.
minor comments (3)
  1. [Eq. (35)] The term e^{−ℓ^2/(4t)}/(12) log t is typographically ambiguous; it should be written as (e^{−ℓ^2/(4t)}/12) log t to avoid being read as e^{−ℓ^2/(4t)}/(12 log t).
  2. [Eqs. (31), (53)] The expression 'Υ + log 2 / 3' is ambiguous; please write (Υ+log 2)/3 or Υ+(log 2)/3 explicitly as intended, and check the resulting numerical constant κ1.
  3. [Sec. 4, Fig. 6] The authors correctly state that the boundary-induced entanglement growth near x = −L/2 is not captured by the hydrodynamic framework. This is an important scope limitation; it should be stated in the conclusions and abstract so that the claimed agreement is understood to hold in the bulk/hydrodynamic regime only.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stochastic-average prediction is obtained by explicit integration over an independently derived contour ensemble and checked against exact lattice numerics.

full rationale

The central claim (25) is not equivalent to its inputs by construction. The per-realization QGHD entropy (23)-(24) is taken from the deterministic literature [19,62,63] as an external benchmark; the new content is the ensemble average (25) over the Fermi-contour distribution (27)-(29), which is derived from the model's Brownian velocity xi_k(t) and not fitted to the target entanglement data. The non-universal Fisher-Hartwig constants in (31) and (33) are imported from independent Toeplitz/Fisher-Hartwig results, and Eq. (33) is explicitly identified with the known non-random result under 2 rho -> t rather than being relabelled. The analytical predictions (35), (37), (39), (54), (55) follow from explicit integration over p_rho and p_phi, with no parameter fitted to the numerical entanglement curves; the numerics are used only as verification. Self-citations to companion works [23,46] are contextual (model definition and motivation) and the stochastic evolution (9)-(12) is rederived here. One caveat, which is a correctness issue rather than a circularity: Eq. (11) contains a factor-2 typo in the Itô diffusion term (2D d^2_x n_k, inconsistent with Eq. (12) and with the exact lattice propagator Eq. (19), which give coefficient D). This is an erratum-level error in the displayed SDE, not a reduction of the prediction to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data: the noise strength D is a model parameter, β is a protocol regulator, and the constants κ_1, Υ are derived from Fisher-Hartwig. The central claim rests on the QGHD requantization assumption and the CFT/Fisher-Hartwig input, both imported from prior literature, plus the formal Itô treatment of step functions.

assumptions (5)
  • domain assumption Hydrodynamic scale-separation: at large space-time scales the system is described by a coarse-grained local occupation function n_k(x,t).
    Invoked in Sec. 3 before Eq. (6); the entire GHD approach rests on this separation of scales.
  • domain assumption Per-realization QGHD requantization: entanglement of a single realization is given by a free massless boson on the Fermi contour with twist fields of dimension h_n = c/24(n−1/n).
    Eq. (23) and Sec. 3; imported from Ruggiero et al. [19] and the CFT work of Calabrese-Cardy [70]. The paper assumes this remains valid for stochastic contours.
  • standard math Fisher-Hartwig conjecture provides the non-universal additive constants κ_1, Υ.
    Used in Eqs. (31) and (53) via Refs. [72-74]; a known conjecture (proved in many cases) for Toeplitz determinant asymptotics.
  • domain assumption Itô formula applied to a step-function occupation n_k yields the stochastic transport equation (11).
    The formal manipulation in Sec. 3 is standard in the GHD literature but is distributional; the factor-2 typo in Eq. (11) is part of this step.
  • standard math Conformal mapping of the initial Fermi contour onto a circle and the twist-field four-point function (Eqs. (46), (51)) from CFT.
    Used in Sec. 4 for the free-expansion protocol; standard CFT results from [70,76].

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Cite this review

Pith. "Pith review of Inhomogeneous quenches and GHD in the $\nu = 1$ QSSEP model." pith.science (2026). https://pith.science/paper/OCX4TXQO

@misc{pith2026260215122,
  author       = {Pith},
  title        = {Pith review of: Inhomogeneous quenches and GHD in the $\nu = 1$ QSSEP model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OCX4TXQO}},
  note         = {Machine review of arXiv:2602.15122}
}
abstract

We investigate the dynamics of the $\nu=1$ Quantum Symmetric Simple Exclusion Process starting from spatially inhomogeneous initial states. This one-dimensional system of free fermions has time-dependent stochastic hopping amplitudes that are uniform in space. We focus on two paradigmatic setups: domain-wall melting and the expansion of a trapped gas. Both are investigated by extending the framework of quantum generalized hydrodynamics to account for the underlying stochastic dynamics. We derive the evolution of the local quasiparticle occupation function, which characterizes the system at large space-time scales, and analyze the resulting entanglement spreading. By incorporating quantum fluctuations of the occupation function and employing conformal field theory techniques, we obtain the exact contribution to the entanglement entropy for each individual noise realization. Averaging over these realizations then yields the full entanglement statistics in the hydrodynamic regime. Our theoretical predictions are confirmed by exact numerical calculations. The results presented here constitute the first application of quantum generalized hydrodynamics to stochastic quantum systems, demonstrating that this framework can be successfully extended beyond purely unitary dynamics to include stochastic effects.

Figures

Figures reproduced from arXiv: 2602.15122 by the authors.

Figure 1
Figure 1. Sketch of the time evolution of the local occupation function nk(x, t) after a domain-wall quench in the ν = 1 QSSEP. The dynamics of nk(x, t) follows the stochastic differential equation (11). Colored regions correspond to nk(x, t) = 1 for single noise realizations at different times. We compute the statistical properties of the out-of￾equilibrium entanglement entropy Sℓ(t) between two intervals A and B, connected … view at source ↗
Figure 2
Figure 2. Left panel: Profile of the average entanglement entropy for the subsystem A = [−L/2, ℓ] at different times t starting from the domain-wall state (3) in the ν = 1 QSSEP. The symbols are the exact average entanglement entropy over ∼ 103 noise realizations, computed numerically in a lattice of size L = 160. The errorbar is estimated from the standard deviation of the mean. The dashed curves correspond to the analytical… view at source ↗
Figure 3
Figure 3. Time evolution of the average half-system entanglement in the ν = 1 QSSEP (1) starting from the domain wall state (3). The symbols correspond to the exact average value computed over ∼ 103 noise realizations for increasing total system size L. The errorbar is estimated from the standard deviation of the mean. The continuous black curve corresponds to the analytical prediction in Eq. (37) derived using QGHD methods. … view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Sketch of the time evolution of the local occupation function nk(x, t) in the free-expansion protocol studied in Sec. 4. The colored regions correspond to nk(x, t) = 1. In the left panel, we show the occupation function of the initial state, corresponding to the ground…
Figure 5
Figure 5. Figure 5: Average half-system entanglement entropy ⟨Sℓ=0(t)⟩ as a function of time in the free expansion protocol, starting from the ground state of the Hamiltonian (4) with β = 0.25. The symbols are exact average entropy over ∼ 103 noise realizations. The corresponding errorbar…
Figure 6
Figure 6. Figure 6: Left panel: Profile of the average entanglement entropy as a function of the system cut ℓ at different times in the free-expansion protocol. We take a system of length L = 240 and an initial confining potential with β = 0.25. The symbols represent the exact average ent…
Figure 7
Figure 7. Figure 7: Half-system entanglement entropy Sℓ=0 as a function of time in the free￾expansion protocol with deterministic unitary dynamics given by Hφ in Eq. (56). The parameter φ controls the density current along the cut of the system. The symbols are the exact entanglement entr…

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