Interacting hydrodynamic modes in spinless fermions with dephasing noise

Fabian H.L. Essler; Patrik Penc

REVIEW

Interacting hydrodynamic modes in spinless fermions with dephasing noise

Fabian H.L. Essler, Patrik Penc

Statistical Mechanics · Submitted 28 Jul 2026 · 47 references

Pith's verdict: Substantial quartic eigenoperator construction, but the paper's own class IV mode makes Eq. (114) miss a conserved N^2 contribution.

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arXiv ID2607.25938 (version 1)

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CitationFabian H.L. Essler, Patrik Penc. "Interacting hydrodynamic modes in spinless fermions with dephasing noise." arXiv:2607.25938 (2026). https://pith.science/paper/LDDALM45

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Editor's note

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Dear colleague,

Two things you should know about this paper. First, it does real technical work: explicit construction of quartic diffusive eigenoperators for the dephasing tight-binding chain, built as scattering states of two k-Λ strings, with an interaction phase, and a hydrodynamic projection formula. That is a genuine step beyond the bilinear analysis in [20], and the physical principle they articulate — that the spatial size of the hydrodynamic eigenoperators, not just symmetry constraints, controls the late-time tails — is worth taking seriously.

Second, the central claim as stated is internally inconsistent. Section 7 asserts that classes II–IV do not contribute to the time evolution of translationally invariant operators. Class IV contradicts that. The class IV eigenoperator, Eqs. (89)–(91), is the exact zero mode (η†)^2|0>, i.e. the double-occupancy part of N^2 = (Σ n_j)^2, with E=0 and zero momentum. The initial operator O = Σ_j n_j n_{j+d} has a nonzero vectorized overlap with this state for d≠0 — of order L. By the paper's own spectral decomposition (26), that gives a time-independent contribution, and Eq. (114) omits it. A norm check confirms the problem: the LHS of (114) has norm squared O(L) while the RHS has norm squared O(L^2/t), so the identity cannot hold uniformly in L. The t^{-1/2} tail from class I may survive after subtracting the conserved N^2 piece, but the claim that classes II–IV vanish, and Eq. (114) as written, are not correct.

There are also secondary cautions: the interaction term in the Takahashi sums is neglected when converting to integrals (Section 7.1), the string deviations are set to zero, and the derivation assumes γ ≥ 2J even though the abstract states the result without that restriction. Those are addressable. The class IV problem is the one that matters.

With the conserved part subtracted and the regime clarified, the paper would be in good shape. As it stands, it deserves a serious referee but only on condition of a substantive revision. I would not cite the main result in its current form.

Best,

Referee report

Significance

If the central result is correct in the intended limit, it is a significant step: it provides an explicit, Bethe-ansatz-based proof that translationally invariant quartic operators in this dissipative model have nonzero diffusive tails, and it gives a concrete formula for the leading amplitude. The interpretation of quartic eigenoperators as interacting pairs of bilinear diffusive modes, with a scattering phase, is conceptually valuable and likely transferable to other integrable Lindblad systems. The paper also makes a useful general point about the role of the spatial structure of hydrodynamic eigenoperators. These strengths are partly offset by the fact that the derivation relies on several unstated or weakly controlled approximations, and by a clear issue with the treatment of exactly conserved operators, detailed below.

Major objections (4)

[§7, Eqs. (103)–(114); §5.4, Eqs. (89)–(91)] The claim that 'contributions of eigenstates in classes II-IV to the time evolution (26) of translationally invariant operators all vanish' is not correct for class IV. The state |N2> = (η†)^2|0>/√(2L(L−1)) has eigenvalue 0, total momentum 0, and its undoing is proportional to N(N−1), where N=Σ_x n_x. A direct calculation gives a nonzero overlap of this state with the vectorized operator Ψ_{d,d,0} for Q=0 and d≠0: in the trace inner product, Tr(N(N−1) Σ_x n_x n_{x+d}) / (2^L √(2L(L−1))) = L/(2√(2L(L−1))) = O(√L). The resulting exact conserved contribution to the time evolution is N(N−1)/(4(L−1)) = O(L), which is absent from Eq. (114). Consequently Eq. (114) is not correct as an operator identity at finite L; it can hold only in a specific double-scaling limit (e.g., L→∞ at fixed t, or t≪L²), which is not stated. For d=0 the problem is even more direct: Ψ = N is exactly conserved and Eq.
[§5.1.1 and abstract] The derivation is restricted to γ ≥ 2J ('we will focus on the first case for simplicity', around Eq. (60)), because only then does the two-particle bound state exist over the full Brillouin zone. The abstract and conclusions, however, state the existence and the t^{-1/2} tail without this restriction. For γ < 2J, Eq. (60) restricts the integration range in Eq. (108), and the amplitude in Eq. (113) would require a separate calculation. The abstract should either state the γ ≥ 2J assumption explicitly or the authors should extend the derivation to the full parameter range.
[§7.1, Eqs. (105)–(106)] The conversion of the sum over Bethe quantum numbers into an integral is performed by dropping the interaction term in the Takahashi equations, with the statement 'Assuming that we therefore can neglect it when turning sums into integrals for large L'. This is a load-bearing approximation: the interaction term is O(1/L) for fixed ν_a, but the momentum integrals that produce the t^{-1/2} tail are dominated by ν_a near ±π, where the rapidities λ_a are large. The resulting correction to the density of states or to the phases could in principle alter the prefactor in Eq. (113). No error estimate or numerical check is provided. Since the paper claims exact results, this step needs to be justified more carefully, for example by keeping the interaction term to next order and showing its contribution vanishes in the L→∞ limit.
[§7.2, Eqs. (110)–(113)] The limit Q→0 is delicate. For any fixed nonzero Q, the eigenvalue E(ν1,ν2) is gapped and the time evolution decays exponentially, so the power-law tail is a property of exactly Q=0. The text says 'we will take the limit Q→0 in the end' (before Eq. (99)) and then sets Q=0 in Eq. (110). This interchange of limits (Q→0 and t→∞) is not discussed. As written, Eq. (113) is the Q=0 result; the crossover for small nonzero Q is not described. The authors should specify the order of limits or show that the Q→0 limit commutes with the late-time limit in the relevant hydrodynamic regime.

Minor objections (6)

[§5.4, Eq. (91)] The undoing of the vectorization of (η†)^2|0> gives N(N−1), not (Σ_j n_j)^2; the difference is O(N) and is negligible in the large-L limit, but it matters for the conserved-sector discussion in the first major comment. Please correct Eq. (91).
[§5.1, Eqs. (52)–(57)] The wave-function expressions contain factors 1/ϵ1ϵ3 whose cancellation with the normalization (80) is not shown explicitly. A few lines indicating how the ϵ factors cancel in the final amplitudes would greatly help the reader.
[§4.1, Eq. (40)] The notation 'P≈π 1' below Eq. (40) appears to contain a typo; presumably it should be 'P≈π'.
[§7, Eq. (99)] The operator Ψ^{(Q)}_{{d,d,0}} is written as Σ_x n_x n_{x+d} e^{iQx}; for d=0 this is exactly the conserved total particle number, which should be explicitly excluded from the class of operators for which Eq. (114) is claimed.
[§7.2, Eq. (114)] The restriction 'L≫x≠y' on the right-hand side is vague. If Eq. (114) is meant as an operator identity in finite volume, it is not correct as discussed. If it is meant in the sense of matrix elements between local excitations in the thermodynamic limit, this should be stated precisely, and the inequality should be replaced by a statement about the order of limits.
[Abstract and §9] The word 'generically' in the abstract and conclusions is stronger than what is proven: the explicit calculation covers density-density operators of the form (99), and the extension to general quartic operators is described only schematically ('can be analyzed in the same way', end of §7.2). Please temper the claim or provide the general argument.

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The paper

The paper claims

Quartic fermion operators in a dephasing chain generically show diffusive t^{-1/2} hydrodynamic tails.

The reading

This paper studies how four-fermion operators decay in a one-dimensional chain of spinless fermions subjected to dephasing noise, described by a Lindblad equation. Using the exact solvability of the model, the authors construct the diffusive eigenoperators quartic in fermions and show they are interacting pairs of the bound-state modes found earlier for bilinears. Their central result is a closed-form expression for the late-time behavior: the translationally invariant density-density operator decays as t^{-1/2} with a known amplitude, proving a conjecture that quartic operators generically have non-vanishing hydrodynamic tails. The physical reason is that the hydrodynamic eigenoperators are small bound states of size comparable to the lattice spacing, so symmetry constraints alone are far too loose to predict the tails. This matters because it shows that operator hydrodynamics cannot be inferred from symmetry or coarse-grained fluctuating hydrodynamics alone; the microscopic structure of the modes decides the tail.

What carries the argument

The central object is the scattering state of two k-Λ strings: in the vectorized picture, quartic diffusive eigenoperators are eigenstates of the non-Hermitian Hubbard Hamiltonian with purely imaginary tunneling, built from two two-particle bound states whose size is set by the lattice spacing. The k-Λ strings are rapidity pairs solving discrete Takahashi equations; their scattering phase, Eq. (98), encodes the interaction between the two bound states. The late-time tail follows from an integral representation (Eq. 108) that is evaluated by expanding around zero momentum, giving the t^{-1/2} decay via a Gaussian integral.

Extended reading notes

Core claim

Translationally invariant operators quartic in fermions generically exhibit non-vanishing diffusive late-time tails, in contrast to bilinears. Explicitly, the density-density operator obeys (Σ_x n_x n_{x+d})(t) ≈ Σ_{x≠y} √(γ/(2π J² t)) n_x n_y at large times (Eq. 114), with the slowest decay coming from the ξ=0 sector of the asymptotic expansion (Eq. 113). This is established by constructing the four-particle eigenstates of the vectorized Lindbladian as scattering states of two k-Λ strings, then converting Bethe-quantum-number sums to integrals and evaluating the late-time integral around the hydrodynamic branch. The result proves the earlier conjecture and demonstrates that hydrodynamic tai

Load-bearing premise

The t^{-1/2} tail rests on neglecting the interaction term when converting sums over Bethe quantum numbers into integrals and on assuming γ≥2J; if that neglected term contributed at leading order in the hydrodynamic limit, the tail exponent could change.

Editorial extensions

If this is right

Density-density correlations in the dephasing fermion chain are predicted to decay as t^{-1/2} with a known amplitude, a concrete signature that could be checked in quantum simulators or numerical simulations.
Hydrodynamic tails of microscopic operators cannot in general be inferred from symmetry constraints or coarse-grained fluctuating hydrodynamics; the effective size of hydrodynamic eigenoperators is decisive.
The same mechanism likely explains tail structures in other models with decoupled BBGKY hierarchies, and may apply to diffusive Floquet systems.
Translationally invariant bilinear operators decay exponentially and are subdominant at late times, so quartic and higher-order operators dictate the asymptotic dynamics.
The explicit closed-form tail (Eq. 114) provides a benchmark against which approximate hydrodynamic theories can be tested.

Reading between the lines

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

If the neglected interaction term in the discrete Takahashi equations contributes at leading order when γ<2J (where the string exists only on part of the Brillouin zone), the t^{-1/2} tail could change; a direct numerical check in that regime would settle whether the result extends beyond the restricted parameter window used in the derivation.
The authors' general principle suggests that in models where hydrodynamic eigenoperators have a large size (≫ lattice spacing), symmetry-based tail predictions might be restored; searching for such models could test the universality of their conclusion.
A natural extension is to sextic operators, which would be scattering states of three bound states; whether the leading tail remains t^{-1/2} or acquires a different exponent is a concrete open question that the same Bethe-ansatz machinery could answer.
The result implies that coarse-grained fluctuating hydrodynamics, which usually assumes spatially extended diffusive modes, may miss operator-specific structures; coupling the microscopic bound-state size to a coarse-graining length could produce a more accurate effective theory.

Original abstract

We study the non-equilibrium dynamics of spinless fermions with dephasing noise in the framework of a many-particle Lindblad equation. Using a mapping to the exactly solvable one-dimensional Hubbard model with purely imaginary tunneling amplitude we analyze the Heisenberg-picture dynamics of operators quartic in fermions and determine their hydrodynamic projections. We construct the relevant diffusive eigenoperators explicitly and show that, in the quartic sector, they can be interpreted as interacting pairs of bilinear hydrodynamic modes. As a consequence, translationally invariant quartic operators generically exhibit non-vanishing diffusive late-time tails, unlike translationally invariant bilinears. Our results show that the hydrodynamic tails of microscopic operators cannot in general be inferred from symmetry constraints or coarse-grained fluctuating hydrodynamics alone; they also depend crucially on the spatial structure and effective size of the hydrodynamic eigenoperators.

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We study the non-equilibrium dynamics of spinless fermions with dephasing noise in the framework of a many-particle Lindblad equation. Using a mapping to the exactly solvable one-dimensional Hubbard model with purely imaginary tunneling amplitude we analyze the Heisenberg-picture dynamics of operators quartic in fermions and determine their hydrodynamic projections. We construct the relevant diffusive eigenoperators explicitly and show that, in the quartic sector, they can be interpreted as interacting pairs of bilinear hydrodynamic modes. As a consequence, translationally invariant quartic operators generically exhibit non-vanishing diffusive late-time tails, unlike translationally invariant bilinears. Our results show that the hydrodynamic tails of microscopic operators cannot in general be inferred from symmetry constraints or coarse-grained fluctuating hydrodynamics alone; they also depend crucially on the spatial structure and effective size of the hydrodynamic eigenoperators.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the non-equilibrium dynamics of spinless fermions with dephasing noise in the framework of a many-particle Lindblad equation.	contribution 0.87	0.93	0.02	asserted 0.89
s1	Using a mapping to the exactly solvable one-dimensional Hubbard model with purely imaginary tunneling amplitude we analyze the Heisenberg-picture dynamics of operators quartic in fermions and determine their hydrodynamic projections.	method 0.99	0.94	0.04	asserted 0.98
s2	We construct the relevant diffusive eigenoperators explicitly and show that, in the quartic sector, they can be interpreted as interacting pairs of bilinear hydrodynamic modes.	result 0.47	0.97	0.19	asserted 1.00
s3	As a consequence, translationally invariant quartic operators generically exhibit non-vanishing diffusive late-time tails, unlike translationally invariant bilinears.	result 0.94	0.91	0.32	asserted 0.99
s4	Our results show that the hydrodynamic tails of microscopic operators cannot in general be inferred from symmetry constraints or coarse-grained fluctuating hydrodynamics alone; they also depend crucially on the spatial structure and effective size of the hydrodynamic eigenoperators.	interpretation 0.98	0.90	0.83	asserted 1.00

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Compares with 2 papers

The late time limit we have considered above applies to the lattice model itself and is very different from continuum scaling limits that have been considered for the QSSEP and similar models [ref] .

Universal classical and quantum fluctuations in the large deviations of current of noisy quantum systems: Quantum symmetric simple exclusion and inclusion processes · Diffusive continuum scaling limit · 0.91
The late time limit we have considered above applies to the lattice model itself and is very different from continuum scaling limits that have been considered for the QSSEP and similar models [ref] .

2602.16544 · Diffusive continuum scaling limit · 0.91

Discussion

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