Rigorous foundation and microscopic interpretation of the spacetime-swap formulas for Rényi-entropy growth in interacting integrable models

After a quantum quench from a low-entangled state, the Rényi-$\alpha$ entropies of a large subsystem in a one-dimensional interacting integrable model grow linearly in time before saturating. Bertini, Klobas, Alba, Lagnese and Calabrese [Phys. Rev. X 12, 031016 (2022)] computed the growth slope exactly in terms of thermodynamic Bethe ansatz (TBA) data by a spacetime duality: the slope of the Rényi entropy equals the stationary entropy density of a dual system obtained by exchanging the roles of space and time, whose state is built from the fixed points of the space transfer matrix. The derivation, however, is heuristic: the slope–stationary-state correspondence is conjectured, the sign structure of the dual TBA equations was originally fixed only by agreement with the exactly solvable Rule 54 automaton, and no algebraic-Bethe-ansatz-level proof exists. Follow-up literature resolved two of the source's open directions (symmetry-resolved Rényi entropies; inhomogeneous quenches) at the same conjecture level and removed the sign ambiguity, but the mathematical proof is explicitly listed as open by the same authors in 2026. The problem asks for: (i) a rigorous proof, or controlled refutation, of the spacetime-swap correspondence and the TBA slope formulas for all TBA-integrable quenches from compatible initial states — whose $\alpha\to 1$ corollary would rigorously validate the 2005 quasiparticle conjecture for entanglement growth; (ii) an emergent microscopic picture reproducing the formulas for $\alpha\neq 1$, replacing the quasiparticle picture that provably fails there; (iii) the leading-order exactness of the conjectured full-time interpolation formula beyond Rule 54; and (iv) the growth of entanglement negativity and higher partial-transpose moments in interacting integrable models beyond the early-time regime.

Background

A quantum quench — the sudden change of a parameter of a closed many-body Hamiltonian — is the cleanest laboratory for far-from-equilibrium quantum dynamics. If a contiguous block $A$ of a one-dimensional chain is described after the quench by the reduced density matrix $\rho_A(t)$, its Rényi entropies $S_A^{(\alpha)}(t)=(1/(1-\alpha))\ln\mathrm{tr}[\rho_A(t)^\alpha]$ generalise the von Neumann entropy (recovered at $\alpha\to1$) and are directly measurable by randomized-measurement protocols, in which many repeated preparations of the system are measured in randomly chosen local bases so that nonlinear functions of the reduced density matrix can be reconstructed from the single-shot outcomes. After a quench from a low-entangled state, $S_A^{(\alpha)}(t)$ grows linearly in time until it saturates at a 'thermodynamic' value fixed by the stationary state; the growth slope $s_\alpha=\lim_{t\to\infty}S_A^{(\alpha)}(t)/(2t)$ (the entanglement production rate; the factor of 2 accounts for the two edges of the block through which $A$ develops correlations with its complement — this per-edge convention is used in all formulas below) carries the key dynamical information.

For non-interacting systems the linear growth has a textbook microscopic explanation, the quasiparticle picture introduced by Calabrese and Cardy in 2005: the quench creates pairs of counter-propagating quasiparticles that are entangled at birth, and at time $t$ every pair with exactly one member inside $A$ contributes to the entropy. For interacting integrable models — one-dimensional systems solvable by the (algebraic) Bethe ansatz, whose scattering is elastic and factorised, and whose thermodynamics is exactly encoded by the thermodynamic Bethe ansatz (TBA) in terms of a rapidity-resolved filling function $\vartheta(\mu)$ and scattering kernel $T$ — Alba and Calabrese (2017) extended the quasiparticle picture to the von Neumann entropy, and it became the standard tool, including inhomogeneous settings through generalised hydrodynamics (GHD), the Euler-scale hydrodynamic theory of integrable transport (Alba, Bertini and Fagotti, 2019). The post-quench stationary state is described by a generalised Gibbs ensemble (GGE), the ensemble fixed by all local conserved charges of the integrable model. A quench is called integrable, and the initial state 'compatible', when the overlaps of the initial state with the Bethe eigenstates are known and factorised, so that the post-quench filling function $\vartheta(\mu)$ is exactly computable.

For $\alpha\neq1$, however, the quasiparticle picture fails in interacting models (it remains valid for free systems). Bertini, Klobas, Alba, Lagnese and Calabrese (Phys. Rev. X 12, 031016 (2022)) solved the long-standing problem of computing $s_\alpha$ there by a spacetime duality argument. Representing $\mathrm{tr}[\rho_A^n]$ as a tensor network of $n$ replicated space-time trajectories and contracting in the spatial rather than the temporal direction, they identified the slope with the stationary entropy of a dual system: writing $M_{L,t}$ and $M_{R,t}$ for the fixed points (leading eigenvectors, also known as influence matrices) of the space transfer matrix, the pseudo density matrix $\tilde{\rho}_{\mathrm{st},t}=M_{L,t}^{\dagger}M_{R,t}$ satisfies

$$s_{\alpha}=\frac{1}{1-\alpha}\lim_{t\to\infty}\frac{1}{t}\ln\mathrm{tr}\left[\left(\tilde{\rho}_{\mathrm{st},t}\right)^{\alpha}\right],$$

i.e. the slope equals the density of Rényi entropy in the stationary state of the dual system. For dual-unitary quantum circuits (brickwork circuits whose gates are unitary both along time and along space) the dual system is again a unitary circuit and the statement can be checked; for interacting integrable models the dual is argued to be an integrable boundary problem, and the TBA machinery yields the slope formula

$$s_{\alpha}=\frac{2}{1-\alpha}\int_{+} d\mu\,\frac{\varepsilon'(\mu)}{2\pi}\,\ln\left[(1-\vartheta(\mu))^{\alpha}+\frac{\vartheta(\mu)^{\alpha}}{y_{\alpha}(\mu)}\right],$$ $$\ln y_{\alpha}(\lambda)=\int_{+} d\mu\,\left[T(\mu,\lambda)-T(\mu,-\lambda)\right]\ln\left[(1-\vartheta(\mu))^{\alpha}+\frac{\vartheta(\mu)^{\alpha}}{y_{\alpha}(\mu)}\right],$$

where $\mu$ runs over positive rapidities (with the sum over quasiparticle species implicit), $\varepsilon'\,d\mu/2\pi$ is the density-of-states factor built from the single-particle energy, $\vartheta$ is the filling function of the post-quench GGE saddle-point state, $T$ is the two-particle scattering kernel, and the prefactor 2 reflects the restriction of the rapidity integral to positive rapidities (for compatible initial states the filling function is reflection-symmetric, so the integral over the full rapidity axis is twice that over positive rapidities). The formula was supported by four nontrivial checks: it reproduces the free-fermion limit, the quasiparticle prediction in the $\alpha\to1$ limit (where, remarkably, the quasiparticle picture re-emerges from the TBA equations), the exact slopes previously known only for the Rule 54 automaton (Klobas and Bertini, 2021), and exact numerics for the XXZ spin-1/2 chain (the anisotropic Heisenberg chain, the paradigmatic interacting Bethe-ansatz-integrable lattice model). Rule 54 is a deterministic, reversible, interacting and integrable one-dimensional cellular automaton that is exactly solvable by transfer-matrix techniques and has effectively a single quasiparticle mode, which is why it serves as the only exactly solvable interacting benchmark.

The derivation nevertheless leaves three unproven steps, which the authors themselves flag. First, the identification of the slope with the stationary entropy of the dual system is conjectured: the fixed points $M_{L,t}$, $M_{R,t}$ generate a generally non-unitary, non-Hermitian dual dynamics whose thermodynamics has no rigorous characterisation outside dual-unitary circuits and Rule 54. Second, the dual single-particle momentum is generically non-monotonic, so the dual TBA equations are blind to the sign of $\varepsilon'(\mu)$; in the original paper this sign ambiguity was fixed only by demanding agreement with the exact Rule 54 result. Third, continuous-time Hamiltonian dynamics is reached only through a Trotter limit of the discrete-time construction (the limit of vanishing discrete time step at fixed real time), and the replica network couples $2n$ replicas through a cyclic shift, requiring the joint control of the limits $L\to\infty$, $t\to\infty$, integer replica index $n$ (continued to real $\alpha$) and the Trotter step. The paper concludes that its 'formula (57) for the slope of Rényi entropies in all TBA-integrable models has not been rigorously proven here', suggests the algebraic Bethe ansatz as the natural framework for such a proof, and notes that a proof 'would also lead to a rigorous validation of the quasiparticle conjecture in the replica limit $\alpha\to1$ — a problem that has been open since 2005'. It further conjectures a full-time interpolation formula, $S^{(\alpha)}_{A,\mathrm{conj}}(t)=\int_+ d\mu\,\min\left(2s_\alpha(\mu)t,\;d_\alpha(\mu)|A|\right)$ (with $d_\alpha(\mu)$ the mode-resolved stationary entropy density), tested only in Rule 54, and poses the multipartite extension: the logarithmic negativity (the mixed-state entanglement monotone $\mathcal{E}=\ln\|\rho_A^{T_1}\|_1$, the logarithm of the trace norm of the partially transposed reduced density matrix) and higher moments of the partial transpose are then understood only in free or dual-unitary settings, where quasiparticle reasoning predicts $\mathcal{E}\approx\tfrac12$ times the Rényi mutual information at $\alpha=1/2$ (Alba and Calabrese, 2019), a relation proven to hold only at times shorter than the subsystem sizes (Bertini, Klobas and Lu, 2022).

The subsequent literature, reviewed below, resolved the symmetry-resolved and inhomogeneous directions at the same conjecture level and removed the sign ambiguity by a rapidity re-parametrisation (Travaglino, Calabrese, Klobas and Bertini, 2026), but explicitly leaves the mathematical proof open; the whole framework is surveyed in the Reviews of Modern Physics article 'Exactly solvable quantum many-body dynamics from space-time duality' (Bertini, Claeys and Prosen, 2026).

Problem Statement

The problem is to complete the theory of Rényi-entropy growth after integrable quenches in interacting integrable models initiated by Bertini, Klobas, Alba, Lagnese and Calabrese (Phys. Rev. X 12, 031016 (2022)), by replacing its heuristic steps with rigorous mathematics and its broken quasiparticle picture with a controlled microscopic mechanism. Concretely:

(1) Rigorous foundation (primary). Prove, or rigorously refute, the spacetime-swap correspondence: for every one-dimensional model treatable by thermodynamic Bethe ansatz, quenched from a compatible integrable initial state, the asymptotic slope of the Rényi-$\alpha$ entropy of a large block, $s_{\alpha}=\lim_{t\to\infty}S_A^{(\alpha)}(t)/(2t)$, with the source's per-edge convention for the slope, is exactly the stationary Rényi-entropy density of the space-time dual system generated by the fixed points of the space transfer matrix (Eq. (16) of the source), and is therefore given by the TBA expressions (57)-(58) of the source (Eq. (50) of the source, the specialisation to diagonal-scattering relativistic quantum field theories, i.e. factorised scattering that does not mix quasiparticle species), together with their generalisations to inhomogeneous quenches and non-symmetric initial states derived in arXiv:2606.20445. This entails an exact, mathematically controlled characterisation of the thermodynamics of the fixed points $M_{L,t}$, $M_{R,t}$ of the space transfer matrix — a generally non-unitary, non-Hermitian dual dynamics with non-monotonic momentum — and a controlled treatment of the replica tensor-network representation of $\mathrm{tr}[\rho_A^{\alpha}]$ (with its cyclic shift of $2\alpha$ replicas), of the joint limits $L\to\infty$, $t\to\infty$, $\alpha$ real, and of the Trotter limit for continuous-time Hamiltonian dynamics, with no step relying on numerical agreement. A valid proof must, as a corollary, rigorously establish the quasiparticle-picture formula for von Neumann entropy growth in the $\alpha\to1$ limit (open since Calabrese and Cardy, 2005) and the exact free-fermion limit.

(2) Emergent microscopic picture for $\alpha\neq1$. Construct an emergent description of Rényi-$\alpha$ entanglement growth in interacting integrable systems — for instance quasiparticles propagating in the multi-replica space, or an equivalent effective description — that reduces to the standard quasiparticle picture at $\alpha=1$ and reproduces the TBA slope formulas (57)-(58) for $\alpha\neq1$.

(3) Full-time regime. Determine to what extent the conjectured full-time interpolation formula $S^{(\alpha)}_{A,\mathrm{conj}}(t)=\int_{+}d\mu\,\min\left(2s_{\alpha}(\mu)t,\,d_{\alpha}(\mu)|A|\right)$ (Eq. (90) of the source) is exact at leading order for large time and subsystem size beyond the single-mode Rule 54 evidence — in particular for multi-species models such as the XXZ spin-1/2 chain — either analytically or through controlled computation with quantified subleading corrections. For this component it suffices to establish leading-order exactness of the formula for one multi-species model (e.g. the XXZ spin-1/2 chain), with quantified subleading corrections, or to exhibit a controlled counterexample; a general multi-species statement beyond this is not required.

(4) Multipartite entanglement. Characterise the growth of the logarithmic negativity and of the higher moments of the partial transpose in interacting integrable models at times larger than the subsystem sizes, and settle whether the quasiparticle-reasoned relation (negativity $\approx$ half the $\alpha=1/2$ Rényi mutual information) survives in that regime.

The common objective of (1)-(4) is a single quantitative theory of Rényi-type entanglement growth in interacting integrable systems after integrable quenches: (1) supplies its rigorous foundation, (2) the microscopic mechanism that replaces the provably broken quasiparticle picture, and (3)-(4) its extension from the initial slope to the full time regime and to multipartite observables. All four components are part of this problem, with (1) the primary result and (2)-(4) also required, not optional. The inhomogeneous and non-symmetric generalisations named in (1) fall within it as well, since their rigorous foundation is the same unproven correspondence. The source's separate symmetry-resolved direction — symmetry-resolved Rényi entropies and charged moments — has been developed at the conjecture level by later work (Bertini, Calabrese, Collura, Klobas and Rylands, Phys. Rev. Lett. 131, 140401 (2023)) and, concerning a different observable, is not part of this problem's open core.

References

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