Space-time duality approach to integrable quenches: exact characterisation of the fixed points of the space evolution and extension to equal-time two-point functions

The space-time duality approach (SDA) maps non-equilibrium dynamics in integrable quantum many-body systems to the equilibrium thermodynamics of a dual channel obtained by exchanging the roles of space and time, and it provides the only known closed-form predictions for the growth of Rényi entanglement entropies and for the full counting statistics of conserved charges after quenches in interacting integrable models, including spatially inhomogeneous (bipartitioning) protocols. A 2026 paper by Travaglino, Calabrese, Klobas and Bertini resolved the long-standing sign ambiguity of the dual thermodynamic Bethe ansatz but posed two open questions, which this problem records: (1) an exact, first-principles characterisation of the thermodynamic properties of the fixed points of the space evolution (the leading eigenoperators of the space transfer matrix), which would upgrade the existing closed-form predictions from conjectural to proven; and (2) whether the approach can be generalised to other probes of non-equilibrium dynamics that are currently beyond exact reach, most prominently equal-time two-point functions of local operators. As of September 2026 both questions are open: the only follow-up literature extends the framework to multi-time correlation functions of extensive charges and to entanglement-Hamiltonian cumulants while still assuming the conjectural fixed-point correspondence.

Background

A quantum quench — the sudden change of a Hamiltonian parameter followed by unitary evolution from a simple initial state — is the paradigmatic protocol of non-equilibrium many-body physics. In one-dimensional integrable models such as the XXZ spin-1/2 chain, the dynamics resolves into stable quasiparticles labeled by a rapidity $\mu$ carrying momentum $p(\mu)$, energy $\varepsilon(\mu)$, and interacting through a scattering kernel $K$ (the logarithmic derivative of the two-body scattering matrix). The thermodynamics of such models is known exactly through the thermodynamic Bethe ansatz (TBA), a set of integral equations for the filling function $\vartheta(\mu)$, the fraction of rapidity-$\mu$ states occupied. Genuine dynamical quantities after a quench are far harder: only the late-time hydrodynamic regime (generalised hydrodynamics, the Euler-scale theory built on local stationary ensembles such as the generalised Gibbs ensemble — the ensemble constructed from all local conserved charges) and the von Neumann entanglement entropy, described by the quasiparticle picture of entangled pairs propagated ballistically (extended to interacting integrable models by Alba, Bertini and Fagotti), are under analytic control.

The space-time duality approach (SDA) produces exact closed-form dynamical predictions for a class of previously inaccessible quantities. Writing expectation values as two-dimensional tensor networks of the doubled (bra-ket) evolution, one may contract the network in the spatial direction instead of the temporal one. In discrete-time models (quantum circuits or Trotter discretisations of Hamiltonians) this is implemented by the space transfer matrix $\mathbb{T}$: a linear superoperator (quantum channel) acting along space on the doubled time lattice. Its leading left and right eigenoperators $L$ and $R$ — the fixed points of the space evolution, unique under mild conditions on the initial state — define a dual stationary state $\tilde{\rho}_{\rm st}=RL$. In the thermodynamic limit, one-point functions, the Rényi entropies $S_\alpha=(1/(1-\alpha))\log\operatorname{Tr}\rho_A^\alpha$ of a half chain (which measure the entanglement between two halves of the system), and the full counting statistics of a conserved charge ($Z_\beta(t)=\langle e^{\beta Q_A(t)}e^{-\beta Q_A}\rangle$, the generating function of charge fluctuations in a subsystem $A$) are all rewritten exactly as equilibrium-like quantities of this dual system.

The working hypothesis of the SDA is the conjectured correspondence $\operatorname{tr}[\tilde{\rho}^{\alpha}_{{\rm st},t}\mathcal{O}]\longleftrightarrow\operatorname{tr}[\rho^{\alpha}_{{\rm st},x}\mathcal{O}]$ under the exchange of space and time ($t\leftrightarrow x$): the thermodynamics of the dual stationary state is that of the regular stationary state $\rho_{{\rm st},x}$ of the time evolution on a subsystem of length $x$, with the filling function preserved and momentum and energy exchanged, $p(\mu)\leftrightarrow\varepsilon(\mu)$. Introduced by Bertini, Klobas, Alba, Lagnese and Calabrese to obtain the asymptotic growth rate of Rényi entropies after quenches in interacting integrable models — where it also demonstrated the breakdown of the quasiparticle picture for $\alpha\neq1$ — the rule was extended to charge fluctuations and symmetry-resolved entanglement, and to entanglement negativity. Subsequent work uncovered an intrinsic ambiguity: since the standard TBA requires a strictly monotonic single-particle momentum $p'(\mu)>0$, the dual equations cannot distinguish insertions of $\operatorname{sgn}(\varepsilon'(\mu))$ (the sign of the bare velocity) from insertions of $\operatorname{sgn}(v^{\rm dr}(\mu))$ (the sign of the dressed, interaction-renormalised quasiparticle velocity). The two choices differ precisely when quasiparticle trajectories are curved, i.e. after quenches from initial states that are not symmetric under the conserved charge or in spatially inhomogeneous settings; fluctuating-hydrodynamics analyses reached the related conclusion that ballistic fluctuation theory (the exact theory of charge fluctuations on translation-invariant states) fails for post-quench full counting statistics in these regimes.

The source paper (Travaglino, Calabrese, Klobas, Bertini, arXiv:2606.20445, June 2026) resolves this ambiguity from first principles. Via a TBA construction valid when the quasiparticle momentum is non-monotonic in the rapidity (an injective but non-monotonic rapidity reparametrisation), it fixes the sign structure uniquely and obtains closed-form expressions for the asymptotic slopes $s_\alpha(\zeta)$ of the Rényi entropies and the full-counting-statistics exponents $z_\beta(\zeta)$, both for homogeneous quenches and for inhomogeneous bipartitioning protocols (two different integrable initial states joined at a point; $\zeta=x/t$ is the ray). The predictions reproduce all known exact results — free-fermion limits, the exact solution of Rule 54 (an interacting integrable quantum cellular automaton, i.e. a discrete-time lattice model whose quench dynamics and fixed points are known exactly), and, at $\alpha\to1$, the quasiparticle-picture formula with curved trajectories — and agree with extensive TEBD simulations (time-evolving block decimation, a matrix-product-state simulation algorithm) of the XXZ chain.

Two gaps remain. First, the predictions are still built on the conjectured correspondence: for Bethe-ansatz interacting models the fixed points $L$ and $R$ of the space evolution have never been characterised from first principles — this is available only in Rule 54, in dual-unitary circuits (a class of quantum circuits exactly solvable in both space and time), and in free models — so the closed forms are physically derived rather than proven. Second, the SDA covers only one-point-like thermodynamic observables (entropies, charge fluctuations); equal-time two-point functions of local operators after interacting integrable quenches, a fundamental probe of correlations, transport and relaxation, are beyond the reach of every existing exact method, especially with spatial inhomogeneity.

Problem Statement

Establish the space-time duality approach (SDA) to (inhomogeneous) integrable quenches as a rigorous and general exact framework, by solving the two problems posed in the conclusions of arXiv:2606.20445.

(1) Fixed points of the space evolution. Develop an exact characterisation, from first principles, of the thermodynamic properties of the fixed points of the space evolution in interacting integrable models — the leading left and right eigenoperators $L$ and $R$ of the space transfer matrix $\mathbb{T}$, whose product $\tilde{\rho}_{\rm st}=RL$ plays the role of the stationary state of the dual (space-time swapped) theory. The characterisation must be sufficient to prove the conjectured correspondence $\operatorname{tr}[\tilde{\rho}^{\alpha}_{{\rm st},t}\mathcal{O}]\longleftrightarrow\operatorname{tr}[\rho^{\alpha}_{{\rm st},x}\mathcal{O}]$ (filling function preserved, $p(\mu)\leftrightarrow\varepsilon(\mu)$), and thereby the closed-form predictions of the source paper for the asymptotic slopes $s_\alpha(\zeta)$ of the Rényi entropies and the full-counting-statistics exponents $z_\beta(\zeta)$, in homogeneous quenches and in inhomogeneous bipartitioning protocols.

(2) Extension to equal-time two-point functions. Determine whether the SDA can be generalised to equal-time two-point functions of local operators after integrable quenches — observables for which no closed-form exact results exist in interacting integrable systems, especially in the presence of spatial inhomogeneity. A complete answer either (i) derives SDA-based closed-form expressions for such correlation functions, identifying their domain of validity (model class, initial-state class, protocol, time regime) and checking consistency with known limits (free models, Rule 54, stationary and Euler-scale correlation results), or (ii) establishes a rigorous obstruction showing that the dual-channel framework does not extend to these observables.

The two parts share one objective: part (1) secures the rigorous status of the existing SDA predictions, part (2) extends the same correspondence and dual stationary state to the next class of nonequilibrium observables; together they address whether the SDA is an exact, general description of interacting integrable dynamics beyond the hydrodynamic regime.

References

1. R. Travaglino, P. Calabrese, K. Klobas, and B. Bertini, 'Space-time duality approach to (inhomogeneous) integrable quenches' (2026), arXiv:2606.20445, DOI: 10.48550/arxiv.2606.20445, https://arxiv.org/abs/2606.20445 (source paper)
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