Hydrodynamic entropy production versus entanglement growth in a trapped one-dimensional Bose gas thermalising via diffusion

Bastianello, De Luca, Doyon and De Nardis [Phys. Rev. Lett. 125, 240604 (2020)] showed that a one-dimensional Bose gas in a smooth trapping potential thermalises because the potential breaks integrability and the diffusive term of generalized hydrodynamics produces Yang–Yang thermodynamic entropy, driving the gas from a prethermal generalized Gibbs ensemble to a Gibbs state at diffusive times $t\sim L^{2}$. In their conclusion the authors pose, as an open question for the future, how this hydrodynamic thermodynamic entropy production relates to the well-studied growth of entanglement entropy after quantum quenches.

Background

The Lieb–Liniger model describes one-dimensional bosons with repulsive contact interactions. It is integrable: it possesses an infinite family of local conserved charges, and its excitations are stable quasiparticles labelled by a rapidity $\theta$ (the quasiparticle momentum in the Bethe-ansatz parametrization). The thermodynamics of a stationary state is encoded in the Yang–Yang entropy, the entropy density of the Bethe-ansatz ensemble of quasiparticle occupancies; after a quench, an integrable system relaxes not to a Gibbs ensemble but to a generalized Gibbs ensemble (GGE) that maximizes this entropy subject to all conserved charges. At large scales, inhomogeneous states of integrable systems are described by generalized hydrodynamics (GHD): the densities $\rho_{\theta}(x,t)$ of quasiparticles of rapidity $\theta$ obey a continuity equation in the joint space-rapidity space, with each species transported at the dressed velocity $v^{\rm eff}_{\theta}$, the group velocity of a quasiparticle shifted from its bare value by interactions with the surrounding quasiparticles. Two refinements matter here. First, a smooth external potential $V(x)$ coupling to the particle density adds a force term $(\partial_{x}V)\partial_{\theta}\rho_{\theta}$; since the potential breaks integrability, it is the only integrability-breaking agent. Second, thermal fluctuations within fluid cells generate a diffusion matrix $\mathfrak{D}$ (a Navier–Stokes-type correction). Bastianello, De Luca, Doyon and De Nardis combined both into the equation

$$\partial_{t}\rho_{\theta}=-\partial_{x}\big(v^{\rm eff}_{\theta}\rho_{\theta}\big)+\partial_{x}\big(\mathfrak{D}^{\alpha}_{\theta}\,\partial_{x}\rho_{\alpha}\big)+(\partial_{x}V)\,\partial_{\theta}\rho_{\theta}$$

(their Eq. (3)), valid up to $O(L^{-3})$ corrections, with $L$ the minimal length scale of spatial variations of the state and of the potential. Diffusion increases the total Yang–Yang entropy at the rate

$$\frac{dS}{dt}=\frac{1}{2}\int dx\sum_{j,k}(\partial_{x}\beta_{j})\,h^{\theta}_{j}\,(\mathfrak{D}C)_{\theta\alpha}\,h^{\alpha}_{k}\,(\partial_{x}\beta_{k})$$

(their Eq. (4)), where $\beta_{j}$ are the Lagrange multipliers of the local GGE, $h^{\theta}_{j}$ the one-particle eigenvalues of the conserved charges, $C$ the susceptibility matrix, and sums over the repeated rapidity indices $\theta,\alpha$ are implied. Because the only null direction of $\mathfrak{D}$ is the constant function of rapidity, the only stationary states of Eq. (3) are Gibbs states: a trapped gas first forms a prethermal GGE at Euler times $t\sim L$ and then thermalises at diffusive times $t\sim L^{2}$, with long polynomial tails caused by the continuum of modes with arbitrarily small diffusion constants. This resolved the decade-old puzzle of the apparent lack of thermalisation of trapped one-dimensional Bose gases (the 'quantum Newton's cradle'). In parallel, the growth of entanglement entropy — the von Neumann entropy of the reduced density matrix of a spatial subregion, the standard measure of its quantum correlations with the rest of the system — after quantum quenches is understood in integrable systems through the quasiparticle picture: entangled quasiparticle pairs created by the quench propagate ballistically, and the entanglement of a region is the entropy of the pairs it shares with its complement. Alba and Calabrese showed that after homogeneous quenches the stationary entanglement entropy of a large subsystem equals the thermodynamic (Yang–Yang) entropy of the GGE, an exact correspondence between quantum entanglement and thermodynamic entropy, and Bertini, Fagotti, Piroli and Calabrese as well as Alba, Bertini and Fagotti extended the quasiparticle picture to inhomogeneous and interacting settings at the Euler scale. These constructions fix the entanglement carried by each quasiparticle pair through the local Yang–Yang entropy, but they describe reversible, entropy-conserving dynamics. What happens to entanglement in the regime where the trap's integrability breaking produces genuine thermodynamic entropy — where quasiparticles diffuse, occupancies drift, and the state thermalises — is not understood. In nearly integrable homogeneous chains, the entanglement entropy is known numerically to relax in two steps, first to the GGE value and then by a slow drift toward the thermal value on timescales inversely proportional to the square of the integrability-breaking strength, but no relation to a hydrodynamic entropy-production functional is known.

Problem Statement

In the setting of Bastianello, De Luca, Doyon and De Nardis, Phys. Rev. Lett. 125, 240604 (2020) — a repulsive Lieb–Liniger gas in a smooth external potential $V(x)$ whose integrability is broken only by the trap — the large-scale dynamics is the convection–diffusion–force equation

$$\partial_{t}\rho_{\theta}=-\partial_{x}\big(v^{\rm eff}_{\theta}\rho_{\theta}\big)+\partial_{x}\big(\mathfrak{D}^{\alpha}_{\theta}\,\partial_{x}\rho_{\alpha}\big)+(\partial_{x}V)\,\partial_{\theta}\rho_{\theta}$$

and the Yang–Yang thermodynamic entropy grows at the rate

$$\frac{dS}{dt}=\frac{1}{2}\int dx\sum_{j,k}(\partial_{x}\beta_{j})\,h^{\theta}_{j}\,(\mathfrak{D}C)_{\theta\alpha}\,h^{\alpha}_{k}\,(\partial_{x}\beta_{k})$$

where $\beta_{j}$ are the Lagrange multipliers of the local generalized Gibbs ensemble, $h^{\theta}_{j}$ the one-particle eigenvalues of the conserved charges, $C$ the susceptibility matrix, and sums over the repeated rapidity indices $\theta,\alpha$ are implied. Determine how this hydrodynamic entropy production relates to the growth of the entanglement entropy of the evolving pure state. Concretely: (a) Derive, or disprove the existence of, a hydrodynamic or quasiparticle-based description of the bipartite entanglement entropy $S_{A}(t)$ of a macroscopic spatial region $A$, valid through the Euler-to-diffusive window in which the local macrostate drifts from the prethermal generalized Gibbs ensemble toward the Gibbs state (i.e. across times $t\sim L$ to $t\sim L^{2}$, within the validity of the equation above). (b) Establish whether $S_{A}(t)$, or its growth rate, is given by, bounded by, or otherwise quantitatively controlled by the Yang–Yang thermodynamic entropy of the evolving macrostate and the entropy-production functional above — that is, whether and how the thermodynamic-entropy/entanglement correspondence established at the Euler scale for homogeneous quenches extends to the thermalising regime of an integrability-broken trapped gas. Together, parts (a) and (b) constitute one scientific objective: the relation between entropy production and entanglement growth. A complete answer consists of an exact or controlled-asymptotic relation (an equality, a bound, or an explicit formula with identified corrections) valid in the stated scaling limit, or a demonstrated failure (no-go) with a concrete mechanism that excludes such a relation generically within the validity of the equation above, rather than only in a specially chosen regime. A demonstrated failure of the alternative offered in part (a) — a proof, valid in the stated scaling limit and not only in a specially chosen regime, that no hydrodynamic or quasiparticle-based description of $S_{A}(t)$ exists there — resolves part (a) and may complete the task only if it also states what that absence implies for part (b): whether a quantitative relation between $S_{A}(t)$ and the Yang–Yang thermodynamic entropy of the evolving macrostate can nevertheless hold. Consistency with known Euler-scale limits — the stationary Alba–Calabrese correspondence and the bipartite entanglement-production rates of inhomogeneous quenches — is part of the target. The paradigmatic case — and the case whose resolution completes the task — is the trapped Lieb–Liniger gas; generalization to other integrable models in smooth external potentials is optional.

References

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