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arXiv:2105.05054 · Journal of Physics A Mathematical and Theoretical

Exact entanglement growth of a one-dimensional hard-core quantum gas during a free expansion

Stefano Scopa, Alexandre Krajenbrink, Pasquale Calabrese, Jérôme Dubail

Published Aug 25, 2021• 41 citations
arXivHTMLPDF

Abstract

Abstract We consider the non-equilibrium dynamics of the entanglement entropy of a one-dimensional quantum gas of hard-core particles, initially confined in a box potential at zero temperature. At t = 0 the right edge of the box is suddenly released and the system is let free to expand. During this expansion, the initially correlated region propagates with a non-homogeneous profile, leading to the growth of entanglement entropy. This setting is investigated in the hydrodynamic regime, with tools stemming from semi-classical Wigner function approach and with recent developments of quantum fluctuating hydrodynamics. Within this framework, the entanglement entropy can be associated to a correlation function of chiral twist-fields of the conformal field theory that lives along the Fermi contour and it can be exactly determined. Our predictions for the entanglement evolution are found in agreement with and generalize previous results in literature based on numerical calculations and heuristic arguments.

DOI
10.1088/1751-8121/ac20ee

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Abstract We consider the non-equilibrium dynamics of the entanglement entropy of a one-dimensional quantum gas of hard-core particles, initially confined in a box potential at zero temperature. At t = 0 the right edge of the box is suddenly released and the system is let free to expand. During this expansion, the initially correlated region propagates with a non-homogeneous profile, leading to the growth of entanglement entropy. This setting is investigated in the hydrodynamic regime, with tools stemming from semi-classical Wigner function approach and with recent developments of quantum fluctuating hydrodynamics. Within this framework, the entanglement entropy can be associated to a correlation function of chiral twist-fields of the conformal field theory that lives along the Fermi contour and it can be exactly determined. Our predictions for the entanglement evolution are found in agreement with and generalize previous results in literature based on numerical calculations and heuristic arguments.
Abstract We study the entanglement dynamics of a one-dimensional hard-core quantum gas initially confined in a box of size L with saturated density ρ = 1. The gas is suddenly released into a region of size 2 L by moving one of the box edges. We show that the analytic prediction for the entanglement entropy obtained from quantum fluctuating hydrodynamics holds quantitatively true even after several reflections of the gas against the box edges. We further investigate the long time limit t / L ≫ 1 where a Floquet picture of the non-equilibrium dynamics emerges and hydrodynamics eventually breaks down.
We investigate the entanglement properties of the nonequilibrium dynamics of one-dimensional noninteracting Fermi gases released from a trap. The gas of NNN particles is initially in the ground state within a hard-wall or a harmonic trap, and then it expands after dropping the trap. We compute the time dependence of the von Neumann and R\'enyi entanglement entropies and the particle fluctuations of spatial intervals around the original trap in the limit of a large number NNN of particles. The results for these observables apply to one-dimensional gases of impenetrable bosons as well. We identify different dynamical regimes at small and large times, depending also on the initial condition, i.e., whether it is that of a hard-wall or a harmonic trap. In particular, we show analytically that the expansion from hard-wall traps is characterized by the asymptotic small-time behavior $S\ensuremath{\approx}(1/3)\mathrm{ln}(1/t)$ of the von Neumann entanglement entropy and the relation $S\ensuremath{\approx}{\ensuremath{\pi}}^{2}V/3$ (where VVV is the particle variance), which are analogous to the equilibrium behaviors whose leading logarithms are essentially determined by the corresponding conformal field theory with central charge c=1c=1c=1. The time dependence of the entanglement entropy of extended regions during the expansion from harmonic traps has a remarkable property in that it can be expressed as a global time-dependent rescaling of the space dependence of the initial equilibrium entanglement entropy.
We consider an analytically tractable model that exhibits the main features of the Page curve characterizing the evolution of entanglement entropy during evaporation of a black hole. Our model is a gas of noninteracting fermions on a lattice that is released from a box into the vacuum. More precisely, our Hamiltonian is a tight-binding model with a defect at the junction between the filled box and the vacuum. In addition to the entanglement entropy we consider several other observables, such as the spatial density profile and current, and show that the semiclassical approach of generalized hydrodynamics provides a remarkably accurate description of the quantum dynamics including that of the entanglement entropy at all times t. Our hydrodynamic results agree closely with those obtained via exact microscopic numerics. We find that the growth of entanglement with time is linear and universal, i.e, independent of the details of the defect. The decay shows 1/t scaling for conformal defect while for nonconformal defects, it is slower. Our analytical results from hydrodynamics enables us to make predictions for long time behaviors of observables that are otherwise highly inaccessible computationally by exact microscopic numerics. Our study shows the power of the semiclassical approach and could be relevant for discussions on the resolution of the black hole information paradox.
A recent work considered the Page curve for entanglement entropy of a freely expanding one-dimensional Fermi gas starting from a finite box (the system) in a zero-temperature initial state that then expands in to a semi-infinite vacuum (the environment). For the case with a single defect at the system-environment junction, it was shown that the Yang-Yang entropy of the system computed from semiclassical hydrodynamics agrees with the microscopically computed entanglement entropy, yielding the Page curve. On the other hand, the environment Yang-Yang entropy showed a monotonic increase. Here, we propose a hydrodynamic description to obtain the eigenvalues of the microscopic correlation matrix for a nonzero temperature freely-expanding Fermi gas. For the system, the phase space density directly gives the correlation matrix eigenvalues and thus, via the Yang-Yang formula, yields the von-Neumann entropy. For the reservoir, a modification of the phase space density is required in the Yang-Yang formula, to incorporate coherences, in order to reproduce accurately the entropy. The hydrodynamic description of correlation matrix eigenvalues also leads to a Levitov-Lesovik type formula for the full-counting-statistics for a finite system in the transient regime. We provide a heuristic derivation of the correlation matrix eigenvalues based on a semiclassical wave-packet picture.
Abstract We apply the theory of quantum generalized hydrodynamics (QGHD) introduced in (2020 Phys. Rev. Lett. 124 140603) to derive asymptotically exact results for the density fluctuations and the entanglement entropy of a one-dimensional trapped Bose gas in the Tonks–Girardeau (TG) or hard-core limit, after a trap quench from a double well to a single well. On the analytical side, the quadratic nature of the theory of QGHD is complemented with the emerging conformal invariance at the TG point to fix the universal part of those quantities. Moreover, the well-known mapping of hard-core bosons to free fermions, allows to use a generalized form of the Fisher–Hartwig conjecture to fix the non-trivial spacetime dependence of the ultraviolet cutoff in the entanglement entropy. The free nature of the TG gas also allows for more accurate results on the numerical side, where a higher number of particles as compared to the interacting case can be simulated. The agreement between analytical and numerical predictions is extremely good. For the density fluctuations, however, one has to average out large Friedel oscillations present in the numerics to recover such agreement.
Abstract We investigate the non-equilibrium dynamics of the symmetry-resolved Rényi entropies in a one-dimensional gas of non-interacting spinless fermions by means of quantum generalised hydrodynamics, which recently allowed to obtain very accurate results for the total entanglement in inhomogeneous quench settings. Although our discussion is valid for any quench setting accessible with quantum generalised hydrodynamics, we focus on the case of a quantum gas initially prepared in a bipartite fashion and subsequently let evolve unitarily with a hopping Hamiltonian. For this system, we characterise the symmetry-resolved Rényi entropies as function of time t and of the entangling position x along the inhomogeneous profile. We observe an asymptotic logarithmic growth of the charged moments at half system and an asymptotic restoration of equipartition of entropy among symmetry sectors with deviations which are proportional to the square of the inverse of the total entropy.