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arXiv:1305.7202 · Physical Review B

Interaction quenches in the one-dimensional Bose gas

Márton Kormos, Aditya Shashi, Yang-Zhi Chou, Jean-Sébastien Caux, Adilet Imambekov

Published Nov 22, 2013• 136 citations
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Abstract

The nonequilibrium dynamics of integrable systems are highly constrained by the conservation of certain charges. There is substantial evidence that after a quantum quench they do not thermalize but their asymptotic steady state can be described by a generalized Gibbs ensemble (GGE) built from the conserved charges. Most of the studies on the GGE so far have focused on models that can be mapped to quadratic systems, while analytic treatment in nonquadratic systems remained elusive. We obtain results on interaction quenches in a nonquadratic continuum system, the one-dimensional (1D) Bose gas described by the integrable Lieb-Liniger model. The direct implementation of the GGE prescription is prohibited by the divergence of the conserved charges, which we conjecture to be endemic to any continuum integrable systems with contact interactions undergoing a sudden quench. We compute local correlators for a noninteracting initial state and arbitrary final interactions as well as two-point functions for quenches to the Tonks-Girardeau regime. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble, allowing for an experimental verification in cold-atom systems.

DOI
10.1103/physrevb.88.205131

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The nonequilibrium dynamics of integrable systems are highly constrained by the conservation of certain charges. There is substantial evidence that after a quantum quench they do not thermalize but their asymptotic steady state can be described by a generalized Gibbs ensemble (GGE) built from the conserved charges. Most of the studies on the GGE so far have focused on models that can be mapped to quadratic systems, while analytic treatment in nonquadratic systems remained elusive. We obtain results on interaction quenches in a nonquadratic continuum system, the one-dimensional (1D) Bose gas described by the integrable Lieb-Liniger model. The direct implementation of the GGE prescription is prohibited by the divergence of the conserved charges, which we conjecture to be endemic to any continuum integrable systems with contact interactions undergoing a sudden quench. We compute local correlators for a noninteracting initial state and arbitrary final interactions as well as two-point functions for quenches to the Tonks-Girardeau regime. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble, allowing for an experimental verification in cold-atom systems.
We obtain exact results on interaction quenches in the 1D Bose gas described by the integrable Lieb-Liniger model. We show that in the long time limit integrability leads to significant deviations from the predictions of the grand canonical ensemble and a description within the generalized Gibbs ensemble (GGE) is needed. For a non-interacting initial state and arbitrary final interactions, we find that the presence of infinitely many conserved charges generates a non-analytic behavior in the equilibrated density of quasimomenta. This manifests itself in a dynamically generated Friedel-like oscillation of the non-local correlation functions with interaction dependent oscillation momenta. We also exactly evaluate local correlations and the generalized chemical potentials within GGE.
One of the fundamental principles of statistical physics is that only partial information about a system's state is required for its macroscopic description. This is not only true for thermal ensembles, but also for the unconventional ensemble, known as Generalized Gibbs Ensemble (GGE), that is expected to describe the relaxation of integrable systems after a quantum quench. By analytically studying the quench dynamics in a prototypical one-dimensional critical model, the massless free bosonic field theory, we find evidence of a novel type of equilibration characterized by the preservation of an enormous amount of memory of the initial state that is accessible by local measurements. In particular, we show that the equilibration retains memory of non-Gaussian initial correlations, in contrast to the case of massive free evolution which erases all such memory. The GGE in its standard form, being a Gaussian ensemble, fails to predict correctly the equilibrium values of local observables, unless the initial state is Gaussian itself. Our findings show that the equilibration of a broad class of quenches whose evolution is described by Luttinger liquid theory with an initial state that is non-Gaussian in terms of the bosonic field, is not correctly captured by the corresponding bosonic GGE, raising doubts about the validity of the latter in general one-dimensional gapless integrable systems such as the Lieb-Liniger model. We also propose that the same experiment by which the GGE was recently observed [Langen et al., Science 348 (2015) 207-211] can also be used to observe its failure, simply by starting from a non-Gaussian initial state.
We investigate the relaxation dynamics of the integrable Lieb-Liniger model of contact-interacting bosons in one dimension following a sudden quench of the collisional interaction strength. The system is initially prepared in its noninteracting ground state and the interaction strength is then abruptly switched to a positive value, corresponding to repulsive interactions between the bosons. We calculate equal-time correlation functions of the nonequilibrium Bose field for small systems of up to five particles via symbolic evaluation of coordinate Bethe-ansatz expressions for operator matrix elements between Lieb-Liniger eigenstates. We characterize the relaxation of the system by comparing the time-evolving correlation functions following the quench to the equilibrium correlations predicted by the diagonal ensemble and relate the behavior of these correlations to that of the quantum fidelity between the many-body wave function and the initial state of the system. Our results for the asymptotic scaling of local second-order correlations with increasing interaction strength agree with the predictions of recent generalized thermodynamic Bethe-ansatz calculations. By contrast, third-order correlations obtained within our approach exhibit a markedly different power-law dependence on the interaction strength as the Tonks-Girardeau limit of infinitely strong interactions is approached.
In the majority of analytical verifications of the conjecture that the Generalized Gibbs Ensemble (GGE) describes the large time asymptotics of local observables in quantum quench problems, both post- and pre-quench Hamiltonians are essentially noninteracting. We test this conjecture studying field correlations in the more general case of an arbitrary pre-quench Hamiltonian, while keeping the post-quench one noninteracting. We first show that, in the previously studied special case of a noninteracting pre-quench Hamiltonian, the validity of the conjecture is a consequence of Wick's theorem. We then show that this conjecture is more generally valid for an arbitrary interacting pre-quench Hamiltonian, but this time as a consequence of the cluster decomposition property of the initial state, which is a fundamental principle for generic physical states. For arbitrary initial states that do not satisfy the cluster decomposition property, the above conjecture is not generally true. As a byproduct of our investigation we obtain an analytical derivation of earlier numerical results for the large time evolution of correlations after a quantum quench of the interaction in the Lieb–Liniger model from a nonzero value to zero.
It is widely believed that the stationary properties after a quantum quench in integrable systems can be described by a generalized Gibbs ensemble (GGE), even if all of the analytical evidence is based on free theories in which the pre- and postquench modes are linearly related. In contrast, we consider the experimentally relevant quench of the one-dimensional Bose gas from zero to infinite interaction, in which the relation between modes is nonlinear, and consequently Wick's theorem does not hold. We provide exact analytical results for the time evolution of the dynamical density-density correlation function at any time after the quench and we prove that its stationary value is described by a GGE in which Wick's theorem is restored.
Abstract We revisit the question of whether and how the steady states arising after non-equilibrium time evolution in integrable models (and in particular in the XXZ spin chain) can be described by the so-called generalized Gibbs ensemble (GGE). Whereas it is known that the micro-canonical ensemble built on a complete set of charges correctly describes the long-time limit of local observables, it has been shown recently by Ilievski et al that the corresponding canonical ensemble is not well defined, and instead a different canonical ensemble was proposed in terms of particle occupation number operators. Here we provide an alternative construction by considering truncated GGEs (tGGEs) that include only a finite number of local and quasi-local conserved operators. It is shown that the tGGEs can approximate the steady states with arbitrary precision, i.e. all physical observables are exactly reproduced in the infinite truncation limit. We trace back the problems encountered in defining an untruncated GGE to the dependence of the associated Lagrange multipliers on the truncation index. Conversely, we show that this problem may be circumvented by considering a new set of (quasi)local charges which are linear combinations of the standard ones, and whose associated Lagrange multipliers are well-defined state functions. Our general arguments are applied to concrete quench situations in the XXZ chain, where the initial states are simple two-site or four-site product states. Depending on the quench we find that numerical results for the local correlators can be obtained with remarkable precision using truncated GGEs with only 10–100 charges.