On generalized Gibbs ensembles with an infinite set of conserved charges

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.