Complete Generalized Gibbs Ensembles in an Interacting Theory

Abstract

In integrable many-particle systems, it is widely believed that the stationary state reached at late times after a quantum quench can be described by a generalized Gibbs ensemble (GGE) constructed from their extensive number of conserved charges. A crucial issue is then to identify a complete set of these charges, enabling the GGE to provide exact steady-state predictions. Here we solve this long-standing problem for the case of the spin-1=2 Heisenberg chain by explicitly constructing a GGE which uniquely fixes the macrostate describing the stationary behavior after a general quantum quench. A crucial ingredient in our method, which readily generalizes to other integrable models, are recently discovered quasilocal charges. As a test, we reproduce the exact postquench steady state of the Néel quench problem obtained previously by means of the Quench Action method. Introduction.-Understanding and describing the equilibration of isolated many-particle systems is one of the main current challenges of quantum physics. The presence of higher conserved charges (above the Hamiltonian) is linked to the absence of full relaxation to a thermalized state; the conjectured appropriate framework to characterize the steady-state properties in such a situation is the generalized Gibbs ensemble (GGE) [1], in which all available charges are ascribed an individual "chemical potential" set by the initial conditions, and the steady state is the maximal entropy state fulfilling all the constraints associated to the conserved charges . The basic idea underlying the GGE is as follows. Let H ≃ H ð1Þ be the Hamiltonian of an integrable model, and fH ðnÞ g a set of conserved charges fulfilling ½H ðnÞ ; H ðmÞ ¼ 0. The situation we are interested in is that of a quantum quench, where we initially prepare our system in the ground state jΨð0Þi of a local Hamiltonian H 0 and then consider unitary time evolution with respect to our integrable Hamiltonian

Overlap distributions for quantum quenches in the anisotropic Heisenberg chain

“…The question of thermalization can then reduce to the question of how narrow in energy the overlap distribution is. Recent work (such as the "quench action" approach [82][83][84][85][86][87][88] and an approach based on linked-cluster expansions [89,90]) has formulated calculations of long-time expectation values directly in the thermodynamic limit, thus avoiding the explicit calculation of overlaps in finite-size systems.…”

General finite-size effects for zero-entropy states in one-dimensional quantum integrable models

“…Important in this respect is the idea that correlations at late times can be computed in a generalized Gibbs ensemble (GGE) [6,7] defined not just by the Hamiltonian, but rather by all [8,9] conserved (quasi) local quantitieŝ…”

“…Equivalently, correlations can be computed on a single representative eigenstate which can be determined by reasonings paralleling (generalizing) the thermodynamic Bethe ansatz (gTBA) or by the Quench Action (QA) method [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,8,9,27]. In the latter, one constructs a free-energy functional straight from the overlaps of the initial state with the eigenstates of the time evolution Hamiltonian H [15,27].…”

Quasilocal charges and progress towards the complete GGE for field theories with nondiagonal scattering

“…In this perspective, it has recently been observed in the context of quantum spin chains that a GGE built out of the traditionally known set of local charges does not completely reproduce the steady-state properties [58][59][60][61][62], and should instead be complemented by newly discovered quasilocal charges [66][67][68][69] whose construction follows from the seminal works of Prosen and collaborators [64,65]. This has further been related to the fact that local charges do not provide any information about the densities of various strings, which is instead contained in quasilocal charges [72][73][74][75].…”