Generalized microcanonical and Gibbs ensembles in classical and quantum integrable dynamics

TL;DR

This paper establishes the validity of the Generalized Gibbs Ensemble (GGE) in classical and quantum integrable systems, extending classical results to infinite degrees of freedom and defining quantum integrability via maximal Hamiltonians.

Contribution

It proves the GGE for classical systems with infinite degrees of freedom and introduces quantum integrability as maximal Hamiltonians, contrasting their dynamics with random matrix models.

Findings

GGE holds for classical integrable systems in the thermodynamic limit.

Quantum integrability characterized by maximal Hamiltonians is established.

Quantum quench dynamics differ from random matrix predictions.

Abstract

We prove two statements about the long time dynamics of integrable Hamiltonian systems. In classical mechanics, we prove the microcanonical version of the Generalized Gibbs Ensemble (GGE) by mapping it to a known theorem and then extend it to the limit of infinite number of degrees of freedom. In quantum mechanics, we prove GGE for maximal Hamiltonians - a class of models stemming from a rigorous notion of quantum integrability understood as the existence of conserved charges with prescribed dependence on a system parameter, e.g. Hubbard U, anisotropy in the XXZ model etc. In analogy with classical integrability, the defining property of these models is that they have the maximum number of independent integrals. We contrast their dynamics induced by quenching the parameter to that of random matrix Hamiltonians.