Approach to Equilibrium of a Restricted Class of Isolated Quantum Systems After a Quench

TL;DR

This paper proves that certain non-ergodic isolated quantum systems with specific algebraic properties approach a generalized Gibbs ensemble after a quench, demonstrating a pathway to equilibrium in non-ergodic quantum systems.

Contribution

It provides a rigorous proof of equilibrium approach for a restricted class of non-ergodic quantum systems using entropy operator evolution and algebraic conditions.

Findings

Systems with degenerate spectra approach a generalized Gibbs ensemble

The approach is demonstrated through entropy operator analysis

Results apply to systems with specific algebraic commutator relations

Abstract

We prove the approach to equilibrium of quenched isolated quantum systems for which the change in the Hamiltonian brought about by the quench satisfies a certain closed commutator algebra with all the extensive integrals of motion of the system before the quench. The proof is carried out by following the exact unitary evolution of the entropy operator, defined as the negative of the logarithm of the nonequilibrium density matrix, and showing that, under the conditions implied by the assumed algebra, this entropy operator becomes, at infinite times, a linear combination of integrals of motion of the perturbed system. That is, we show how the nonequilibrium density matrix approaches a generalized Gibbs ensemble. The restricted class of systems for which the present results apply turn out to have degenerate spectra in general, as opposed to some generic systems for which a kind of ergodicity is expected, with a nontrivial dynamics believed to find instances in one-dimensional infinite super-integrable systems. Our findings constitute a direct demonstration of how a non-ergodic isolated quantum system may get to statistical equilibrium.