Exact relaxation in a class of non-equilibrium quantum lattice systems

TL;DR

This paper demonstrates that in a specific quantum lattice system, the Bose-Hubbard model, the system relaxes exactly to a non-thermal steady state after a quench, with relaxation properties proven for infinite and large finite systems.

Contribution

It provides an exact relaxation result for the Bose-Hubbard model after a quantum quench, including finite-size effects and relaxation for periodic initial states.

Findings

Local relaxation to a maximum entropy state with fixed second moments.

Exact relaxation occurs for all large times in the infinite system limit.

Finite systems exhibit a well-defined relaxation time interval.

Abstract

A reasonable physical intuition in the study of interacting quantum systems says that, independent of the initial state, the system will tend to equilibrate. In this work we study a setting where relaxation to a steady state is exact, namely for the Bose-Hubbard model where the system is quenched from a Mott quantum phase to the strong superfluid regime. We find that the evolving state locally relaxes to a steady state with maximum entropy constrained by second moments, maximizing the entanglement, to a state which is different from the thermal state of the new Hamiltonian. Remarkably, in the infinite system limit this relaxation is true for all large times, and no time average is necessary. For large but finite system size we give a time interval for which the system locally "looks relaxed" up to a prescribed error. Our argument includes a central limit theorem for harmonic systems and exploits the finite speed of sound. Additionally, we show that for all periodic initial configurations, reminiscent of charge density waves, the system relaxes locally. We sketch experimentally accessible signatures in optical lattices as well as implications for the foundations of quantum statistical mechanics.