Emergent statistical mechanics from properties of disordered random matrix product states

TL;DR

This paper investigates the properties of disordered random matrix product states, revealing their equilibration behavior and entanglement entropy characteristics, and introduces a statistical mechanics perspective for understanding their non-equilibrium properties.

Contribution

It provides a rigorous analysis of disordered random matrix product states, connecting their properties to classical statistical models and revealing emergent statistical mechanics behavior.

Findings

Disordered random MPS equilibrate exponentially well under certain Hamiltonians.

Entanglement entropy is extensive for disconnected subsystems.

Small connected systems exhibit nearly maximal entropy at large bond dimensions.

Abstract

The study of generic properties of quantum states has led to an abundance of insightful results. A meaningful set of states that can be efficiently prepared in experiments are ground states of gapped local Hamiltonians, which are well approximated by matrix product states. In this work, we introduce a picture of generic states within the trivial phase of matter with respect to their non-equilibrium and entropic properties: We do so by rigorously exploring non-translation-invariant matrix product states drawn from a local i.i.d. Haar-measure. We arrive at these results by exploiting techniques for computing moments of random unitary matrices and by exploiting a mapping to partition functions of classical statistical models, a method that has lead to valuable insights on local random quantum circuits. Specifically, we prove that such disordered random matrix product states equilibrate exponentially well with overwhelming probability under the time evolution of Hamiltonians featuring a non-degenerate spectrum. Moreover, we prove two results about the entanglement Renyi entropy: The entropy with respect to sufficiently disconnected subsystems is generically extensive in the system-size, and for small connected systems the entropy is almost maximal for sufficiently large bond dimensions.