Emergent Statistical Mechanics from Properties of Disordered Random Matrix Product States

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 nonequilibrium and entropic properties. We do so by rigorously exploring nontranslation-invariant matrix product states drawn from a local independent and identically distributed 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 nondegenerate spectrum. Moreover, we prove two results about the entanglement Rényi 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.