Emergent Statistical Mechanics from Properties of Disordered Random Matrix Product States

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.

Popular Summary

Randomness has turned out as an extraordinarily powerful tool to capture properties of complex quantum systems. Indeed, suitable random ensembles of quantum states allow us to capture a number of generic properties that typical many-body systems exhibit in nature. Random matrix theory in particular makes predictions for a wide range of properties of condensed matter systems, prominently including notions of conductance. The generic properties of random ensembles can be analytically calculated with modern mathematical techniques even when making statements about individual instances becomes infeasible. The upshot of such approaches is, on a high level, that random ensembles capture a number of physical phenomena well, even if less important details can be averaged away.

A key difficulty here lies in the choice of a suitable random model that already captures the essence of a physical problem while still being amenable to analytical calculations. An important such property is that of the locality of physical interactions. In this work, we derive important properties, such as equilibration and entropies, of generic quantum many-body systems using random ensembles of quantum states that exhibit a one-dimensional local structure, bridging a previous gap between mathematical techniques and realistic physical assumptions. This is made feasible by exploiting new mappings of expectation values over certain matrix ensembles to partition functions of classical statistical models.