Correlations in Disordered Solvable Tensor Network States

TL;DR

This paper analytically investigates how disorder affects correlation functions in solvable tensor network states, revealing that average correlations vanish but their fluctuations remain nonzero, using advanced mathematical tools.

Contribution

It provides an exact analytical expression for the average moments of correlation functions in disordered tensor network states using Weingarten calculus.

Findings

Average correlation functions vanish in disordered states.

Covariance of correlation functions remains nonzero.

Complexity of the expression scales factorially with the moment order.

Abstract

Solvable matrix product and projected entangled pair states evolved by dual and ternary-unitary quantum circuits have analytically accessible correlation functions. Here, we investigate the influence of disorder. Specifically, we compute the average behavior of a physically motivated two-point equal-time correlation function with respect to random disordered solvable tensor network states arising from the Haar measure on the unitary group. By employing the Weingarten calculus, we provide an exact analytical expression for the average of the $k$th moment of the correlation function. The complexity of the expression scales with $k!$ and is independent of the complexity of the underlying tensor network state. Our result implies that the correlation function vanishes on average, while its covariance is nonzero.