The absence of the selfaveraging property of the entanglement entropy of disordered free fermions

TL;DR

This paper demonstrates that in one-dimensional disordered free fermion systems, the entanglement entropy does not self-average, meaning its variance remains significant even for large system sizes, unlike higher-dimensional cases.

Contribution

It analytically and numerically shows the absence of self-averaging of entanglement entropy in 1D disordered free fermions, contrasting with higher-dimensional behaviors.

Findings

Variance of entanglement entropy remains bounded away from zero as system size increases.

Entanglement entropy's distribution is essential for complete description in 1D.

Contrasts with higher dimensions where variance vanishes, ensuring mean representativity.

Abstract

We consider the macroscopic system of free lattice fermions in one dimension assuming that the one-body Hamiltonian of the system is the one dimensional discrete Schr\"odinger operator with independent identically distributed random potential. We show analytically and numerically that the variance of the entanglement entropy of the segment $[-M,M]$ of the system is bounded away from zero as $M\rightarrow \infty $. This manifests the absence of the selfaveraging property of the entanglement entropy in our model, meaning that in the one-dimensional case the complete description of the entanglement entropy is provided by its whole probability distribution. This also may be contrasted the case of dimension two or more, where the variance of the entanglement entropy per unit surface area vanishes as $% M\rightarrow \infty $ \cite{El-Co:17}, thereby guaranteeing the representativity of its mean for large $M$ in the multidimensional case.