# The absence of the selfaveraging property of the entanglement entropy of disordered free fermions in one dimension

## TL;DR

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

## Contribution

It proves the absence of the self-averaging property of entanglement entropy in 1D disordered free fermions, contrasting with higher-dimensional systems where self-averaging occurs.

## Key findings

- Variance of entanglement entropy remains bounded away from zero as system size increases.
- Entanglement entropy's distribution is non-trivial and not characterized by its mean in 1D.
- Contrasts with higher dimensions where variance vanishes, indicating self-averaging.

## Abstract

We consider the macroscopic system of free lattice fermions in one dimensions 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 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 guaranteing the representativity of its mean for large $M$ in the multidimensional case.