Multifractality and self-averaging at the many-body localization transition

TL;DR

This paper investigates the lack of self-averaging in many-body localization studies, showing that considering large ensembles reveals that eigenstate properties near the transition are well described by an analytical model from a non-interacting system.

Contribution

It highlights the importance of large sample sizes in MBL studies and introduces an analytical expression for eigenstate dimensions near the transition, linking many-body and non-interacting systems.

Findings

Eigenstate dimensions are well described by the analytical model.

Large ensembles are crucial for accurate MBL analysis.

The model from the Fibonacci lattice applies to the disordered Heisenberg chain.

Abstract

Finite-size effects have been a major and justifiable source of concern for studies of many-body localization, and several works have been dedicated to the subject. In this paper, however, we discuss yet another crucial problem that has received much less attention, that of the lack of self-averaging and the consequent danger of reducing the number of random realizations as the system size increases. By taking this into account and considering ensembles with a large number of samples for all system sizes analyzed, we find that the generalized dimensions of the eigenstates of the disordered Heisenberg spin-1/2 chain close to the transition point to localization are described remarkably well by an exact analytical expression derived for the non-interacting Fibonacci lattice, thus providing an additional tool for studies of many-body localization.