# Multifractal Scalings across the Many-Body Localization Transition

## TL;DR

This paper investigates the many-body localization transition by analyzing multifractal properties of eigenstates in large Hilbert spaces, revealing basis-dependent multifractality and a non-universal transition signature.

## Contribution

It provides the first detailed multifractal analysis of eigenstates across the MBL transition in large systems using exact diagonalization.

## Key findings

- Eigenstates are fully ergodic in the delocalized phase.
- MBL regime exhibits basis-dependent multifractal, delocalized but non-ergodic states.
- The MBL transition shows a non-universal jump in multifractal dimensions.

## Abstract

In contrast with Anderson localization where a genuine localization is observed in real space, the many-body localization (MBL) problem is much less understood in the Hilbert space, support of the eigenstates. In this work, using exact diagonalization techniques we address the ergodicity properties in the underlying ${\cal{N}}$-dimensional complex networks spanned by various computational bases for up to $L=24$ spin-1/2 particles (i.e. Hilbert space of size ${\cal{N}}\simeq 2.7\,10^6$). We report fully ergodic eigenstates in the delocalized phase (irrespective of the computational basis), while the MBL regime features a generically (basis-dependent) multifractal behavior, delocalized but non-ergodic. The MBL transition is signaled by a non-universal jump of the multifractal dimensions.