# Hilbert-space fragmentation, multifractality, and many-body localization

## TL;DR

This paper introduces a decimation scheme to analyze many-body localization by fragmenting the Hilbert space, enabling the study of larger systems and providing a geometric interpretation of MBL multifractality.

## Contribution

It presents a novel decimation method that efficiently fragments the Hilbert space, improving the analysis of MBL and predicting the transition quantitatively.

## Key findings

- Hilbert space fragmentation enables larger system analysis at strong disorder.
- The MBL transition is quantitatively predicted.
- Multifractality is interpreted as Hilbert space shattering.

## Abstract

Investigating many-body localization (MBL) using exact numerical methods is limited by the exponentialgrowth of the Hilbert space. However, localized eigenstates display multifractality and only extend over a vanishing fraction of the Hilbert space. Here, building on this remarkable property, we develop a simple yet efficient decimation scheme to discard the irrelevant parts of the Hilbert space of the random-field Heisenberg chain. This leads to an Hilbert space fragmentation in small clusters, allowing to access larger systems at strong disorder. The MBL transition is quantitatively predicted, together with a geometrical interpretation of MBL multifractality as a shattering of the Hilbert space.