# Percolation in Fock space as a proxy for many-body localisation

## TL;DR

This paper investigates classical percolation models in Fock space as proxies for the many-body localisation transition, revealing a transition characterized by cluster size statistics and local observables, with implications for understanding ergodicity breaking.

## Contribution

It introduces a novel percolation framework in Fock space to analyze the MBL transition, including a mapping to kinetically constrained models for larger system sizes.

## Key findings

- Identification of a percolation transition in Fock space linked to MBL.
- Finite-size scaling yields consistent critical properties.
- Local observables reflect the transition similarly to eigenstate measures.

## Abstract

We study classical percolation models in Fock space as proxies for the quantum many-body localisation (MBL) transition. Percolation rules are defined for two models of disordered quantum spin-chains using their microscopic quantum Hamiltonians and the topologies of the associated Fock-space graphs. The percolation transition is revealed by the statistics of Fock-space cluster sizes, obtained by exact enumeration for finite-sized systems. As a function of disorder strength, the typical cluster size shows a transition from a volume law in Fock space to sub-volume law, directly analogous to the behaviour of eigenstate participation entropies across the MBL transition. Finite-size scaling analyses for several diagnostics of cluster size statistics yield mutually consistent critical properties. We show further that local observables averaged over Fock-space clusters also carry signatures of the transition, with their behaviour across it in direct analogy to that of corresponding eigenstate expectation values across the MBL transition. The Fock-space clusters can be explored under a mapping to kinetically constrained models. Dynamics within this framework likewise show the ergodicity-breaking transition via Monte Carlo averaged local observables, and yield critical properties consistent with those obtained from both exact cluster enumeration and analytic results derived in our recent work [arXiv:1812.05115]. This mapping allows access to system sizes two orders of magnitude larger than those accessible in exact enumerations. Simple physical pictures based on freezing of local real-space segments of spins are also presented, and shown to give values for the critical disorder strength and correlation length exponent $\nu$ consistent with numerical studies.