# Exact solution of a percolation analogue for the many-body localisation transition

## TL;DR

This paper introduces an exactly solvable classical percolation model on Fock space that mimics the quantum many-body localisation transition, providing precise critical exponents and insights into the phase transition.

## Contribution

It constructs and solves a classical percolation analogue for the many-body localisation transition, deriving an exact localisation length exponent of 2.

## Key findings

- Exact value of the localisation length exponent $
u=2$
- Cluster enumeration confirms the phase transition behavior
- Model acts as a classical proxy for quantum MBL transition

## Abstract

We construct and solve a classical percolation model with a phase transition that we argue acts as a proxy for the quantum many-body localisation transition. The classical model is defined on a graph in the Fock space of a disordered, interacting quantum spin chain, using a convenient choice of basis. Edges of the graph represent matrix elements of the spin Hamiltonian between pairs of basis states that are expected to hybridise strongly. At weak disorder, all nodes are connected, forming a single cluster. Many separate clusters appear above a critical disorder strength, each typically having a size that is exponentially large in the number of spins but a vanishing fraction of the Fock-space dimension. We formulate a transfer matrix approach that yields an exact value $\nu=2$ for the localisation length exponent, and also use complete enumeration of clusters to study the transition numerically in finite-sized systems.