Exact solution of a percolation analogue for the many-body localisation
transition
Phys. Rev. B 99 220201-220201
Authors:
Sthitadhi Roy, David E Logan, JT Chalker
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.
