Exact solution of a percolation analog for the many-body localization transition

We construct and solve a classical percolation model with a phase transition that we argue acts as a proxy for the quantum many-body localization 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 hybridize 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 localization length exponent, and also use complete enumeration of clusters to study the transition numerically in finite-sized systems.