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

5 pages, 4 figures + supplementary material

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• Sub-diffusive Thouless time scaling in the Anderson model on random regular graphs
• Multifractal phase in the weighted adjacency matrices of random Erdös-Rényi graphs
• The Fock-space landscape of many-body localisation
• Probability transport on the Fock space of a disordered quantum spin chain
• Structural properties of local integrals of motion across the many-body localization transition via a fast and efficient method for their construction
• Localization crossover and subdiffusive transport in a classical facilitated network model of a disordered, interacting quantum spin chain
• Anatomy of localisation protected quantum order on Hilbert space
• Analog charged black hole formation via percolation: Exploring cosmic censorship and Hoop conjecture
• Heat percolation in many-body flatband localizing systems
• Orthogonal projections of hypercubes
• Information Bounds on phase transitions in disordered systems