Quantum statistics on graphs

Quantum graphs are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter. We consider quantum statistics for indistinguishable spinless particles on a graph, concentrating on the simplest case of abelian statistics for two particles. In spite of the fact that graphs are locally one-dimensional, anyon statistics emerge in a generalized form. A given graph may support a family of independent anyon phases associated with topologically inequivalent exchange processes. In addition, for sufficiently complex graphs, there appear new discrete-valued phases. Our analysis is simplified by considering combinatorial rather than metric graphs -- equivalently, a many-particle tight-binding model. The results demonstrate that graphs provide an arena in which to study new manifestations of quantum statistics. Possible applications include topological quantum computing, topological insulators, the fractional quantum Hall effect, superconductivity and molecular physics.