n-particle quantum statistics on graphs

TL;DR

This paper characterizes abelian quantum statistics on graphs, relating the number of anyon phases to graph connectivity, and distinguishes possible particle statistics based on graph planarity and connectivity.

Contribution

It provides a complete classification of abelian quantum statistics on graphs, linking graph topology to particle statistics and offering new proofs for existing topological results.

Findings

Number of anyon phases depends on graph connectivity.

For 2-connected graphs, quantum statistics are independent of particle number.

Non-planar 3-connected graphs admit only bosons and fermions.

Abstract

We develop a full characterization of abelian quantum statistics on graphs. We explain how the number of anyon phases is related to connectivity. For 2-connected graphs the independence of quantum statistics with respect to the number of particles is proven. For non-planar 3-connected graphs we identify bosons and fermions as the only possible statistics, whereas for planar 3-connected graphs we show that one anyon phase exists. Our approach also yields an alternative proof of the structure theorem for the first homology group of n-particle graph configuration spaces. Finally, we determine the topological gauge potentials for 2-connected graphs.