Non-abelian Quantum Statistics on Graphs
arXiv:1806.02846 · doi:10.1007/s00220-019-03583-5
Abstract
We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space . The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.
arXiv comment50 pages, v3: updated to reflect the published version. Commun. Math. Phys. (2019)
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