Non-abelian Quantum Statistics on Graphs
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 X. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of X 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.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- November 2019
- DOI:
- 10.1007/s00220-019-03583-5
- arXiv:
- arXiv:1806.02846
- Bibcode:
- 2019CMaPh.371..921M
- Keywords:
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- Mathematical Physics;
- Mathematics - Algebraic Topology;
- Quantum Physics
- E-Print:
- 50 pages, v3: updated to reflect the published version. Commun. Math. Phys. (2019)