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.
Highlights
In classical mechanics, particles are considered distinguishable
The classical configuration space of n indistinguishable particles is the orbit space Cn(M) = (M×n − Δ)/Sn, where Δ corresponds to the configurations for which at least two particle are at the same point in M, and Sn is the permutation group
One can either consider the one-dimensional Schrödinger operator acting on the edges, with matching conditions for the wave functions at the vertices, or a discrete Schrödinger operator acting on connected vertices
Summary
Particles are considered distinguishable. the n-particle configuration space is the Cartesian product, M×n, where M is the one-particle configuration space. One can either consider the one-dimensional Schrödinger operator acting on the edges, with matching conditions for the wave functions at the vertices, or a discrete Schrödinger operator acting on connected vertices (i.e., a tight-binding model on the graph) Such systems are of considerable independent interest and their single-particle quantum mechanics has been studied extensively in recent years [8]. Making extensive use of discrete Morse theory and some graph invariants, Ko and Park [12] extended the results of [11] to an arbitrary graph Γ Their approach relies on a suite of relatively elaborate techniques—mostly connected to a proper ordering of vertices and choices of trees to reduce the number of critical cells—and the relationship to, and consequences for, the physics of quantum statistics are not identified. 4 we develop a full characterization of the first homology n-Particle Quantum Statistics on Graphs group for 2-particle graph configuration spaces. The last part of the paper is devoted to the characterization of topological gauge potentials for 2-connected graphs
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
