Quantum statistics on graphs

Abstract

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.

21 pages, 6 figures

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• Kirchhoff's Rule for Quantum Wires
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Cited by in corpus (21)

• Quantum walks of correlated photon pairs in two-dimensional waveguide arrays
• Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermions
• Universal sign-control of coupling in tight-binding lattices
• n-particle quantum statistics on graphs
• Quantum graphs with singular two-particle interactions
• Topological Exchange Statistics in One Dimension
• Non-abelian Quantum Statistics on Graphs
• Particle creation and annihilation at interior boundaries: One-dimensional models
• A Generalization of the One-Dimensional Boson-Fermion Duality Through the Path-Integral Formalism
• State selection in frustrated magnets
• Effective theories for quantum spin clusters: Geometric phases and state selection by singularity
• Order by singularity in Kitaev clusters
• Geometric presentations of braid groups for particles on a graph
• Topological classification of time-asymmetry in unitary quantum processes
• Discrete Morse functions for graph configuration spaces
• A quantum-network approach to spin interferometry driven by Abelian and non-Abelian fields
• Universal properties of anyon braiding on one-dimensional wire networks
• Novel quantum phases on graphs using abelian gauge theory
• Quantum Walk on Orbit Spaces
• Approximate Hofstadter- and Kapit-Mueller-like parent Hamiltonians for Laughlin states on fractals
• Extending the planar theory of anyons to quantum wire networks