PDEA Seminar: Anyons on many-particle quantum graphs

Abstract: Particles in 2 dimensions may have quantum statistics between those of bosons or fermions. In the abelian case, these statistics are characterised by a change of complex phase of the wavefunction under exchange of particles. It is known that anyons on graphs may have even more general statistics under which topologically distinct exchanges on the graph may give rise to different exchange phases. We study quantum mechanics for anyons on graphs using a generalisation of quantum graph models to many-particle systems, with the Hamiltonian obtained through the (self-adjoint) Friedrichs extension. We see that the boundary conditions induced by this Hamiltonian can describe the Kirchoff conditions as a particle passes through a vertex, as well as local two-particle interactions and anyonic exchanges. Using (our own implementations of) numerical finite-difference methods, we are able to study two-particle eigenfunctions and spectra. Theoretical and numerical methods are used to analyse the regularity of these solutions.

This is a seminar in the PDEs and Applications seminar series.