Douglas Lundholm

Exchange and exclusion in the non-abelian anyon gas

Anyons are effective quantum particles that emerge in certain planar systems and have properties intermediate to bosons and fermions. Roughly speaking, they may be defined by allowing their Schrödinger wave function to be multivalued and representing a continuous exchange of coordinates by an arbitrary complex phase (abelian anyons) or, more generally, a unitary matrix (non-abelian anyons). This yields a probabilistic realization of the braid group, with links to knot invariants, conformal field theory and Chern-Simons theory, and furthermore, anyons have been proposed as potential building blocks for topologically protected quantum computing.
A fundamental problem concerns the relationship between exchange phases and exclusion (cf. Pauli's exclusion principle for fermions). I will discuss how to extend methods of exchange, statistics repulsion and local exclusion (Poincaré, Hardy and Lieb-Thirring inequalities) from fermions to non-abelian anyons, focusing on two of the most popular such models: the Fibonacci and the Ising anyon models. This is joint work with Viktor Qvarfordt.