QM Research Seminar: Computing the category of superselection sectors of 2d lattice spin systems

Speaker: Boris Kjær (​University of Copenhagen).

Abstract:
In recent years, the analysis of topologically ordered ground states of 2d quantum lattice systems in the thermodynamic limit has been developed to provide a rigorous invariant of gapped phases. The methodology is based on the approach of Doplicher, Haag, and Roberts in the context of algebraic quantum field theory, and derives a braided C*-tensor category which represents the anyonic excitations above the ground state and the fusion and braiding among them. 

We present the computation of this category for the class of Levin-Wen models, which yields the Drinfeld center Z(C) of an arbitrary unitary fusion category C up to braided monoidal equivalence. Previous methods, applied to quantum double models based on finite groups, relied on defining string operators on the quasi-local observable algebra implementing an action of Z(Vec_G). But there are obstructions in K-theory that limit this approach to group-like models with integer quantum dimensions. Our approach requires detailed understanding of the local ground state spaces of the model in terms of topological quantum field theory based on skein modules. Based on work joint with Alex Bols.