Superselection sector theory of quantum spin systems

The low energy effective theory of a quantum spin system is governed by a topological field theory leading to a conjectural classification of gapped phases in two dimensions. The infinite lattice provides a setting for the rigorous analysis of this problem, with the key invariant provided by the superselection sector theory of the vacuum representation of a given phase. This is an adaptation of the category of localised and transportable endomorphisms introduced by Doplicher, Haag, and Roberts in the setting of relativistic quantum field theories which captures the gauge symmetry from the structure of local fields. In this talk we review the phenomenon of topological order in spin systems and the operator algebraic approach to the classification of phases in 2D. We touch upon connections to state-sum TQFTs and conformal nets and discuss recent advances.