Abstract

Symmetry is a central concept for classical and quantum field theory, usually, symmetry is described by a finite group or Lie group. In this talk, we introduce the weak Hopf algebra extension of symmetry, which arises naturally in anyonic quantum systems; and we establish the weak Hopf symmetry breaking theory based on the fusion closed set of anyons. As a concrete example, we implement a thorough investigation of the quantum double model based on a given weak Hopf algebra and show that the vacuum sector of the model has weak Hopf symmetry. The gapped boundary and domain wall theories are also established, and the microscopic lattice constructions of the gapped boundary and domain wall are discussed. We also introduce the weak Hopf tensor network states, via which we solve the weak Hopf quantum double lattice models on closed and open surfaces. This is a joint work with Z. Jia, D. Kaszlikowski and L. Chang.