On weak Hopf symmetry and weak Hopf quantum double model

TL;DR

This paper introduces weak Hopf algebra symmetry in quantum systems, explores its breaking, and applies it to quantum double models, revealing new topological phases, boundary conditions, and tensor network states.

Contribution

It extends symmetry concepts to weak Hopf algebras, develops a symmetry breaking theory, and constructs weak Hopf quantum double models with detailed boundary and domain wall analysis.

Findings

Weak Hopf symmetry is present in quantum double models.

Gapped boundaries are characterized by comodule algebras.

Weak Hopf tensor network states effectively model these systems.

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 work, we introduce the weak Hopf algebra extension of symmetry, which arises naturally in anyonic quantum systems; and we establish 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 topological excitations and ribbon operators are discussed in detail. The gapped boundary and domain wall theories are also established, we show that the gapped boundary is algebraically determined by a comodule algebra, or equivalently, a module algebra; and the gapped domain wall is determined by the bicomodule algebra, or equivalently, a bimodule algebra. The microscopic lattice constructions of the gapped boundary and domain wall are discussed in detail. 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. The duality of the quantum double phases is discussed in the last part.