The breaking of quantum double symmetries by defect condensation

TL;DR

This paper develops a generalized criterion for quantum double symmetry breaking using Hopf algebras, enabling a comprehensive classification of defect and ordinary symmetry breaking patterns in two-dimensional systems.

Contribution

It introduces a more general formalism for quantum double symmetry breaking, extending existing criteria and applicable to various symmetries and physical systems.

Findings

Reproduces known results for electric condensates

Derives new formulae for defect condensates involving symmetry restoration

Provides a unified framework for classifying symmetry breaking patterns

Abstract

In this paper, we study the phenomenon of Hopf or more specifically quantum double symmetry breaking. We devise a criterion for this type of symmetry breaking which is more general than the one existing in the literature, and therefore extends the number of possible breaking patterns that can be described consistently. We start by recalling why the extended symmetry notion of quantum double algebras is an optimal tool when analyzing a wide variety of two dimensional physical systems including quantum fluids, crystals and liquid crystals. The power of this approach stems from the fact that one may characterize both ordinary and topological modes as representations of a single (generally non-Abelian) Hopf symmetry. In principle a full classification of defect mediated as well as ordinary symmetry breaking patterns and subsequent confinement phenomena can be given. The formalism applies equally well to systems exhibiting global, local, internal and/or external (i.e. spatial) symmetries. The subtle differences in interpretation for the various situations are pointed out. We show that the Hopf symmetry breaking formalism reproduces the known results for ordinary (electric) condensates, and we derive formulae for defect (magnetic) condensates which also involve the phenomenon of symmetry restoration. These results are applied in two papers which will be published in parallel.