Hopf symmetry breaking and confinement in (2+1)-dimensional gauge theory

Abstract

Gauge theories in 2+1 dimensions whose gauge symmetry is spontaneously broken to a finite group enjoy a quantum group symmetry which includes the residual gauge symmetry. This symmetry provides a framework in which fundamental excitations (electric charges) and topological excitations (magnetic fluxes) can be treated on equal footing. In order to study symmetry breaking by both electric and magnetic condensates we develop a theory of symmetry breaking which is applicable to models whose symmetry is described by a quantum group (quasitriangular Hopf algebra). Using this general framework we investigate the symmetry breaking and confinement phenomena which occur in (2+1)-dimensional gauge theories. Confinement of particles is linked to the formation of string-like defects. Symmetry breaking by an electric condensate leads to magnetic confinement and vice-versa. We illustrate the general formalism with examples where the symmetry is broken by electric, magnetic and dyonic condensates.The irreps of our transformation group algebras are then described by the following theorem, which is a simple consequence of theorem 3.9 in [29]

Condensate-induced transitions between topologically ordered phases

“…In earlier work, 16,17 we have developed a theory of quantum group symmetry breaking and applied it to discrete gauge theories. This theory was later refined and applied to phase transitions in quantum nematics and other systems.…”

“…However, even for the restricted class of orbifolds we deal with here, namely, those which are obtained from topologically trivial CFTs, more complicated behavior than we have shown is possible. For example, on condensation of a purely magnetic particle with charge of the form ⌸ 1 A , we will end up with a theory described by the quantum double of a quotient group of G, rather than a subgroup of G. 17 Even more complicated phenomena emerge when one starts from orbifolds of topologically nontrivial CFTs.…”

Entanglement phases as holographic duals of anyon condensates

“…The formalism of anyon condensation allows to construct "simpler" anyon models from more rich ones, and suggests to think of the "condensate fraction" of the condensed anyon as an order parameter for a Landau-like description of the phase transition. Yet, it is a priori not clear how such an order parameter should be measured, and existing approaches describe anyon condensation as a breaking of the global symmetry of the quantum group or tensor category underlying the model [2][3][4][5].…”

Josephson-coupled Moore-Read states

“…Therefore, the natural starting point for the CFT is a combination of two Ising theories with primary fields denoted as {1,σ ↑ ,ψ ↑ },{1,σ ↓ ,ψ ↓ }, to represent the neutral sector, and a chiral Bose field U(1) 4 [read: U(1) level 4] with primary fields {1,e iφ c /2 ,e −iφ c /2 ,e iφ c }, to account for the charge sector of this state. Here, the specified "level" N = 4, which represents the compactification of U(1) N with radius R = √ N = 2, is related to the quantization of charge, with the smallest possible charge in the system being e/2 where e is the unit charge of electrons.…”

“…↑ ×Ising ↓ to U(1) 4 We will now further investigate the effects of the confinement of σ ↑/↓ operators on the topological order of the system. We will see that the topological order is in fact Abelian and we will find an alternative description of the electron and quasihole operators in terms of the primary fields of a U(1)×U(1) CFT.…”