Citation: Bais, F. Alexander, Bernd J. Schroers, and Joost K. Slingerland. "Broken quantum symmetry and confinement phases in planar physics." Physical review letters 89.18 (2002): 181601.
Web: https://arxiv.org/abs/hep-th/0205117
Tags: Phase-transition, Quantum-groups
In this paper, the authors introduce a formalism to describe symmetry breaking under quantum group (quasi-triangular Hopf algebra) symmetries. The idea is to generalize the breaking of a group to its subgroup by breaking a quantum group to a sub-quantum group.
The way they do this is as follows. They imagine that you are given a state which globally transforms in some given way by a quantum group. That is, a quantum group acts on it by some irreducible representation. This state will not be symmetric under the whole Hopf algebra. It will only be left invariant by some subalgebra of the Hopf algebra. That is, there is some subalgebra of the Hopf algebra on which the representation acts by the counit. In this case, the symmetry has been spontaneously broken to this subalgebra!
The authors then describe what confinement and condensation look like in this picture. The paper overall is very short. It was later elaborated on; it is the first in a sequence of three papers, the latter two are below:
> Bais, Alexander F., Bernd J. Schroers, and Joost K. Slingerland. "Hopf symmetry breaking and confinement in (2+ 1)-dimensional gauge theory." Journal of High Energy Physics 2003.05 (2003): 068.
> Bais, F. Alexander, and J. K. Slingerland. "Condensate-induced transitions between topologically ordered phases." Physical Review B—Condensed Matter and Materials Physics 79.4 (2009): 045316.
These three works form the original foundational treatment of anyon condensation. A later work in this field, which is more relevant to the way I think about things, is
> Kong, Liang. "Anyon condensation and tensor categories." Nuclear Physics B 886 (2014): 436-482.