Remarks on quantum symmetry

William Martin McClain

Yuan K Ha

Two-dimensional quantum field theories are important in many problems in physics because they contain exact symmetries and are often completely integrable. We demonstrate the power of bosonization in elucidating the structure of a multi-component fermion theory with chiral and internal symmetries. Many remarkable features and quantum symmetries in such a system are most readily uncovered in the bosonized form of the original fermion theory.

Isamu Ohnishi

Anthony Brinton Wolbarst

Heinz Dietrich Doebner, V. K. Dobrev

Yuan-Ming Lu, Lukasz Fidkowski

An exotic feature of the fractional quantum Hall effect is the emergence of anyons, which are quasiparticle excitations with fractional statistics. In the presence of a symmetry, such as $U(1)$ charge conservation, it is well known that anyons can carry fractional symmetry quantum numbers. In this paper we reveal a different class of symmetry realizations, i.e., anyons can ``breed'' in multiples under symmetry operation. We focus on the global Ising (${\mathbb{Z}}_{2}$) symmetry and show examples of these unconventional symmetry realizations in Laughlin-type fractional quantum Hall states. One remarkable consequence of such an Ising symmetry is the emergence of anyons on the Ising symmetry domain walls. We also provide a mathematical framework which generalizes this phenomenon to any Abelian topological orders.