In this thesis we will focus on a particular set of physically-inspired models for anyons that correspond to the spectrum of electric and magnetic charges of a quantum gauge theory with discrete symmetry group. Mathematically, these correspond to representations of Drinfeld’s quantum double of the symmetry group.
Physically, such models arise when a regular continuous-group quantum gauge theory has its symmetry broken, via the Higgs mechanism, to a discrete group G. In such a field theory all the gauge particles are massive, and hence do not mediate long-range interactions. A set of electric and magnetic charged particles remain unscreened, however, and such charges can be detected via Aharonov-Bohm interactions.
Abelian anyons have already been observed in the fractional quantum Hall effect, and non-abelian anyons are conjectured to exist at certain levels as well [NW96,RR99]. Unfortunately, such anyons do not belong to the anyon model discussed above, but rather to the one analyzed in [Fre00, FKLW01].
Though no system is currently known that supports the model of discrete-group gauge theory anyons discussed herein, it is possible that such a system could be engineered.
Recent proposals include optical lattices [DDL02] and Josephson-junction arrays [DIV03]. In the latter case, an explicit array is constructed that simulates on a lattice the gauge theory with group S3 , which is the smallest group for which our construction works.
-- Ph.D. Thesis 2005