Disorder variables and para-fermions in two-dimensional statistical mechanics

It is shown that “clock” type models in two-dimensional statistical mechanics possess order and disorder variables φ_n and χ_m with n and m falling in the range 1,2, …,p. These variables respectively describe abelian analogs to charged fields and the fields of 't Hooft monopoles with charges q = n/p and topological quantum number m. They are related to one another by a dual symmetry. Products of these operators generate, via a short-distance expansion, para-fermion operators in which rotational symmetry and the internal symmetry group are tied together. The clock models in two dimensions are shown to be an ideal laboratory where these ideas have a very simple realization.