Operator formalism for discretely holomorphic parafermions of the two-color Ashkin-Teller, loop $\mathrm{O} ( 1 )$, staggered eight-vertex, odd eight-vertex, and abelian sandpile models

Abstract: We extend an operator formalism, developed by Hongler, Kyt\"ol\"a, and Zahabi in 2012 for the Ising model, to the Ashkin-Teller and loop $\mathrm{O} ( n ) $ models. The formalism is primarily dependent upon notions of massive, and massless, s-holomorphicity of the Ising model, which are respectively satisfied at, and below, the critical temperature, hence allowing for rigorous analyses of the transfer matrix, correlation functions, and lattice fermion operators. From results of another paper, by Tanhayi-Ahari and Rouhani in 2012, which demonstrates that parafermionic observables for the staggered eight-vertex model coincide with those of the Ashkin-Teller model, as well as a 2003 paper, by Wu and Kunz, which establishes a correspondence between the staggered eight-vertex and odd eight-vertex models, we establish further associations.