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

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