Solving models with generalized free fermions: Algebras and eigenstates.

We study quantum spin chains solvable via hidden free fermionic structures. We review the algebras behind such models, establishing connections to the mathematical literature of the so-called "graph-Clifford" or "quasi-Clifford" algebras. We also introduce the "defining representation" for such algebras, and show that this representation actually coincides with the terms of the Hamiltonian in two relevant models: the XY model and the "free fermions in disguise" model of Fendley. Afterwards we study a particular antisymmetric combination of commuting Hamiltonians; this is performed in a model-independent way. We show that for this combination there exists a reference state, and few-body eigenstates can be created by the fermionic operators. Concrete application is presented in the case of the "free fermions in disguise" model.