Solving models with generalized free fermions I: Algebras and eigenstates

Abstract: We study quantum spin chains solvable via hidden free fermionic structures. We study the algebras behind such models, establishing connections to the mathematical literature of the so-called graph-Clifford'' orquasi-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 thefree fermions in disguise'' model of Fendley. Afterwards we study a particular anti-symmetric 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.