Integrable Free and Interacting Fermions

Integrability conditions on local Hamiltonians for one-dimensional quantum systems to be free and interacting fermions are introduced. The definition of free fermion is the simultaneous satisfaction of the Yang-Baxter equation and Shastry's decorated star-triangle relation by the $R$-matrix, which is more general than the previous `free-fermion algebra' by Maassarani and more special than free fermions as in the context of exactly solvable quantum models or integrable classical two-dimensional vertex models dual to quantum spin chains. Free fermionic $R$-matrices are of the difference form and have a conjugation symmetry. These free Hamiltonians may sometimes be deformed by the conjugation operator to describe an integrable interacting system with non-relativistic $R$-matrices, as are the cases of the Hubbard model and the XY model in a longitudinal field. A further criterion is obtain on precisely when such deformations remain integrable. A practical procedure is proposed to iteratively solve the free fermionic $R$-matrices from local Hamiltonians, which can be used to construct non-relativistic $R$-matrices if the conditions are met.