Solving models with generalized free fermions I: Algebras and eigenstates

TL;DR

This paper explores algebraic structures behind quantum spin chains solvable via free fermions, establishing connections to graph-Clifford algebras, and demonstrates how certain Hamiltonians admit eigenstates constructed from fermionic operators.

Contribution

It introduces the defining representation of graph-Clifford algebras and links it to models like XY and free fermions in disguise, providing a new algebraic perspective.

Findings

Identified algebraic structures underlying solvable quantum spin chains.

Established the equivalence of the defining representation with Hamiltonian terms in key models.

Showed existence of reference states and fermionic eigenstates for specific Hamiltonian combinations.

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'' 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 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.