Solving models with generalized free fermions I: Algebras and eigenstates

TL;DR

This work develops a unifying algebraic framework for a class of quantum spin chains with hidden free-fermion structures by leveraging graph-Clifford (quasi-Clifford) algebras. The authors introduce the defining representation on a spin-chain, showing how the XY model and FFD model Hamiltonians arise from this representation, and study the antisymmetric combination $H_A=H- ilde{H}$, which admits an exact reference state and allows construction of few-body eigenstates via fermionic operators. They establish that claw-free, even-hole-free frustration graphs guarantee a rich set of commuting charges and a tractable transfer matrix with an inversion relation, enabling a systematic free-fermion solution. In the FFD case, they characterize the algebraic structure, central elements, and edge operators, and demonstrate that under certain conditions the defining representation covers the full operator algebra, allowing explicit eigenstate construction and analysis of symmetry operations and entanglement. The results bridge novel graph-theoretic algebras with traditional Jordan-Wigner solvability and set the stage for explicit eigenstate constructions in disguised free-fermion models and beyond.

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.

Figures (5)

• Figure 1: Frustration graph for the XY and Ising models with periodic boundary conditions.
• Figure 2: Frustration graph for the FFD model with $M=6$.
• Figure 3: A "claw" graph.
• Figure 4: The four states associated with the eigenmode with single-particle energy $\varepsilon_k$. They split into two fermionic doublets, and the zero energy level is doubly occupied.
• Figure 5: Short summary of the notations for the different families of fermions. In the "spectrum generating" families the positive indices take values $k=1\dots S$, whereas for the auxiliary fermions we have $k=1\dots S'$.

Theorems & Definitions (22)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Theorem 5
• proof