A Unified Graph-Theoretic Framework for Free-Fermion Solvability
May 26, 2023
Adrian Chapman, Samuel Elman, and I have just uploaded to arXiv our paper "A Unified Graph-Theoretic Framework for Free-Fermion Solvability". In this paper we show that a quantum spin system has an exact description by free fermions if its frustration graph is claw-free and contains a simplicial clique. Previously, it was shown that a free-fermion solution exists if the frustration graph is either a line graph, or (even-hole, claw)-free. Our characterisation unifies these two conditions.
We consider quantum spin systems on \(n\) qubits with Hamiltonians written in the Pauli basis
$$H := \sum_{j \in V}b_j\sigma^j,$$where \(V\subseteq\{I,x,y,z\}^{\times n}\) is a set of strings labelling n-qubit Pauli operators \(\{\sigma_j\}_{j \in V}\) and \(\{b_j\}_{j \in V}\) are non-zero real numbers.
The frustration graph \(G=(V,E)\) of \(H\) is the graph with vertex set \(V\) with edges between any two vertices if the corresponding Pauli terms anticommute. A graph is claw-free if it does not include the claw (\(K_{1,3}\)) as an induced subgraph. A simplicial clique is a clique \(K_s\), such that for every \(v \in K_s\), the neighbourhood of \(v\) in \(V \setminus Ks\) induces a clique.
Our main result may be stated informally as follows.
