Partition Functions of Strongly Correlated Electron Systems as "Fermionants"

TL;DR

This paper introduces the 'fermionant', a new mathematical object generalizing determinants, which can represent partition functions of complex strongly correlated electron systems and potentially address fermion sign problems.

Contribution

The paper defines the fermionant, relates it to physical models, and suggests its computation could solve fermion sign problems in quantum many-body physics.

Findings

Fermionants generalize determinants for N>1.

Partition functions of key models are expressible as fermionants.

Polynomial-time computation of fermionants could resolve fermion sign issues.

Abstract

We introduce a new mathematical object, the "fermionant" ${\mathrm{Ferm}}_N(G)$, of type $N$ of an $n \times n$ matrix $G$. It represents certain $n$-point functions involving $N$ species of free fermions. When N=1, the fermionant reduces to the determinant. The partition function of the repulsive Hubbard model, of geometrically frustrated quantum antiferromagnets, and of Kondo lattice models can be expressed as fermionants of type N=2, which naturally incorporates infinite on-site repulsion. A computation of the fermionant in polynomial time would solve many interesting fermion sign problems.