Equivalence of the staggered fermion Hamiltonan and the discrete Hodge-Dirac operator on square lattices

TL;DR

This paper demonstrates that the free massless staggered fermion Hamiltonian on square lattices is mathematically equivalent to a discrete Hodge-Dirac operator, unifying two formulations in lattice field theory.

Contribution

It establishes an exact operator equivalence between staggered fermion Hamiltonians and discrete Hodge-Dirac operators on square lattices, connecting different mathematical frameworks.

Findings

Proves the operator equivalence on $ ext{h}\mathbb{Z}^d$ lattices.

Identifies the operators as identical matrices under suitable representations.

Utilizes recent formulations of staggered fermions and discrete cohomology.

Abstract

We show that the free massless staggered fermion (or the KS-fermion) Hamiltonian is equivalent to a discrete Hodge-Dirac operator on the $d$-dimensional square lattice $h\mathbb{Z}^d$. In fact, they are identical operator valued matrices under suitable choices of their representations on $\ell^2(2h\mathbb{Z}^d)\otimes\mathbb{C}^{2^d}$. We employ the formulations of the staggered fermion by Nakamura (2024), and the discrete cohomology structure on the square lattices by Miranda-Parra (2023).