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

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).

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