  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).
