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

Shu Nakamura

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).

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 -dimensional square lattice . In fact, they are identical operator valued matrices under suitable choices of their representations on . We employ the formulations of the staggered fermion by Nakamura (2024), and the discrete cohomology structure on the square lattices by Miranda-Parra (2023).
Paper Structure (5 sections, 1 theorem, 24 equations)

This paper contains 5 sections, 1 theorem, 24 equations.

Table of Contents

  1. Introduction
  2. Proof
  3. Notations
  4. The staggered fermion model
  5. The discrete Hodge-Dirac operators

Key Result

Theorem 1

There is a unitary transform $U$ : $\bigoplus_{k=0}^d\bigwedge^k \mathbb{C}^d\simeq \mathbb{C}^{2^d} \to \mathbb{C}^{2^d}$ such that

Theorems & Definitions (1)

  • Theorem 1