Taste-splitting mass and edge modes in $3+1$ D staggered fermions

TL;DR

This paper explores the symmetry and anomaly structure of 3+1D staggered fermions, revealing how boundary flavor symmetries and anomalies originate from the lattice Hamiltonian through domain-wall constructions.

Contribution

It identifies specific mass terms and their symmetries in the staggered fermion Hamiltonian, demonstrating the realization of boundary anomalies from the ultraviolet lattice theory.

Findings

A larger residual symmetry is preserved by a one-link mass term.

Massless Dirac fermions localize on domain walls with specific symmetry properties.

The boundary exhibits a parity anomaly linked to the bulk staggered fermion structure.

Abstract

We investigate the symmetry structure of the $3+1$ D staggered fermion Hamiltonian and its implications for anomalies. Since the spin and flavor degrees of freedom of Dirac fermions are distributed over the lattice, in addition to the standard on-site mass term, the staggered fermion system also admits one-, two-, and three-link bilinear terms within a unit cube as local, charge conserving mass terms with different spin and flavor dependence. We identify the spin flavor structures of all those bilinear mass terms and determine the symmetries preserved by each of them. Among them, one of the one-link mass terms preserves a larger residual symmetry associated with conserved charges that generate the Onsager algebra. Motivated by this structure, we consider a kink profile of the one-link mass and analyze the resulting domain-wall system. In the low-energy limit, the $3+1$ D bulk becomes gapped, while two-flavor massless Dirac fermions appear as localized modes on the $2+1$ D domain wall. We show that the bulk conserved charges act on the wall as generators of a flavor $\mathrm{SU}(2)$ symmetry, and that no symmetric mass gap is allowed for the boundary theory when this $\mathrm{SU}(2)$ symmetry and space reflection symmetry are both imposed. This realizes the parity anomaly of the boundary theory and shows that the boundary flavor symmetry and anomaly descend from the ultraviolet staggered-fermion Hamiltonian rather than emerging only in the infrared.