Search for Fermion Actions on Hyperdiamond Lattices

TL;DR

This paper explores four-dimensional hyperdiamond lattice actions for fermions, aiming to preserve chiral symmetry and minimize fermion doubling, while analyzing the symmetry properties necessary to avoid fine-tuning.

Contribution

It constructs hyperdiamond lattice actions with sufficient symmetries to prevent fine tuning, and examines the conditions under which minimal doubling can be achieved.

Findings

Constructed symmetric hyperdiamond lattice actions that exclude fine tuning.

Found that minimal doubling actions lack the necessary lattice symmetries.

Identified the trade-off between symmetry and minimal fermion doubling.

Abstract

Fermions moving in a two-dimensional honeycomb lattice (graphene) have, at low energies, chiral symmetry. Generalizing this construction to four dimensions potentially provides fermions with chiral symmetry and only the minimal fermion doubling demanded by the Nielsen-Ninomiya no-go theorem. The practical usefulness of such fermions hinges on whether the action has a necessary set of discrete symmetries of the lattice. If this is the case, one avoids the generation of dimension three and four operators which require fine tuning. We construct hyperdiamond lattice actions with enough symmetries to exclude fine tuning; however, they produce multiple doublings. The limit where the actions exhibit minimal doubling does not possess the requisite symmetry.