Search for fermion actions on hyperdiamond lattices

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 operators which require fine-tuning. We construct hyperdiamond lattice actions with enough symmetries to exclude such fine-tuning; however, they produce multiple doublings. Constraining the actions to exhibit minimal doubling breaks the requisite symmetry.