Topological Aspects of Fermions on Hyperdiamond

TL;DR

This paper calculates the topological index of Dirac fermions on a family of high-dimensional lattices generalizing graphene, linking lattice properties with topological and anomaly phenomena in quantum field theories.

Contribution

It introduces a method to compute the Dirac index on L_{2N} lattices, connecting lattice topologies with SU(2N+1) weight lattices and topological invariants.

Findings

Index matches chiral anomalies in graphene and QCD_4

Explicit solutions of the gauged Dirac equation on L_{2N}

Topological charges computed on 2-cycles of 2N-dimensional supercells

Abstract

Motivated by recent results on the index of the Dirac operator D={\gamma}^{{\mu}}D_{{\mu}} of QCD on lattice and also by results on topological features of electrons and holes of 2-dimensional graphene, we compute in this paper the Index of D for fermions living on a family of even dimensional lattices denoted as L_{2N} and describing the 2N-dimensional generalization of the graphene honeycomb. The calculation of this topological Index is done by using the direct method based on solving explicitly the gauged Dirac equation and also by using specific properties of the lattices L_{2N} which are shown to be intimately linked with the weight lattices of SU(2N+1). The Index associated with the two leading N=1 and N=2 elements of this family describe precisely the chiral anomalies of graphene and QCD_4. Comments on the method using the spectral flow approach as well as the computation of the topological charges on 2-cycles of 2N-dimensional compact supercell in L_{2N} and applications to QCD_4 are also given.