Topological aspects of fermions on hyperdiamond

Abstract

Motivated by recent results on the index of the Dirac operator D = γ µ D µ 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.

Mathematical structure of the three-dimensional (3D) Ising model

“…The open problems in this field are recent study on topological aspects of fermions on hyperdiamond might be helpful for understanding the 3D-honeycomb Ising lattices. [111] Then the third problem would be the only open question remaining for a rigorous proof of the simple orthorhombic Ising lattice. One might need to develop novel mathematical tools to obtain a rigorous proof of the equivalence between the transfer matrixes V and V' (before/after the unitary transformation), which are respectively with and without the internal factors, and meanwhile, one has to give a detailed analysis of the topologic factors.…”

Higher-order topological matter and fractional chiral states

“…These 8 corners are particularly interesting for us as they host the gapless states of HOT matter we are studying in this paper. Being gapless, these corner states obey massless Dirac-like equation Dψ corner = 0 where D the Dirac-like operator which in our study is given by eqs (47) and (50). The solutions of this equation is studied in the main text.…”

“…The T ν matrices generate the 63 directions in the space of traceless 8×8 matrices; they encode the algebraic structure of H k ; for that we start by studying them with some details and turn after to F ν . First of all, notice that there are different ways to deal with the T ν matrix generators; one of them is given by the algebra of su (8) matrices having seven charge operators as basic observables known as Cartan charge operators [50]; but an interesting realisation of these T ν 's is that given by using 8×8 Dirac matrices Γ A and their normal products [51]…”