The Dirac materials, such as graphene and three-dimensional topological insulators, have attracted much attention because they exhibit novel quantum phenomena with their low energy electrons governed by the relativistic Dirac equations. One particular interest is to generate Dirac cone anisotropy so that the electrons can propagate differently from one direction to the other, creating an additi...

The fermionic topological charge of lattice gauge fields, given in terms of a spectral flow of the Hermitian Wilson–Dirac operator, or equivalently, as the index of Neuberger’s lattice Dirac operator, is shown to have analogous properties to Lüscher’s geometrical lattice topological charge. The main new result is that it reduces to the continuum topological charge in the classical continuum lim...

We study the asymptotic of the spectrum of the spin Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin–Uribe [13, Theorem 2], which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin [10].

For a compact spin manifold M isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the Dirac operators, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues.

It is a well-known fact that on a bounded spectral interval the Dirac spectrum can described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function Z → R and endow the space of all spectra with an arsinh-uniform metric. We prove that the sp...

Let M be a compact Riemannian manifold with an action by isometries of a compact Lie group G. Suppose that this action could be lifted to an action by isometries on a Clifford bundle E over M. We use the method of the Witten deformation to compute the virtual representation-valued index of a transversally elliptic Dirac operator on E. We express the multiplicities of the associated representati...

casimir forces of massive ferrnionic dirac fields are calculated for parallel plates geometry in spatial space with dimension d and imposing bag model boundary conditions. it is shown that in the range of ma>>l where m is mass of fields quanta and a is the separation distance of the plates, it is equal to massive bosonic fields casimir force for each degree of freedom. we argue this equality ex...

Using a coset version of the cubic Dirac operators for affine Lie algebras, we give an algebraic construction of the Dirac induction homomorphism for loop group representations. With this we prove a homogeneous generalization of the Weyl–Kac character formula and show compatibility with Dirac induction for compact Lie groups.