Generalized unparticles, zeros of the Green function, and momentum space topology of the lattice model with overlap fermions

Abstract

The definition of topological invariants $\tilde{\cal N}_4, \tilde{\cal N}_5$ suggested in \cite{VZ2012} is extended to the case, when there are zeros and poles of the Green function in momentum space. It is shown how to extend the index theorem suggested in \cite{VZ2012} to this case. The non - analytical exceptional points of the Green function appear in the intermediate vacuum, which exists at the transition line between the massive vacua with different values of topological invariants. Their number is related to the jump $\Delta\tilde{\cal N}_4$ across the transition. The given construction is illustrated by momentum space topology of the lattice model with overlap fermions. In the vicinities of the given points the fermion excitations appear that cannot be considered as usual fermion particles. We, therefore, feel this appropriate to call them generalized unparticles. This notion is, in general case different from the Georgi's unparticle. However, in the case of lattice overlap fermions the propagator of such excitations is indeed that of the fermionic unparticle suggested in \cite{fermion_unparticle}.