Wilson fermion doubling phenomenon on irregular lattice: the similarity and difference with the case of regular lattice

TL;DR

This paper demonstrates that Wilson fermion doubling occurs on irregular lattices, with the doubled modes being exponentially suppressed, differing from regular lattices where they are propagating particles.

Contribution

It proves the existence of Wilson fermion doubling on irregular lattices using the Atiyah-Singer index theorem and highlights the fundamental difference in the nature of doubled quanta.

Findings

Wilson fermion doubling exists on irregular lattices

Doubled modes decay exponentially on irregular lattices

Difference between regular and irregular lattice doubling phenomena

Abstract

It is shown that the Wilson fermion doubling phenomenon on irregular lattices (simplicial complexes) does exist. This means that the irregular (not smooth) zero or soft modes exist. The statement is proved on 4 Dimensional lattice by means of the Atiyah-Singer index theorem, then it is extended easily into the cases $D<4$. But there is a fundamental difference between doubled quanta on regular and irregular lattices: in the latter case the propagator decreases exponentially. This means that the doubled quanta on irregular lattice are "bad" quasiparticles.