Wilson fermion doubling phenomenon on an irregular lattice: Similarity and difference with the case of a regular lattice

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 in the case when the "naive fermions" are introduced. The statement is proved on a four-dimensional lattice by means of the Atiyah-Singer index theorem, and 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 lattices are "bad" quasiparticles.