Spectrum of Relativistic Fermions in a 2d Doped Lattice

Motivated by some previous work on fermions on random lattices and by suggestions that impurities could trigger parity breaking in 2d crystals, we have analyzed the spectrum of the Dirac equation on a two dimensional square lattice where sites have been removed randomly — a doped lattice. We have found that the system is well described by a sine-Gordon action. The solitons of this model are the lattice fermions, which pick a quartic interaction due to the doping and become Thirring fermions. They also get an effective mass different from the lagrangian mass. The system seems to exhibit spontaneous symmetry breaking, exactly as it happens for a randomly triangulated lattice. The associated “Goldstone boson” is the sine-Gordon scalar. We argue, however, that the peculiar behaviour of 〈ψ̄ψ〉 is due to finite size effects.