Disorder effects in two-dimensional Fermi systems with conical spectrum: exact results for the density of states

Abstract

The influence of weak non-magnetic disorder on the single-particle density of states ϱ( ω) of two-dimensional electron systems with a conical spectrum is studied. We use a non-perturbative approach, based on the replica trick with subsequent mapping of the effective action onto a one-dimensional model of interacting fermions, the latter being treated by abelian and non-abelian bosonization methods. Specifically, we consider a weakly disordered p- or d-wave superconductor, in which case the problem reduces to a model of (2+l)-dimensional massless Dirac fermions coupled to random, static, generally non-abelian gauge fields. It is shown that the density of states of a two-dimensional p- or d-wave superconductor, averaged over randomness, follows a non-trivial power-law behavior near the Fermi energy: ϱ( ω) ∼ 1 | ω| α . The exponent α > 0 is exactly calculated for several types of disorder. We demonstrate that the property ϱ(0) = 0 is a direct consequence of a continuous symmetry of the effective fermoic model, whose breakdown is forbidden in two dimensions. As a counter example, we also discuss another model with a conical spectrum - a two-dimensional orbital antiferromagnet, where static disorder leads to a finite ϱ(0) due to the breakdown of a discrete (particle-hole) symmetry.