Dirac Fermions with Disorder in Two Dimensions: Exact Result

TL;DR

This paper derives an exact analytical result for disordered Dirac fermions in two dimensions, linking Green's functions with different types of disorder and comparing lattice and continuum approaches.

Contribution

It presents an exact solution for the Green's function of disordered 2D Dirac fermions, connecting lattice disorder models with continuum renormalization techniques.

Findings

Imaginary part of Green's function expressed as another Green's function with poles only in the lower half-plane.

Exact results obtained for Lorentzian disorder distribution using Cauchy integration.

Comparison made between lattice model results and continuum renormalization group calculations.

Abstract

Dirac fermions on a two-dimensional lattice with disorder are considered. The Dirac mass, which controls the gap between the two bands of the fermions, is subject to random fluctuations. Another type of disorder is discussed presented by a random vector potential. It is shown that the imaginary part of the one-particle Green's function can be written as the imaginary part of another Green's function which has only poles on the lower half-plane. Therefore, it is possible to perform a Cauchy integration for a Lorentzian distribution in analogy with the Lloyd model. The results are compared with calculations performed in the continuum limit based on renormalization group and bosonization methods.