Ultraviolet divergence and Ward-Takahashi identity in a two-dimensional Dirac electron system with short-range impurities

Dirac electrons in two-dimensional continuum space subject to $\delta$-function impurities, which have inherent ultraviolet divergence, fail to be gauge-invariant if regularized by the standard momentum (Brillouin-zone) cutoff. A regularization scheme which respects gauge invariance is proposed here based on the cutoff in the impurity-potential range. The relation to a lattice model, which has no ultraviolet divergence, is also discussed.

Figure 1

Three types of extended impurities on a honeycomb lattice. A unit cell is indicated by a dashed rectangle. The primitive translation vectors, ${\mathit{e}}_{1,2}=(\pm 1/2,\sqrt{3}/2)$, are also shown.

Figure 2

Typical diagrams expressing $K_{ij}$ (a) and $K_i$ (b). The solid line represents the electron Green's function [Eq. (4) or (16)], and the dashed line with a cross represents scattering from impurities. Ladder corrections are considered for the vertices consistent with the present self-consistent Born approximation. Expansion of $K_{ij}$ ($K_i$) with respect to $\omega$ ($q_j$) gives ${\sigma }_{ij}^{\left(1\right)}$ (${\sigma }_{ij}^{\left(2\right)}$).