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Static transport properties of random alloys: vertex corrections in conserving approximations

The theoretical formulation and numerical evaluation of the vertex corrections in multiorbital techniques of theories of electronic properties of random alloys are analyzed. It is shown that current approaches to static transport properties within the so-called conserving approximations lead to the inversion of a singular matrix as a direct consequence of the Ward identity relating the vertex corrections to one-particle self-energies. We propose a simple removal of the singularity for quantities (operators) with vanishing average values for electron states at the Fermi energy, such as the velocity or the spin torque; the proposed scheme is worked out in details in the self-consistent Born approximation and the coherent potential approximation. Applications involve calculations of the residual resistivity for various random alloys, including spin-polarized and relativistic systems, treated on an ab initio level, with particular attention paid to the role of different symmetries (inversion of space and time).