Ward identity and effective medium approximation for conductivity of liquid metals

The self-energy to define the ensemble average of a one-electron Green function for a disordered system is related to the vertex correction for the average of the product of two Green functions by the generalized optical theorem or by the so-called Ward identity. Using this relationship, we evaluate the conductivity of liquid metals both by the Ishida-Yomezawa approximation and by the effective medium approximation from the self-energy forms corresponding to respective approximations. The conductivity formulations thus obtained are shown to be identical with those derived from the diagram method.