Finite-temperature Fermi-liquid theory of electrical conductivity

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Abstract
The Fermi-liquid transport theory of electrical conductivity formulated by Eliashberg [Zh. Eksp. Teor. Fiz. 41, 1241 (1961)] is further developed, using the analytic continuation of the finite-temperature Ward-Takahashi relations obtained by the present author [Ann. Phys. (N.Y.) 173, 226 (1987)] to maintain the theoretical consistency of approximation schemes for solving the Bethe-Salpeter equation. In particular, the finite-temperature Ward-Takahashi relations corresponding to the local conservation of the electron number density and to the conservation of the electron momentum are considered. A general expression for the electrical conductivity in terms of the proper vertex part is also obtained. In the formula, cancellation of the quasiparticle renormalization factors at the Fermi surface is explicitly shown in a general way without assuming a particular model. In the case that the electrons are scattered by phonons, the formula gives the Grüneisen-Bloch formula for the on-shell phonons. For the impurity resistivity, the theory of Edwards [Philos. Mag. 3, 1020 (1958)] is developed to finite temperature by introducing a hypothetical boson field that replaces the impurity averaging.