Electronic Transport at Low Temperatures: Diagrammatic Approach

TL;DR

This paper demonstrates that a diagrammatic approach to electronic transport at low temperatures aligns with Boltzmann theory, clarifying the validity of approximations and providing exact solutions for specific scattering mechanisms.

Contribution

It proves the consistency of the diagrammatic evaluation with Boltzmann results and shows that certain uncontrolled approximations are unnecessary for accurate low-temperature behavior.

Findings

Diagrammatic evaluation matches Boltzmann results for low-temperature conductivity.

Exact solutions of the integral equation confirm the asymptotic behavior.

Results apply to electron scattering from phonons and helimagnons.

Abstract

We prove that a diagrammatic evaluation of the Kubo formula for the electronic transport conductivity due the exchange of bosonic excitations, in the usual conserving ladder approximation, yields a result consistent with the Boltzmann equation. In particular, we show that an uncontrolled approximation that has been used to solve the integral equation for the vertex function is unnecessary. An exact solution of the integral equation yields the same asymptotic low-temperature behavior as the approximate one, albeit with a different prefactor, and agrees with the temperature dependence of the Boltzmann solution. Examples considered are electron scattering from acoustic phonons, and from helimagnons in helimagnets.