Quantum Transport Framework for Highly Correlated Electrons

Abstract

Before discussing the quantum transport framework, which is the main topic of this chapter, we would like to underline the important points of the conventional treatment. Firstly, it should be noted that in the quantum transport theory the description of the electrical conductivity, even for the simplest cases, presents a substantially harder problem then in the kinetic equation method. The exact methods are usually based on evaluation of the current—current correlation function entering Kubo’s formula. With this formula chosen as a starting point for the Feynman diagram treatment, in order to obtain even the well-known DC conductivity results of the semiclassical kinetic equation method, one needs to include the self-energy effects and vertex corrections, collecting contributions from the infinite series of diagrams (for examples, see [38]). The conductivity is found in an integral form containing the ratio Λ(e)/Γ(e), where Λ(e) is the vertex function and Γ(e) = −2ImΣ(e) = ħτ −1 is the scattering rate (the inverse average time between scattering events). Physically, the vertex function transforms the collision rate Γ(e) into the transport relaxation rate Γ (tr) (e), by introducing a factor of 1 — cosθ which favors scattering events at high-momentum transfer. Nevertheless, the accurate form must be found by solving an integral equation. Disregarding the vertex corrections and replacing Λ(e) → 1 is often a serious mistake. Therefore it seems that one cannot feel comfortable in this field without advanced knowledge and skills in the Feynman diagram technique.