General Transport Theory for Weak Inhomogeneities and Quantum Solids in High External Fields

Abstract

The exact transport equation obtained in a recent work on new renormalization methods for single-time Green′s functions is applied to weakly inhomogeneous quantum systems. The initial density matrix and the external fields are assumed to vary slowly over microscopic length scales like the de Broglie wavelength λ F of the particles, the range λ V , of the interaction and the lattice constant a. In addition, the Hamiltonian without the external fields has to fulfil (quasi-)momentum conservation. The final result is given by a set of local and nonlinear integral equations for the Weyl transforms of all one-particle distribution functions occurring in the system. As an application a quantum solid of Bloch electrons, phonons, and impurities in arbitrarily time-dependent and weakly inhomogeneous electric and magnetic fields is studied. The fields can be moderately high in the sense that r L ⪢ max {λ F , λ V , a} and eE max{λ F , λ V , a} ⪡ ϵ F . Here r L denotes the radius of the Landau orbits, E is the electric field, and ϵ F is the fermi energy of the electrons. In Born approximation, a set of generalized Boltzmann equations is obtained which include interband transition terms from scattering and Zeener tunneling as well as collisional broadening and intracollisional field effects.