Coherent state path integral and Langevin equation of interacting fermions

Abstract

Interacting fermions, electrons and holes in a semiconductor, are coupled to a thermal reservoir of bosons which yield the fluctuating noise. We use a coherent state path integral formulation on the time contour for non-equilibrium systems in terms of anticommuting variables which replace the fermionic creation- and annihilation operators in the time development operator. An auxiliary commuting field σ x ( t p ), defined on the time contour, is introduced by a Hubbard–Stratonovich transformation. In terms of this new field, a Langevin equation is derived which is similar to a saddle-point equation with a random force f x ( t). In comparison to the bosonic case of excitons in a semiconductor previously described, one obtains a different expression for the Langevin equation and, especially, a different relation for the probability distribution of the noise term f x ( t). The calculation for density matrix elements with the Langevin approach can also be interpreted as an average over modified non-equilibrium Green functions with the appropriately derived probability distribution of the noise.