Theory of a two-level system interacting with a degenerate electron gas. I. Partition function

The problem of the two-level system (TLS) interacting with a free-electron gas is considered. In a previous work Yu and Anderson treated that problem by taking into account the screening by the conduction electrons described by an arbitrarily large phase shift. The present paper is devoted to a generalization of their result by including weak electron-assisted TLS transition processes. Generalizing the Yuval-Anderson technique, the partition function is derived, which has the form of the partition function of a one-dimensional Coulomb gas with logarithmic interactions. The charges introduced depend on the phase shifts and on the electron spin and orbital quantum numbers corresponding to the incoming and outgoing electrons in the related electron-TLS scattering processes. Additionally charged dipoles are introduced to describe those processes in which the scattered electrons do not change their quantum number. Special care is paid to the formal divergencies occurring in these processes if the long-time asymptotic expressions are used for the conduction-electron Green’s functions. In this way, going beyond the charge-charge interaction, charge-dipole and dipole-dipole interactions are introduced. The next paper deals with the derivation of the scaling equations, which are derived by eliminating the short-time behavior.