Theory of a two-level system interacting with a degenerate electron gas. II. Scaling theory of a one-dimensional Coulomb gas

A one-dimensional Coulomb gas is considered in which the particles represent both charges and dipoles. The values of the charges and dipole moments and the fugacities of that gas have been obtained in the preceding paper by mapping the dynamics of a two-level system (TLS) interacting with a degenerate electron gas onto the present problem. The scaling procedure of Anderson, Yuval, and Hamann is applied to eliminate the close pair which corresponds to the short-time behavior of the TLS. The main contribution arises from pairs of zero total charge, but the role of other pairs is considered as well. The scaling equations show that the electron-assisted transitions of the TLS play a very important role if their value is different from zero for the starting Hamiltonian. In that case localization cannot occur in the case of a spin-(1/2 fermionic system. Localization may occur only if the fermion spin has an artificially large value with spin degeneracy $N_s$>2. That clearly demonstrates that a particle moving in a double potential well may have different dissipative behavior depending upon whether it is coupled to a fermionic or a bosonic heat bath. The screening of the TLS by electrons is described by an arbitrarily large phase shift δ which renormalizes towards the value δ=π/2$N_s$, but the scaling equations lose their validity as the fugacities corresponding to the TLS transitions become of the order of unity. The method developed is a very general one, and thus it can be applied to other similar one-dimensional gases as well.