The Effects of Phonon Dynamics on the Singular Behavior of a One-Dimensional Fermi Gas.

Phonon dynamical effects are investigated for two model systems of fermion-phonon interactions. The first model is a Tomonaga-Luttinger model generalized to include forward scattering fermion-phonon interactions. For a certain range of the coupling constants, the model is solved exactly. The method of solution is a direct diagonalization of the Hamiltonian in the space of the Tomonaga bosons. Bosonization of Fermi fields is used to obtain the single-particle propagator, and the charge-density wave (CDW) and superconducting (SUP) response functions. A CDW occurs for regions of repulsive direct fermion-fermions interactions. The bulk of phase space for which the SUP instability occurs lies within the region of attractive fermion-fermion interactions. The second model studied is the Frohlich Hamiltonian which includes backscattering of fermions by phonons. This model is investigated by employing a low-type renormalization group. The scaling equations are obtained in the parquet (one-loop) approximation, and are found to be identical to that of the nonretarded model. The phonon dynamics, however, affect the correlation functions, which are also obtained in the parquet approximation.