Path-integral method for a heavy particle moving in a periodic potential and screened by a light degenerate Fermi gas

We study a heavy particle moving in a periodic lattice of arbitrary dimension and interacting with a degenerate fermionic heat bath. We assume that the Fermi energy of the screening particles is large compared to the tight-binding bandwidth of the particle. We develop a path-integral method for the partition function of the heavy particle where we integrate out the electronic degrees of freedom. We perform also a summation over the paths between the screening-interaction points that makes the method capable of treating band motion and hopping. Following the method of Anderson and Yuval, we derive scaling equations by eliminating that part of the phase space of the light particles which is far from the Fermi energy. We find that the screening strength is not renormalized; the bandwidth of the heavy particle is, however, essentially reduced. That reduction coincides with the reduction in the two-site model. Extrapolating the result for larger values of the screening strength we find band motion and localization at zero temperature depending on the coupling strength; thus the behavior is of Ohmic type. We show that the results obtained can be combined with the classical kinetic equation to get the diffusion coefficient. We believe that the present work based on the path-integral method helps to clarify the connection between results obtained by different methods. The relation of the model to other models, especially to the Caldeira-Leggett model, is discussed in detail.