Holstein Model In Infinite Dimensions

Patrizia Benedetti, R. Zeyher

The normal state of the Holstein model is studied at half-filling in infinite dimensions and in the adiabatic regime. The dynamical mean-field equations are solved using perturbation expansions around the extremal paths of the effective action for the atoms. We find that the Migdal-Eliashberg expansion breaks down in the metallic state if the electron-phonon coupling $\ensuremath{\lambda}$ exceeds a value of about 1.3 in spite of the fact that the formal expansion parameter $\ensuremath{\lambda}{\ensuremath{\omega}}_{0}{/E}_{F} ({\ensuremath{\omega}}_{0}$ is the phonon frequency, ${E}_{F}$ the Fermi energy) is much smaller than 1. The breakdown is due to the appearance of more than one extremal path of the action. We present numerical results which illustrate in detail the evolution of the local Green's function, the self-energy, and the effective atomic potential as a function of $\ensuremath{\lambda}.$

Chunli Zhang

Florian Gebhard

James K. Freericks

A strong-coupling expansion for the Holstein (electron-phonon) model is carried out through fourth order. In the large-frequency limit the expansion agrees with known results for the attractive Hubbard model. Mean-field theory is employed to determine transition temperatures of the effective (pseudospin) Hamiltonian and the results are compared with recent Monte Carlo simulations of these models in infinite dimensions.

Walter Metzner, D. Vollhardt

Glen L. Goodvin, Mona Berciu

We investigate the existence of end points in the dispersion of Holstein polarons in various dimensions, using the Momentum Average (MA) approximation which has proved to be very accurate for this model. An end point separates momenta for which the lowest-energy state is a discrete level, i.e. , an infinitely-lived polaron, from those where the lowest-energy feature is a continuum in which the "polaron" is signalled by a resonance with a finite lifetime. While such end points are known to not appear in 1D, we show here that they are generic in 3D if the particle-boson coupling is not too strong. The 2D case is "critical": a pure 2D Holstein model has no end points, like the 1D case. However, any amount of interlayer hopping leads to 3D-like behavior. As a result, such end points are expected to appear in the spectra of layered, quasi-2D systems described by Holstein models. Generalizations to other models are also briefly discussed.