Numerical study of the crossover from free electrons to small polarons

Abstract

The electron-phonon interaction is one of the fundamental interactions in almost all condensed matter materials. In conventional superconductors, the electron-phonon interaction is the glue that attracts two electrons to one another to form a pair. A strong electron-phonon interaction leads to the concept of a polaron, which is an electron with lattice distortions around it. The small polaron is a polaron with spatial extent comparable to an interatomic dimension of the solid. Evidence for polarons has been identified in many experiments in superconductors and semiconductors. In this thesis we present exact calculations of the polaron. Specifically we have refined Trugman's method to solve the ground state of an electron-phonon coupled system in the whole parameter regime, and we also generalized this method to treat spin-orbit coupled systems. The most difficult regimes, which is the strong-coupling regime and the small phonon frequency limit, have been solved by these refinements. There are three representative kinds of electron-phonon interaction, the Holstein model, the Fr"ohlich model and the BLF-SSH model. In this thesis we have addressed the first and the third one. The second one, the Fr"ohlich model, is very similar to the Holstein model but the interaction is nonlocal. For the Holstein model we have observed the expected smooth crossover from free electrons to small polarons, while for the BLF-SSH model, we have studied the weak coupling regime with perturbation theory and derived a new analytical result for the one-dimensional problem.