Several Electrons in the One-Dimensional Deformable Continuum

Abstract

Strongly coupled electron-phonon systems have attracted interest for more than fifty years. It was shown that an electron interacting with acoustic phonons through the deformation potential can be trapped by the self-induced lattice distortion. I) The condition for the self-trapping depends on the dimensionality of the system. ),3) There is a certain critical value of the coupling constant in three dimensions, while in one dimension the self-trapping occurs, however small the coupling constant may be. It was pointed out that two electrons attract each other through the phonon-mediated interaction and form a bipolaron when the phonon mediated attraction overcomes the inter-electronic repulsion.) Recently, intrinsic defect states in polyacetylene have aroused renewed attention. Explicit solutions have been obtained exactly for the soliton,S) the polaron) and the bipolaron),8) in the continuum model of polyacetylene.S) This success is indebted to the application of the inverse scattering method (ISM). In the model of polyacetylene, the object of the ISM is a two component Dirac equation of one dimension which arises as the Bogoliubov-de Gennes equation. On the other hand, the electronic state is determined by a Schrodinger equation in the molecular-crystal model,g) where electrons in a single energy band interact with an acoustic phonon field. Although the ISM for Schrodinger equations is much simpler than that for Dirac equations, the ISM has not applied to the molecular-crystal model yet. The essence of the problem in the molecular-crystal model within the adiabatic approximation is in determining the electronic state and the lattice distortion selfconsistently. In this paper, the ISM is used to represent the lattice distortion in the one-dimensional molecular-crystal model by the scattering data of the electron. As a consequence, one can automatically satisfy the self-consistency condition and one needs to manage only the electronic freedom. Let us consider several electrons interacting with an acoustic phonon field in a one-dimensional lattice. Since the main subject of this paper is to study the effect of electron-phonon interaction, the inter-electronic interaction is completely neglected. Replacing the phonon field by an elastic continuum with dilation 0(x), one can write the Schrodinger equation which determines the electronic states as