The Peierls transition and electron localization by a random potential in a one-dimensional conductor

The influence of a random impurity potential on a simple model of a one-dimensional metal undergoing a Peierls transition is considered. The following properties are studied: the order parameter, the transition temperature, the state density. Contrary to the "kinetic" approach, where only nonintersecting impurity lines are taken into account, it appears that the energy gap is absent at any impurity concentration, although at small concentrations the state density at the Fermi level decreases exponentially with concentration. The static conductivity of a finite sample is calculated at various temperatures. The formation of a charge density wave enhances localization: the localization length decreases with increasing order parameter. The conductivity at small but finite frequency is obtained. Close to the critical temperature it decreases rapidly withT_c-T.