Luttinger Liquid with One Impurity: Duality, Universality and Qualitative Description of the Phenomenon

Abstract

A new approach describing Luttinger Liquid with point-like impurity as field theory for the phase of scattering is developed. It is based on a matching of the electron wave functions at an impurity position point. As a result of the approach, an expression for non-local action has been taken. The non-locality of the theory leads to the convergence of the observed values in an ultraviolet region. It allows studying conductance of the channel up to electron–electron interaction strength of the order of unit. Expansion of the non-local action in small frequency powers makes possible to develop a new approach to the renormalization group analysis of the problem. This method differs from the “poor man’s” approach widely used in solid-state physics. We have shown, that in the Luttinger liquid “poor man’s” approach breaks already in two-loop approximation. We analyse the reason of this discrepancy. The qualitative description of the phenomenon is discussed in detail.