Representation of the Luttinger Liquid with Single Point-like Impurity as a Field Theory for the Phase of Scattering

Abstract

A new approach describing Luttinger Liquid with point-like impurity as field theory for the phase of scattering is developed. It based on a matching of the electron wave functions at 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 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, 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.