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

TL;DR

This paper introduces a novel field theory approach for Luttinger Liquids with a point impurity, enabling better analysis of conductance and renormalization effects beyond traditional methods.

Contribution

It develops a non-local action-based field theory for the impurity problem, allowing for improved renormalization group analysis and revealing limitations of the traditional 'poor man's' approach.

Findings

The new approach converges in the ultraviolet region.

It extends analysis to electron-electron interactions of order unity.

The 'poor man's' approach breaks down at two-loop approximation.

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.