Breaking a "poor man" RG approach in the Luttinger liquid with one impurity

TL;DR

This paper develops a new RG approach based on Gell-Mann-Low formulation to analyze conductance in the Luttinger liquid with an impurity, revealing limitations of the traditional 'poor man's' scaling and providing a method for more accurate calculations.

Contribution

It introduces a novel RG analysis method for the Luttinger model with impurity, highlighting the scheme dependence of the Gell-Mann-Low function and improving conductance calculations beyond leading order.

Findings

'Poor man's' scaling is only valid at leading order.

Corrections to 'poor man's' approximation differ from previous results.

Method for conductance calculation in post-leading approximations is formulated.

Abstract

Using derived previously effective theory we explore conductance in the Luttinger model with one impurity. A new approach to the renormalization group (RG) analysis of this model is developed. It is based on the original Gell-Mann-Low formulation of RG. We sum up infrared logarithmic contibutions to conductance in the leading and few subsequent approximations. We analyze the validity of widely used ``poor man's'' scaling approach and find that it is applicable only in the leading approximation. Our results for corrections to this approximation are different from results obtained in other papers. It should be expected beforehand, as Gell-Mann-Low function of the model is not regularization scheme invariant. For this reason the observed quantity (e.g., conductance) can not satisfy the Gell-Mann-Low equation beyond the "leading-log"\ approximation as it is supposed in the "poor man's" approach. We formulate the method to calculate the conductance from renormalized hamiltonian in the post-leading approximations and match results to the case of weak impurity where the answer is known in any order in electron-electron interaction.