Conductance through a potential barrier embedded in a Luttinger liquid: nonuniversal scaling at strong coupling

TL;DR

This paper investigates the conductance of a Luttinger liquid with a potential barrier, deriving RG equations for both weak and strong interactions, revealing nonuniversal scaling behaviors at strong coupling.

Contribution

It introduces a comprehensive RG analysis of conductance in Luttinger liquids with arbitrary interactions and barrier strengths, highlighting nonuniversal effects at strong coupling.

Findings

Derived RG equations for arbitrary interaction strength and barrier

Identified universal and nonuniversal contributions to the beta-function

Provided analytical solutions for temperature-dependent conductance

Abstract

We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g_2, and subject to a potential barrier of arbitrary strength, as a function of temperature. We map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. First the renormalization group (RG) equation for weak interaction is derived in the current operator language both using the operator product expansion and the equation of motion method. To access the strong coupling regime, two methods of deducing the RG equation from perturbation theory, based on the scaling hypothesis and on the Callan-Symanzik formulation, are discussed. The important role of scale independent terms is emphasized. The latter depend on the regulaization scheme used (length versus temperature cutoff). Analyzing the perturbation theory in the fermionic representation, the diagrams contributing to the renormalization group beta-function are identified. A universal part of the beta-function is given by a ladder series and summed to all orders in g_2. First non-universal corrections beyond the ladder series are discussed and are shown to differ from the exact solutions obtained within conformal field theory which use a different regularization scheme. The RG equation for the temperature dependent conductance is solved analytically. Our result agrees with known limiting cases.