Resonant tunnelling between Luttinger liquids: solvable case

TL;DR

This paper analyzes the conductance properties of a Luttinger liquid with a quantum dot, revealing a solvable model that provides exact expressions for current-voltage characteristics and conductance scaling.

Contribution

It introduces a solvable model of resonant tunneling in Luttinger liquids using bosonisation and re-fermionisation, mapping it to a Kondo-type problem with an exact solution.

Findings

Derived an analytic expression for non-linear current-voltage characteristics.

Obtained the exact scaling function for linear conductance.

Analyzed the effects of bias, gate voltages, temperature, and asymmetry on conductance.

Abstract

We discuss the conductance of a Luttinger liquid interrupted by a quantum dot containing a single resonant level. Using bosonisation and re-fermionisation methods, we find a mapping to a Kondo-type problem which possesses a non-trivial Toulouse-type solvable point. At this point, we obtain an analytic expression for the non-linear current-voltage characteristics and analyse the differential conductance and the width of the resonance peak as functions of bias and gate voltages, temperature, and barrier asymmetry. We also determine the exact scaling function for the linear conductance.