Duality between different geometries of a resonant level in a Luttinger liquid

TL;DR

This paper establishes an exact duality between two geometries of a quantum dot in a Luttinger liquid, revealing how transport properties like conductance are affected by interactions and bias voltage.

Contribution

It introduces a novel duality mapping between side-coupled and embedded quantum dot geometries in Luttinger liquids, valid even under finite bias.

Findings

Conductance exhibits an antiresonance with a width that vanishes or diverges depending on the interaction parameter g.

The duality maps the Luttinger parameter g to its inverse, relating transmittance and reflectance.

Transport properties are significantly altered by interactions, with conductance behavior depending on the value of g.

Abstract

We prove an exact duality between the side-coupled and embedded geometries of a single level quantum dot attached to a quantum wire in a Luttinger liquid phase by a tunneling term and interactions. This is valid even in the presence of a finite bias voltage. Under this relation the Luttinger liquid parameter g goes into its inverse, and transmittance maps onto reflectance. We then demonstrate how this duality is revealed by the transport properties of the side-coupled case. Conductance is found to exhibit an antiresonance as a function of the level energy, whose width vanishes (enhancing transport) as a power law for low temperature and bias voltage whenever g>1, and diverges (suppressing transport) for g<1. On resonance transmission is always destroyed, unless g is large enough.