Quantum Dot in Interacting Environments

TL;DR

This paper uses Bethe Ansatz to exactly solve models of a quantum dot coupled to an interacting quantum wire in two geometries, revealing how interactions influence the dot's behavior and its role as an impurity.

Contribution

It provides exact solutions for the eigenstates, spectrum, and ground state occupation of quantum dots in Luttinger liquids, elucidating the effects of interactions and geometry.

Findings

At low energies, the dot becomes fully hybridized.

The dot acts as a backscattering impurity or tunnel junction depending on geometry.

Interaction sign determines the relation between geometries.

Abstract

A quantum impurity attached to an interacting quantum wire gives rise to an array of of new phenomena. Using Bethe Ansatz we solve exactly models describing two geometries of a quantum dot coupled to an interacting quantum wire: a quantum dot that is (i) side-coupled and (ii) embedded in a Luttinger liquid. We find the eigenstates and determine the spectrum through the Bethe Ansatz equations. Using this we derive exact expressions for the ground state dot occupation. The thermodynamics are then studied using the thermodynamics Bethe Ansatz equations. It is shown that at low energies the dot becomes fully hybridized and acts as a backscattering impurity or tunnel junction depending on the geometry and furthermore that the two geometries are related by changing the sign of the interactions. Although remaining strongly coupled for all values of the interaction in the wire, there exists competition between the tunneling and backscattering leading to a suppression or enhancement of the dot occupation depending on the sign of the bulk interactions.