Exact Boundary Modes in an Interacting Quantum Wire

TL;DR

This paper exactly analyzes boundary modes in an interacting quantum wire with pairing, revealing a phase transition and bound states at edges, governed by interaction strength and a self-duality symmetry.

Contribution

It provides an exact Bethe Ansatz solution for boundary modes in an interacting quantum wire with pairing, uncovering a phase transition and bound state behavior.

Findings

Bound states localized at edges for attractive interactions

Absence of bound states at edges for repulsive interactions

Sharp avoided level crossing due to self-duality

Abstract

The boundary modes of one dimensional quantum systems can play host to a variety of remarkable phenomena. They can be used to describe the physics of impurities in higher dimensional systems, such as the ubiquitous Kondo effect or can support Majorana bound states which play a crucial role in the realm of quantum computation. In this work we examine the boundary modes in an interacting quantum wire with a proximity induced pairing term. We solve the system exactly by Bethe Ansatz and show that for certain boundary conditions the spectrum contains bound states localized about either edge. The model is shown to exhibit a first order phase transition as a function of the interaction strength such that for attractive interactions the ground state has bound states at both ends of the wire while for repulsive interactions they are absent. In addition we see that the bound state energy lies within the gap for all values of the interaction strength but undergoes a sharp avoided level crossing for sufficiently strong interaction, thereby preventing its decay. This avoided crossing is shown to occur as a consequence of an exact self-duality which is present in the model.