Spontaneous symmetry breaking in the non-linear Schrodinger hierarchy with defect

TL;DR

This paper develops an exact quantum solution for a multi-component non-linear Schrödinger equation with a defect, revealing how the defect induces spontaneous symmetry breaking and classifying boundary conditions via RT algebras.

Contribution

It introduces a quantum inverse scattering approach for the NLS equation with a defect, classifies all boundary conditions, and demonstrates spontaneous symmetry breaking caused by the defect.

Findings

Explicit classification of reflection and transmission matrices.

Identification of unbroken symmetry generators.

Demonstration of defect-induced spontaneous symmetry breaking.

Abstract

We introduce and solve the one-dimensional quantum non-linear Schrodinger (NLS) equation for an N-component field defined on the real line with a defect sitting at the origin. The quantum solution is constructed using the quantum inverse scattering method based on the concept of Reflection-Transmission (RT) algebras recently introduced. The symmetry of the model is generated by the reflection and transmission defect generators defining a defect subalgebra. We classify all the corresponding reflection and transmission matrices. This provides the possible boundary conditions obeyed by the canonical field and we compute these boundary conditions explicitly. Finally, we exhibit a phenomenon of spontaneous symmetry breaking induced by the defect and identify the unbroken generators as well as the exact remaining symmetry.