Exact results for the one-dimensional many-body problem with contact interaction: Including a tunable impurity

TL;DR

This paper derives exact solutions for a one-dimensional many-body quantum system with contact interactions and a tunable impurity, revealing how the impurity influences energy spectra, degeneracies, and bound states, with potential condensed matter applications.

Contribution

It extends the coordinate Bethe ansatz to include a tunable impurity, providing explicit eigenfunctions and Bethe equations for this class of systems.

Findings

Impurity can lift degeneracies in energy levels.

Tunable impurity can create bound states with attractive interactions.

Impurity enables asymmetric confinement of stationary states.

Abstract

The one-dimensional problem of $N$ particles with contact interaction in the presence of a tunable transmitting and reflecting impurity is investigated along the lines of the coordinate Bethe ansatz. As a result, the system is shown to be exactly solvable by determining the eigenfunctions and the energy spectrum. The latter is given by the solutions of the Bethe ansatz equations which we establish for different boundary conditions in the presence of the impurity. These impurity Bethe equations contain as special cases well-known Bethe equations for systems on the half-line. We briefly study them on their own through the toy-examples of one and two particles. It turns out that the impurity can be tuned to lift degeneracies in the energies and can create bound states when it is sufficiently attractive. The example of an impurity sitting at the center of a box and breaking parity invariance shows that such an impurity can be used to confine asymmetrically a stationary state. This could have interesting applications in condensed matter physics.