Exact ground state of finite Bose-Einstein condensates on a ring

TL;DR

This paper provides exact solutions for the ground state of finite Bose-Einstein condensates on a ring, demonstrating the applicability of Bethe-ansatz for both repulsive and attractive interactions and comparing finite N results with thermodynamic limits.

Contribution

It presents the exact ground state energies for finite N bosons on a ring, extending Lieb and Liniger's solutions and analyzing the transition between repulsive and attractive interactions.

Findings

Exact ground state energies are obtained for up to fifty particles.

The Bethe-ansatz applies smoothly across zero interaction strength.

Finite N energies can significantly differ from thermodynamic limit results.

Abstract

The exact ground state of the many-body Schr\"odinger equation for $N$ bosons on a one-dimensional ring interacting via pairwise $\delta$-function interaction is presented for up to fifty particles. The solutions are obtained by solving Lieb and Liniger's system of coupled transcendental equations for finite $N$. The ground state energies for repulsive and attractive interaction are shown to be smoothly connected at the point of zero interaction strength, implying that the \emph{Bethe-ansatz} can be used also for attractive interaction for all cases studied. For repulsive interaction the exact energies are compared to (i) Lieb and Liniger's thermodynamic limit solution and (ii) the Tonks-Girardeau gas limit. It is found that the energy of the thermodynamic limit solution can differ substantially from that of the exact solution for finite $N$ when the interaction is weak or when $N$ is small. A simple relation between the Tonks-Girardeau gas limit and the solution for finite interaction strength is revealed. For attractive interaction we find that the true ground state energy is given to a good approximation by the energy of the system of $N$ attractive bosons on an infinite line, provided the interaction is stronger than the critical interaction strength of mean-field theory.