Excitation spectrum of bosons in a finite one-dimensional circular waveguide via the Bethe ansatz

TL;DR

This paper numerically analyzes the excitation spectrum of finite one-dimensional bosonic systems using the Bethe ansatz, revealing novel finite-size effects and phase transition characteristics relevant to ultra-cold atom experiments.

Contribution

It provides the first detailed numerical solutions of Bethe ansatz equations for finite-size attractive bosonic systems, highlighting deviations from string solutions and exploring phase transitions.

Findings

Finite-size solutions differ from classical string solutions.

Excited state string solutions are characterized in the strong interaction limit.

Quantum phase transition features are identified as a function of interaction strength.

Abstract

The exactly solvable Lieb-Liniger model of interacting bosons in one-dimension has attracted renewed interest as current experiments with ultra-cold atoms begin to probe this regime. Here we numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss the novel features of the solutions, and how they deviate from the well-known string solutions [H. B. Thacker, Rev. Mod. Phys.\ \textbf{53}, 253 (1981)] at finite densities. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Finally we compare our results to those of exact diagonalization of the many-body Hamiltonian in a truncated basis. We also present excited state solutions and the excitation spectrum for the repulsive 1D Bose gas on a ring.