Exact crystalline solution for a one-dimensional few-boson system with point interaction

TL;DR

This paper provides exact solutions for a one-dimensional system of few spinless bosons with point interactions, analyzing density profiles and wave functions to distinguish between liquid and crystalline states based on quantum numbers and interaction strength.

Contribution

It offers a detailed analysis of the ground and excited states of a few-boson system using Bethe ansatz, identifying conditions for crystalline and liquid phases.

Findings

Ideal crystal state occurs at specific quantum numbers and coupling strength.

Ground state remains a liquid with a nodeless wave function for all interaction strengths.

Wave functions of crystal and liquid states exhibit distinct nodal structures.

Abstract

We study the exact solutions for a one-dimensional system of $N=2; 3$ spinless point bosons for zero boundary conditions. In this case, we are based on M. Gaudin's formulae obtained with the help of Bethe ansatz. We find the density profile $\rho(x)$ and the nodal structure of a wave function for a set of the lowest states of the system for different values of the coupling constant $\gamma\geq 0$. The analysis shows that the ideal crystal corresponds to the quantum numbers (from Gaudin's equations) $n_{1}=\ldots =n_{N}=N$ and to the coupling constant $\gamma \leq 1$. We also find that the ground state of the system ($n_{1}=\ldots =n_{N}=1$) corresponds to a liquid for any $\gamma$ and any $N\gg 1$. In this case, the wave function of the ground state is nodeless, and the wave function of the ideal crystal has nodes.