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arXiv:2108.06494 · Journal of Physics A Mathematical and Theoretical

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

Maksim Tomchenko

Published Feb 15, 2022• 2 citations
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Abstract

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 ρ ( 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 γ ⩾ 0. The analysis shows that the ideal crystal corresponds to the quantum numbers (from Gaudin’s equations) n 1 = ⋯ = n N = N and to the coupling constant γ ≲ 1. We also find that the ground state (GS) of the system ( n 1 = ⋯ = n N = 1) corresponds to a liquid for any γ and any N ≫ 1. In this case, the wave function of the GS is nodeless, and the wave function of the ideal crystal has nodes.

DOI
10.1088/1751-8121/ac552b

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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 ρ ( 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 γ ⩾ 0. The analysis shows that the ideal crystal corresponds to the quantum numbers (from Gaudin’s equations) n 1 = ⋯ = n N = N and to the coupling constant γ ≲ 1. We also find that the ground state (GS) of the system ( n 1 = ⋯ = n N = 1) corresponds to a liquid for any γ and any N ≫ 1. In this case, the wave function of the GS is nodeless, and the wave function of the ideal crystal has nodes.
As is known, the ground state (GS) of a system of spinless bosons must be non-degenerate and must be described by a nodeless wave function. With the help of the general quantum mechanical analysis we show that any \textit{anisotropic} state of a system of spinless bosons is degenerate. We prove this for a two-dimensional (2D) system, infinite or finite circular, and for a three-dimensional (3D) system, infinite or finite ball-shaped. It is natural to expect that this is valid for finite 2D and 3D systems of any shape. Hence, GS of a Bose system of any density is isotropic and, therefore, corresponds to a liquid or gas. Therefore, the lowest state of a 2D or 3D natural crystal consisting of spinless bosons should be described by a wave function with nodes. This leads to nontrivial experimental predictions. We propose a possible ansatz for the wave function of the lowest state of a 3D Bose crystal and discuss possible experimental consequences.
As is known, an elementary excitation of a many-particle system with boundaries is not characterized by a definite momentum. We obtain the formula for the quasimomentum of an elementary excitation for a one-dimensional system of N spinless point bosons under zero boundary conditions (BCs). In this case, we use Gaudin's solutions obtained with the help of the Bethe ansatz. We have also found the dispersion laws of the particle-like and hole-like excita tions under zero BCs. They coincide with the known dispersion laws obtained under periodic BCs.
We perform the symmetry analysis of the properties of the ground state of a finite system of interacting spinless bosons for the three most symmetric boundary conditions (BCs): zero BCs with spherical and circular symmetries, as well as periodic BCs. The symmetry of the system can lead to interesting properties. For instance, the density of a periodic Bose system is an exact constant: ρ(r) = const. Moreover, in the case of perfect spherical symmetry of BCs, the crystalline state cannot produce the Bragg peaks. The main result of the article is that symmetry properties and general quantum-mechanical theorems admit equally both crystalline and liquid ground state for a Bose system of any density.
We study bosons in a one-dimensional hard-wall box potential. In the case of contact interaction, the system is exactly solvable by the Bethe ansatz, as first shown by Gaudin in 1971. Although contained in the exact solution, the boundary energy in the thermodynamic limit for this problem is only approximately calculated by Gaudin, who found the leading order result at weak repulsion. Here we derive an exact integral equation that enables one to calculate the boundary energy in the thermodynamic limit at an arbitrary interaction. We then solve such an equation and find the asymptotic results for the boundary energy at weak and strong interactions. The analytical results obtained from the Bethe ansatz are in agreement with the ones found by other complementary methods, including quantum Monte Carlo simulations. We study the universality of the boundary energy in the regime of a small gas parameter by making a comparison with the exact solution for the hard rod gas.
It is usually assumed that the Bose crystal at T=0T=0T=0 corresponds to the genuine ground state of a Bose system, i.e., this state is non-degenerate and is described by the wave function without nodes. By means of symmetry analysis we show that the ground state of a Bose system of any density should correspond to a liquid or gas, but not to a crystal. The main point is that any anisotropic state of a system of spinless bosons is degenerate. We prove this for an infinite three-dimensional (3D) system and a finite ball-shaped 3D system. One can expect that it is true also for a finite system of any form. Therefore, the anisotropic state cannot be the genuine ground state. Hence, a zero-temperature natural 3D crystal should correspond to an excited state of a Bose system. The wave function Ψ0cΨ^{c}_{0}Ψ0c​ of a zero-temperature 3D Bose crystal is proposed for zero boundary conditions. Apparently, such Ψ0cΨ^{c}_{0}Ψ0c​ corresponds to a local minimum of energy (absolute minimum corresponds to a liquid). Those properties yield the possibility of existence of superfluid liquid H2_{2}2​, Ne, Ar, and other inert elements. We propose several possible experimental ways of obtaining them.