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arXiv:2112.14641 · Low Temperature Physics

Symmetry properties of the ground state of the system of interacting spinless bosons

Maksim Tomchenko

Published Sep 1, 2022• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1063/10.0013277

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•Jan 1, 2002•15 citations
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.
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.
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.
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.
We investigate the properties of a dilute boson-fermion mixture at zero temperature. The ground-state energy of the system is calculated to second order in the Bose-Fermi coupling constant. The Green's function formalism is applied to obtain results for the effect of Bose-Fermi interactions on the spectrum of phonon excitations, the momentum distribution, the condensate fraction, and the superfluid density of the bosonic component. A quantitative discussion of these effects is presented in the case of mixtures with attractive interspecies interaction.
PACS. 05.30.-d – Quantum statistical mechanics. PACS. 67.40.Db – Quantum statistical theory, ground state, elementary excitations. PACS. 05.30.Jp – Boson systems. Abstract. – The ground state properties and low-lying excitations of a (quasi) one-dimensional system of longitudinally confined interacting bosons are studied. This is achieved by extending Haldane’s harmonic-fluid description to open boundary conditions. The boson density, oneparticle density matrix, and momentum distribution are obtained accounting for finite-size and boundary effects. Friedel oscillations are found in the density. Finite-size scaling of the momentum distribution at zero momentum is proposed as a method to obtain from the experiment the exponent that governs phase correlations. The strong correlations between bosons induced by reduced dimensionality and interactions are displayed by a Bijl-Jastrow wave function for the ground state, which is also derived. The names of Tomonaga and Luttinger are commonly associated with interacting electron systems in one-dimension (1d). The so-called Tomonaga-Luttinger liquids (TLL’s) have the remarkable property that their low-energy spectrum is completely exhausted by gapless