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arXiv:2108.03633

On the nature of the lowest state of a Bose crystal

Maksim Tomchenko

Published Aug 8, 2021• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2108.03633

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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.
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.
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.
Abstract It is generally accepted that phonons in a superfluid Bose gas are Goldstone bosons. This is justified by spontaneous symmetry breaking (SSB), which is usually defined as follows: the Hamiltonian of the system is invariant under the U ( 1 ) transformation Ψ ^ ( r , t ) → e i α Ψ ^ ( r , t ) , whereas the order parameter Ψ ( r , t ) is not. However, the strict definition of SSB is different: the Hamiltonian and the boundary conditions are invariant under a symmetry transformation, while the ground state is not. Based on the latter criterion, we study a finite system of spinless, weakly interacting bosons using three approaches: the standard Bogoliubov method, the particle-number-conserving Bogoliubov method, and the approach based on the exact ground-state wave function. Our results show that the answer to the question in the title is ‘no’. Thus, phonons in a real-world (finite) superfluid Bose gas are similar to sound in a classical gas: they are not Goldstone bosons, but quantized collective vibrational modes arising from the interaction between atoms. In the case of an infinite Bose gas, however, the picture becomes paradoxical: the ground state can be regarded as either infinitely degenerate or non-degenerate, making the phonon both similar to a Goldstone boson and different from it.
A microscopic description of the zero-energy two-body ground state and many-body static properties of anisotropic homogeneous gases of bosonic dipoles in two dimensions at low densities is presented and discussed. By changing the polarization angle with respect to the plane, we study the impact of the anisotropy, present in the dipole-dipole interaction, on the energy per particle, comparing the results with mean-field predictions. We restrict the analysis to the regime where the interaction is always repulsive, although the strength of the repulsion depends on the orientation with respect to the polarization field. We present a series expansion of the solution of the zero-energy two-body problem, which allows us to find the scattering length of the interaction and to build a suitable Jastrow factor that we use as a trial wave function for both a variational and diffusion Monte Carlo simulation of the infinite system. We find that the anisotropy has an almost negligible impact on the ground-state properties of the many-body system in the universal regime where the scattering length governs the physics of the system. We also show that scaling in the gas parameter persists in the dipolar case up to values where other isotropic interactions with the same scattering length yield different predictions.
We present the exact solution for the many-body wavefunction of a one-dimensional mixture of bosons and spin-polarized fermions with equal masses and infinitely strong repulsive interactions under external confinement. Such a model displays a large degeneracy of the ground state. Using a generalized Bose-Fermi mapping, we find the solution for the whole set of ground-state wave functions of the degenerate manifold and we characterize them according to group-symmetry considerations. We find that the density profile and the momentum distribution depends on the symmetry of the solution. By combining the wave functions of the degenerate manifold with suitable symmetry and guided by the strong-coupling form of the Bethe-Ansatz solution for the homogeneous system, we propose an analytic expression for the many-body wave function of the inhomogeneous system which well describes the ground state at finite, large, and equal interaction strengths, as validated by numerical simulations.
An exact expression for the single-particle reduced density matrix of a translationally invariant system of bosons is obtained, as an average of a functional of the density and current operators. An approximation scheme, based on small fluctuations in density and current, is used in its lowest order, to approximate the single-particle reduced density matrix and the condensate density by a functional of the structure factor, which is a measurable quantity. This is done for the ground state and for finite temperatures. The resulting expressions for the condensate density are discussed. For the ideal gas the result is correct below the condensation temperature. For the case of an interacting liquid, Bose-Einstein condensation exists, in the ground states for two- and three-dimensional systems, and for finite temperatures it does exist only in three dimensions. The present experimental knowledge of the structure factor enables only a crude estimate of the condensed fraction which is found to be 0.1-0.2.