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数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu

Boson Gas Mean Field Model Trapped by Weak Harmonic Potentials in Mesoscopic Scaling (Applications of Renormalization Group Methods in Mathematical Sciences)

Hiroshi Tamura

Published Dec 1, 2010• 0 citations
OpenAlex

Abstract

A model of a mean-field interacting boson gas trapped by a weak harmonic potential is considered by means of the theory of Random Point Field [RPF].In the previous work, the weak potential limit of the RPF which describes the position distribution of the constituent particles of the gas has been obtained.The limiting RPF concerns the distribution of the gas near the bottom of the harmonic potentials.In this note, we deal with the same model in the different scale to consider the large scale distribution of the system. §1. Introduction and Main ResultsWe consider the quantum statistical mechanical models of boson gas in \mathbb{R}^{d}(d> 2) equipped with a $\kappa$ -parameterized family of one-particle Hamiltonians of harmonic oscillators:(which are self-adjoint operators in the Hilbert space \mathfrak{H}:=L^{2}() .In this model, the limit $\kappa$\rightarrow\infty (weak harmonic limit [WHL]) can be considered as a kind of thermodynamic limit.In the previous paper [15], the position distribution of the constituent bosons of the model and its WHL is studied by means of random point

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A model of a mean-field interacting boson gas trapped by a weak harmonic potential is considered by means of the theory of Random Point Field [RPF].In the previous work, the weak potential limit of the RPF which describes the position distribution of the constituent particles of the gas has been obtained.The limiting RPF concerns the distribution of the gas near the bottom of the harmonic potentials.In this note, we deal with the same model in the different scale to consider the large scale distribution of the system. §1. Introduction and Main ResultsWe consider the quantum statistical mechanical models of boson gas in \mathbb{R}^{d}(d> 2) equipped with a κ\kappaκ -parameterized family of one-particle Hamiltonians of harmonic oscillators:(which are self-adjoint operators in the Hilbert space \mathfrak{H}:=L^{2}() .In this model, the limit κ\kappaκ\rightarrow\infty (weak harmonic limit [WHL]) can be considered as a kind of thermodynamic limit.In the previous paper [15], the position distribution of the constituent bosons of the model and its WHL is studied by means of random point
A model of the mean-field interacting boson gas trapped by a weak harmonic potential is considered by the boson random point field methods. We prove that in the weak harmonic trap limit there are two phases distinguished by the boson condensation and by a different behavior of the local particle density. For chemical potentials less than a certain critical value, the resulting random point field (RPF) coincides with the usual boson RPF, which corresponds to a noninteracting (ideal) boson gas. For the chemical potentials greater than the critical value, the boson RPF describes a divergent (local) density, which is due to localization of the macroscopic number of condensed particles. Notice that it is this kind of transition that observed in experiments producing the Bose–Einstein condensation in traps.
We study the condensation of a homogeneous interacting Bose gas in the presence of a superimposed weak periodic potential. In contrast to, e.g., a harmonic trap, the periodic potential does not reduce the significance of long-wavelength fluctuations and thus influences the phase transition in a more profound way. We present general thermodynamic descriptions of the system above condensation in terms of mean-field as well as renormalization-group theory. Our approaches are based on describing the interplay between the effects of the interactions and the potential by means of a renormalization of the potential amplitude. These theories are then used to study the critical chemical potential and particle density and their dependence on the properties of the applied potential. A main result of our investigations concerns the characteristic nonperturbative momentum scale kc{k}_{c}kc​ of the critical interacting Bose gas. It is shown that the existence of this momentum scale becomes macroscopically manifest in the behavior of the critical thermodynamic properties.
It is a well-established fact that statistical properties of energy-level spectra are the most efficient tool to characterize nonintegrable quantum systems. The statistical behavior of different systems such as complex atoms, atomic nuclei, two-dimensional Hamiltonians, quantum billiards, and noninteracting many bosons has been studied. The study of statistical properties and spectral fluctuations in interacting many-boson systems has developed interest in this direction. We are especially interested in weakly interacting trapped bosons in the context of Bose-Einstein condensation (BEC) as the energy spectrum shows a transition from a collective nature to a single-particle nature with an increase in the number of levels. However this has received less attention as it is believed that the system may exhibit Poisson-like fluctuations due to the existence of an external harmonic trap. Here we compute numerically the energy levels of the zero-temperature many-boson systems which are weakly interacting through the van der Waals potential and are confined in the three-dimensional harmonic potential. We study the nearest-neighbor spacing distribution and the spectral rigidity by unfolding the spectrum. It is found that an increase in the number of energy levels for repulsive BEC induces a transition from a Wigner-like form displaying level repulsion to the Poisson distribution for P(s)P(s)P(s). It does not follow the Gaussian orthogonal ensemble prediction. For repulsive interaction, the lower levels are correlated and manifest level-repulsion. For intermediate levels P(s)P(s)P(s) shows mixed statistics, which clearly signifies the existence of two energy scales: external trap and interatomic interaction, whereas for very high levels the trapping potential dominates, generating a Poisson distribution. Comparison with mean-field results for lower levels are also presented. For attractive BEC near the critical point we observe the Shnirelman-like peak near s=0s=0s=0, which signifies the presence of a large number of quasidegenerate states.
We present a simple renormalization-group method of investigating ground-state properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows investigation of properties unavailable through mean-field approximations, such as one- and two-particle reduced density matrices of the ground state. As an example, we model a weakly interacting one-dimensional Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the ground-state one-particle density matrix. We have also obtained the Hall-Post lower bounds for the ground-state energy of the gas. All results have been obtained by straightforward numerical diagonalization of the Hamiltonian matrix.
We calculate the quantum corrections of the thermodynamic quantities of a system of confined bosons at finite temperature. Systematically quantum corrections are written in a series of $\ensuremath{\Elzxh},$ which is convergent when kT\mathrm{kT}kT is much larger than the spacing between energy levels of the system. We apply this method to the different physical systems. First we calculate the thermodynamic potential of spin-1 bosons confined in a general power-law potential, and calculate the paramagnetic susceptibility of the gas including the quantum corrections. Local-density approximation for charged bosons in the weak magnetic field is improved by the semiclassical expansion. Diamagnetic susceptibility of a system of confined charged bosons is calculated using this method. Finally we apply this method to calculate analytically the thermodynamic potential of a weakly interacting Bose-gas confined in at three-dimensional harmonic-oscillator potential. For a large number of particles, quantum corrections become small, and contribute to the finite-size corrections to scaling.