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Ukrainian Journal of Physics

The Effective Mass of an Impurity Atom in the Bose Liquid with a Deformed Heisenberg Algebra

I. O. Vakarchuk, G. Panochko

Published Feb 1, 2017• 8 citations
Publisher

Abstract

We consider the movement of a 3He impurity atom in the Bose liquid. We suggest to describe the many-particle correlations between atoms of the Bose liquid, by using a deformed Heisenberg algebra. As generalized coordinates, we choose the collective variables that are the Fourier components of fluctuations of the density of Bose particles. The wave function of the investigated system in the zeroth approximation is the product of the wave function of the liquid helium-4 within deformed commutation relations between generalized coordinates and momenta and the plane wave of the impurity atom. We calculate the correction to the ground-state energy of the system “Bose liquid plus impurity” and the effective mass of the impurity atom 3He, by assuming that the boson-impurity interaction is a small perturbation.

DOI
10.15407/ujpe62.02.0123

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We consider the movement of a 3He impurity atom in the Bose liquid. We suggest to describe the many-particle correlations between atoms of the Bose liquid, by using a deformed Heisenberg algebra. As generalized coordinates, we choose the collective variables that are the Fourier components of fluctuations of the density of Bose particles. The wave function of the investigated system in the zeroth approximation is the product of the wave function of the liquid helium-4 within deformed commutation relations between generalized coordinates and momenta and the plane wave of the impurity atom. We calculate the correction to the ground-state energy of the system “Bose liquid plus impurity” and the effective mass of the impurity atom 3He, by assuming that the boson-impurity interaction is a small perturbation.
We propose to consider nonlinear fluctuations in the theory of liquid 4 He deforming the commutation relations between the generalized coordinates and momenta.Generalized coordinates are coefficients of density fluctuations of Bose particles.The deformation parameter takes into account the effects of three-and four-particle correlations in the behavior of a system.This parameter is defined from the experimental values of the elementary excitation spectrum and the structure factor extrapolated to T = 0 K.The numerical estimation of the ground state energy and the Bose condensate fraction is made.The elementary excitation spectrum and the potential of interaction between the helium atoms are recovered.
The description of the properties of liquid Helium is a challenge for any microscopic many-body theory. In this context, we study the ground state and the excitation spectrum of one 3 He impurity in liquid 4 He at T=0 with the aim of illustrating the power of the correlated basis function formalism in describing heavily correlated systems. The strong interatomic interaction and the large density require the theory to be pushed to a high degree of sophistication. A many-body correlation operator containing explicit two- and three-particle correlation functions is needed to obtain a realistic ground state wave function, whereas a perturbative expansion including up to two phonon correlated states must be enforced to study the impurity excitation energies. The theory describes accurately the experimental spectrum along all the available momentum range. As empirically shown by the experiments, a marked deviation from the quadratic Landau-Pomeranchuck behavior is found and the momentum dependent effective mass of the impurity increases of ~50% at q~1.7 Å-1 with respect to its q=0 value. Although the main emphasis is given to the correlated basis function theory, we present also comparisons with other methods, as diffusion Monte Carlo, variational Monte Carlo with shadow wave functions and time dependent correlations.
In order to obtain a quantitative estimate of the filling of the Bose condensate in the ground state of a Bose liquid the authors formulate a self-consistent perturbation theory with respect to the exactly solvable hard-spheres model. This has made it possible to take into account the contribution of the two-particle correlations to the erosion of the condensate and obtain a simple expression for the relative number of condensate atoms in terms of the liquid structure factor, which is taken as the initial information. The authors use the Bogolyubov-Zubarev expression for the ground-state energy of a many-boson system in the approximation of two sums over the wave vector in order to calculate the contributions of the three- and four-particle correlations. For this, they calculate the kinetic energy and from it recover the single particle density matrix. The results of the theory are used to calculate the filling of the Bose condensate in the ground state of liquid He/sup 4/ as a function of the liquid density.
By introducing a number of auxiliary variables, quantum-mechanical collective and internal motions in a Bose system are described. In a collective representation obtained here, the wave functions are expressed as functions of the auxiliary variables and the internal coordinates. The energy for the ground state and the excitation energies of collective motion and internal motion are obtained for the case of a long-range interaction potential. The lowest energy obtained proves to be smaller than that afforded by the Hartree-Fock approximation. The excitations of the collective motion with smaller wave numbers than a cut-off wave number kc correspond to Landau's phonons. The excitations of the internal motion resemble Landau's rotons; but they are forbidden for the smaller wave numbers K ≦ kc. When one applies this method to the problem of liquid helium, it becomes necessary to take account of the two-body correlation of individual particles. It is pointed out that the phonon energy depends upon the zero-point motion of the individual particles.
The Green function is derived for a small concentration of fermions in a weakly interacting Bose gas using mass operator perturbation theory. For small momentum it is found that the spectral weight function of the system consists of a quasi-particle peak associated with an effective mass m* as well as some weight at higher energies The momentum dependence of both parts of the spectral weight function is determined. The implications of the weight at higher energies in the case of dilute 3He-4He mixtures are discussed.