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arXiv:1503.04746 · Condensed Matter Physics

The theory of a many boson system with the deformed Heisenberg algebra

I. O. Vakarchuk, G. Panochko

Published Sep 1, 2015• 2 citations
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Abstract

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.

DOI
10.5488/cmp.18.33002

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Physical Review A•Oct 1, 1974•25 citations
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.
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.
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.
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.
The quantum field Hamiltonian expressed in terms of density and current density variables has been employed together with the equal-time commutation relations among these variables to find the ground state energy and the density fluctuation excitation spectrum of a system of interacting bosons at T = 0°K. The approximation involved consists in assuming that the density fluctuation in space is small compared with the average density. The results easily obtained in the lowest-order approximation agree with those of Bogoliubov. However, in our treatment no condensation of particles in zero-momentum state is assumed or apparent. A connection between the present treatment and the quantum hydrodynamic approach to the irrotational flow of a Bose liquid has been made.
The paradoxes and disparities in the contemporary microscopic theory of superfluid helium (He–II) are discussed along with possible ways of resolving them by taking pair correlations of He4 atoms into consideration. It is shown that most paradoxes are associated with the commonly accepted initial assumption concerning the dominating role of single-particle Bose condensate (SPBC) in the quantum microstructure of the superfluid component ρs. The existence of intensive SPBC leads to a strong hybridization of the elementary excitation branches and to a common dispersion law for all boson branches, which is identified with the quasiparticle spectrum E(p) observed experimentally from slow neutron scattering in liquid helium. However, the stability of this spectrum during a transition through the λ-point and the large value of the gap in the vicinity of the “rotonic” minimum contradict both the Landau theoretical criterion of superfluidity and the small value of experimentally measured critical velocity. At the same time, a strong interaction between particles in the Bose liquid He4 strongly suppresses the SPBC which amounts to less than 1% of all He4 atoms and hence cannot be the main constituent of the superfluid component, unlike the case of a weakly nonideal Bose gas. Moreover, for a quite strong attraction between particles in a certain region of the momentum space, bound pairs of bosons can be formed in the superfluid Bose liquid, and a coherent pair condensate (CPC) analogous to the Cooper pair condensate in superconductors may appear. Such a strong CPC may completely suppress the weak SPBC. In this case, the one-particle spectrum ε(p) of elementary excitations does not hybridize with the collective (two-particle) spectrum and does not appear in the structure of the dynamic form factor S(p,ε), i.e., does not coincide with the spectrum measured from neutron scattering. The dispersion of one-particle spectrum is defined by the momentum dependence of the pair order parameter Ψ̃(p) and may have a minimum or a point of inflection at p≠0. This peculiarity in the one-particle spectrum of a Bose liquid with CPC but without SPBC vanishes together with Ψ̃(p) at the temperature Tc=Tλ of the phase transition from the superfluid to the normal state (unlike the rotonic minimum in the collective spectrum), while the corresponding critical velocity vc=min[ε(p)/p] vanishes at the λ-point in accordance with the Landau criterion and the experimental data. The assumption that the strong “Cooper-like” CPC is responsible for the quantum structure of the superfluid component ρs is confirmed indirectly by the successful application of the Justrow approximation (based on strong pair correlations) for describing the properties of liquid He4 and quantum liquid mixtures He3–4He on one hand, and by an anomalously large effective mass of He3 impurity atoms in He4, which is approximately equal to total mass of He3 and He4 atoms, thus pointing to the existence of helium atoms in superfluid liquid He–II. The value of the superfluid velocity circulation quantum in the Onsager–Feynman vortices in a Bose liquid with CPC but without SPBC is discussed as well as the critical velocities of superfluid He4 in ultrathin films and channels in which the creation and motion of quantum vortices are ruled out, and the quasiparticle spectrum undergoes dimensional quantization.
It is now well established that current and density may be used as coordinates in describing a nonrelativistic many-body system. The current and density operators form a closed algebra as they obey a set of equal-time commutation relations. A particular choice of the algebra containing only the longitudinal component of the current describes a fluid in a simple way. For Bose systems, we show that there exists a special choice of this longitudinal current which is an irreducible representation of the algebra, but expressed in two different ways leading to the formulations of Bogoliubov and Zubarev (BZ) and that of Sunakawa and his coworkers (S). This demonstration proves the formal equivalence of the two formulations. The BZ formalism leads to a non-Hermitian Hamiltonian for which use of a certain mathematical technique recently proposed by us yields completely divergence-free results for Bose systems. A temperature-dependent matrix Green's-function theory is developed and the self-energy matrix calculated. From this, we also deduce the structure factor in a straightforward way. These results are applied here to compute numerically the excitation spectrum of the superfluid liquid helium using the experimental structure factor as the only input into the computation. Comparison of this with other calculations and with experiment is discussed in detail in this paper. The problem of a charged Bose gas is considered as a testing ground for all theories of Bose fluids since certain exact results are known for this system. We have applied our method using the BZ Hamiltonian to this problem and we are in complete agreement with the exact results. It thus appears that the current-algebra approach can be quite successfully applied in elucidating properties of interacting Bose systems.