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Journal of Physics Conference Series

Degenerate Bose gas without anomalous averages

Viktor Borisovich Bobrov, S. A. Trigger, Pieter P. J. M. Schram

Published Nov 1, 2016• 0 citations
Publisher

Abstract

Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. Instead of this assumption, we use the assumption on the C-number nature of the density operator N 0 = a † 0 a 0 with zero momentum. It is shown that the pole of the "density-density" Green function (DDGF), as well as the pole of the single-particle Green function (SPGF), exactly coincide with the Bogoliubov phonon-roton spectrum of excitations for both assumptions. This spectrum, as is known confirmed by many neutron and x-ray experimental measurements of the dynamic structure factor in He II, is straightly related to the DDGF. At the same time, we show that in the other case under consideration, when neither N 0 nor a † 0 and a 0 are C-numbers, a gap can exist in SPGF. This gap in SPGF excitations is straightly related to the density of particles in the "condensate". Therefore, the spectra of excitations for the DDGF and SPGF in the last case under consideration are different, in contrast to the Bogoliubov-type theory where these spectra are identical.

DOI
10.1088/1742-6596/774/1/012149

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Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. Instead of this assumption, we use the assumption on the C-number nature of the density operator N 0 = a † 0 a 0 with zero momentum. It is shown that the pole of the "density-density" Green function (DDGF), as well as the pole of the single-particle Green function (SPGF), exactly coincide with the Bogoliubov phonon-roton spectrum of excitations for both assumptions. This spectrum, as is known confirmed by many neutron and x-ray experimental measurements of the dynamic structure factor in He II, is straightly related to the DDGF. At the same time, we show that in the other case under consideration, when neither N 0 nor a † 0 and a 0 are C-numbers, a gap can exist in SPGF. This gap in SPGF excitations is straightly related to the density of particles in the "condensate". Therefore, the spectra of excitations for the DDGF and SPGF in the last case under consideration are different, in contrast to the Bogoliubov-type theory where these spectra are identical.
Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. It is shown that the pole of the "density-density" Green's function exactly coincides with the Bogolybov's phonon-roton spectrum of excitations. At the same time, a gap exists in the one-particle excitation spectrum . This gap is related to the density of particles in the "condensate".
Model of the degenerated weakly non-ideal Bose gas is considered without suggestion on C-number representation of the creation and annihilation operators with zero momentum. The "density-density" correlation function and the one-particle Green function are calculated on basis of the suggestion about C-number representation of the operator of particle density in the Bose condensate. It is shown, that the pole in the "density-density" Green function determines the Bogolyubov's spectrum of the collective excitations, which have the phonon-roton form. At the same time the spectrum of the one-particle excitations possesses a gap, whose value is connected with the density of particles in the Bose condensate.
The Bogoliubov treatment of a weakly interacting Bose gas is extended by allowing macroscopic occupation of many single-particle quantum states. This generalization of having more than a single condensed mode gives rise to a nonuniform condensate and is being considered as a possible mechanism for the description of a dense strongly interacting Bose gas at low temperatures. The model with a repulsive $\ensuremath{\delta}$-function interparticle potential gives an energy spectrum of elementary excitations or quasiparticles with both phonon and roton features. The values obtained for the roton minimum energy $\ensuremath{\Delta}$ and the speed of (first) sound agree reasonably well with inelastic-neutron-scattering data.
A unified theory of repulsive bose gas is constructed. The theory is based on the universal feature of the excitation spectra - linear and quadratic for small and large values of momenta, respectively. The ensemble of bosons is treated as a mixture of condensates and noncondensates. The measurable amount of condensates is determined by imposing that they coexist at equilibrium. This scenario of coexistence gives a natural explanation why the excitation spectra are gapless. Calculations made, based on this theory, yield the established results for one and three dimensions. For the first time, a new result is obtained in two dimensions indicating a strong dimensional crossover. The theory is consistent with vanishing of the anomalous self-energy for zero momentum and energy for dimensions less than or equal to three.
We obtain ab-initio estimations of the dynamic structure factor, S ( q ,ω), of Bose gases at zero temperature. More precisely, we use the Genetic Inversion via Falsification of Theories (GIFT) algorithm to perform analytic continuations of imaginary time correlation functions computed via an exact Path Integral projector method. Using the hard-sphere potential to model the two-body interactions between the atoms, we compute S ( q ,ω) changing the gas parameter from the dilute regime ( na 3 = 10 −4 ) up to the density corresponding to superfluid 4 He at equilibrium ( na 3 = 0.2138). With increasing density, we observe the emergence of a broad multiphonon contribution accompanying the quasiparticle peak and a crossover of the dispersion of elementary excitations from a Bogoliubov-like spectrum to a phonon-maxon- roton curve. Apart from the low wave vector region, for na 3 = 0.2138 the energy-momentum dispersion relation and the static density response function, χ( q ), turns out to be in good agreement with the superfluid 4 He experimental data at equilibrium density.