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The excitation spectrum for a Bose gas with repulsive and attractive interactions

Gerald V. Brown

Published Oct 1, 1968• 0 citations
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Abstract

The correct qualitative features of the helium II elementary excitation spectrum a r e derived microscopically for a realistic interatomic potential.The strong repulsive core is included by using a reaction matrix in the Hamiltonian.The attractive well i s success fully included by assuming a generalized Bose-Einstein condensation.The pair Hamilton ian is diagonalized by the thermodynamically equivalent Hamiltonian method.Numerical solutions yield spectra with phonon and roton regions.The spectrum energies a r e too high for the Yntema-Schneider potential.Another potential, constructed to fit virial co efficient data classically, gives better results.Spectra a r e presented for a series of at tractive well strengths.

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Excitation Spectrum for a Bose Gas with Repulsive Core and Attractive Well

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The correct qualitative features of the helium II elementary excitation spectrum are derived microscopically for a realistic interparticle potential using a form of pair Hamiltonian. The strong repulsive core is included by using reaction matrix elements in the Hamiltonian, and the attractive well is successfully included by assuming a generalized Bose-Einstein condensation. The pair Hamiltonian is diagonalized by the thermodynamically equivalent Hamiltonian method of Wentzel. Numerical solution of the equations yields spectra with phonon and roton regions. Since the energies in the spectrum are too high when the Yntema-Schneider potential is used, another potential is constructed which still fits the second-virial-coefficient data and gives better results. Spectra calculated for a series of attractive well strengths show the effect of weaker wells.

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•Sep 1, 1973•0 citations
The correct qualitative features of the helium II elementary excitation spectrum a r e derived microscopically for a realistic interatomic potential.The strong repulsive core is included by using a reaction matrix in the Hamiltonian.The attractive well i s success fully included by assuming a generalized Bose-Einstein condensation.The pair Hamilton ian is diagonalized by the thermodynamically equivalent Hamiltonian method.Numerical solutions yield spectra with phonon and roton regions.The spectrum energies a r e too high for the Yntema-Schneider potential.Another potential, constructed to fit virial co efficient data classically, gives better results.Spectra a r e presented for a series of at tractive well strengths.
The correct qualitative features of the helium II elementary excitation spectrum are derived microscopically for a realistic interparticle potential using a form of pair Hamiltonian. The strong repulsive core is included by using reaction matrix elements in the Hamiltonian, and the attractive well is successfully included by assuming a generalized Bose-Einstein condensation. The pair Hamiltonian is diagonalized by the thermodynamically equivalent Hamiltonian method of Wentzel. Numerical solution of the equations yields spectra with phonon and roton regions. Since the energies in the spectrum are too high when the Yntema-Schneider potential is used, another potential is constructed which still fits the second-virial-coefficient data and gives better results. Spectra calculated for a series of attractive well strengths show the effect of weaker wells.
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.
We continue the analysis of the one-dimensional gas of Bose particles interacting via a repulsive delta function potential by considering the excitation spectrum. Among other things we show that: (i) the elementary excitations are most naturally thought of as a double spectrum, not a single one; (ii) the velocity of sound derived from the macroscopic compressibility is shown to agree with the velocity of sound derived from microscopic considerations, i.e., from the phonon spectrum. We also introduce a distinction between elementary excitations and quasiparticles, on the basis of which we give some heuristic reasons for expecting the double spectrum to be a general feature, even in three dimensions, and not an exception.
The properties of a Bose-Einstein gas with repulsive interparticle interactions are determined at low temperature. It is shown that the contribution to the energy arising from particle excitation with small momentum transfers computed in conventional perturbation theory is divergent, but that this difficulty can be avoided by an alternative procedure. An exact method is developed for dealing with that part of the Hamiltonian which gives rise to the perturbation divergence and an exact solution is obtained. The treatment of the region of large momentum transfers is then carried out by introducing the concept of the scattering length of the interaction. The energy is given by the well-known result of the optical theorem corrected by a series in powers of ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$, where $\ensuremath{\rho}$ is the density and aaa the scattering length. Examination of the processes contributing to the energy shows that 95.8% of the correction of order ${(\ensuremath{\rho}{a}^{3})}^{\frac{1}{2}}$ results from a simple alteration of the perturbation energy denominators which takes into account the interaction energy of excited pairs with the unexcited pairs of the medium. The remaining 4.2% arises from multiple particle excitation. These methods also predict the existence of phonon excitation with phonon energies that are linear in the phonon momentum for small excitation and approach q22M\frac{{q}^{2}}{2M}2Mq2​ for large momentum. The detailed consideration of this spectrum forms part of the following paper.The results are compared with the theory of fermion systems with strong interactions, where again the largest correction to the first-order "optical" energy arises from alteration of the two-particle propagator which is required to take into account interactions with the many unexcited particles of the medium. The propagator corrections in the fermion and boson systems are similar except for a characteristic difference arising from the statistics. The theory for bosons is shown to be essentially identical with that given by Brueckner and Levinson in their general formulation of the theory of many-body systems with strong interactions.Systems with attractive but saturating interactions are also considered and some new difficulties are shown to arise which cannot be treated by a straightforward application of the methods of this paper.
The properties of a hard-sphere, Bose–Einstein gas are investigated numerically in the t-matrix, ladder approximation, using both the coordinate-space treatment of Brueckner and Sawada, as modified by Parry and ter Haar, and the pseudopotential method of Liu and Wong. For each treatment, the form of the excitation spectrum and the thermodynamic properties of the gas at low temperatures are compared with the corresponding properties of liquid helium. The pseudopotential method gives better agreement for a reasonable choice of the radius of the "hard-core" interaction. This result is not surprising since the actual potential corresponding to the pseudopotential of Liu and Wong is intermediate between the "soft-core" potential of Brueckner and Sawada and a true "hard-core" potential.
The spectrum of elementary excitations in the system of interacting Bose- and Fermi-particles was obtained at a temperature of absolute zero (Dose- condensation being taken into account by means of the Bogolyabov canonic transformations. In terms of a model of permeable spheres'' with finite repulsion it(is shown that the boson spectrum omega (p) in a definite region of parameter has the minimum at p = 0. By selecting parameters one succeeds in obtaining a quite satisfactory quantitative agreement of the theoretical and experimental spectra for superfluid helium. In the case when the Fermi energy is considerably higher than the bosorins maximum energy, a gap characterizing bond energy of the copper pairs can appear in the fermion speetrum due to exchange of virtual supercondensate'' bosons. If in this case the interaction between fermions at infinity is of attraction character, then under certain conditions sustained longitudinal collective excitations of Pose-type with a finite gap'' in the energy spectrum may exist in the superfluid fermion subsystem. Possibility of appearance of similar excitations in superfluid nuclear matter of neutron stars (pulsars) due to gravitational interaction between nucleons is discussed. (auth)