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arXiv:cond-mat/9707044

λλλ-Type phase transition for a weakly interacting bose gas

I. Cabrera, Hugo Pérez Rojas

Published Jul 3, 1997• 0 citations
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Abstract

A mechanism generating a $λ$-type behavior in the specific heat of a Bose gas near the critical temperature $T_c$ is discussed. It is shown to work for a general class of quasiparticle spectra, and especially, for the Bogoliubov's model of a weakly interacting Bose gas, where a temperature-dependent quasi-particle spectrum is obtained.

DOI
10.48550/arxiv.cond-mat/9707044

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A mechanism generating a λλλ-type behavior in the specific heat of a Bose gas near the critical temperature TcT_cTc​ is discussed. It is shown to work for a general class of quasiparticle spectra, and especially, for the Bogoliubov's model of a weakly interacting Bose gas, where a temperature-dependent quasi-particle spectrum is obtained.
We discuss the standard approach to the problem of the low momentum limit of the spectrum for a weakly interacting Bose gas. The Bogoliubov's spectrum is shown to be obtained as a Goldstone mode thanks to the introduction of a chemical potential μμμ. This procedure has, however, difficulties since the breaking of the gauge symmetry implies that the corresponding chemical potential must be taken as zero, unless it is introduced before breaking the symmetry. But if this is done, after the symmetry breaking μμμ loses its meaning as a chemical potential. An alternative two-mode solution is suggested having two modes, one of them being the free-particle quadratic in momentum spectrum, the second bearing a gap. This gap leads to a λλλ-type behavior of the specific heat near the critical temperature.
The low-temperature properties of the interacting Bose system at high density are investigated under the random phase approximation and the self-consistent Hartree approximation based on the “normal-mode analysis”. To treat rigorously the tempereture-dependent “depletion” effect of the particles in the zero-momentum level, a new approach is proposed. As a result, the critical temperature, Tc, at which the zero-momentum particles disappear is determined. In the region below Tc, the temperature-dependent energy-spectrum of the quasi-particles, which has at the zero-temperature an intimate connection with that of Bogoliubov, is obtained, and the property of superfluidity which disappears at Tc is discussed. It is shown that the quasi-particles are transformed into the normal Bose particles at Tc, and the temperature-dependent energy-spectrum of the normal particles in the region above Tc is determined. It is noteworthy that the phase transition of the second-order occurs at Tc. In so far as the interaction gives the real and positive value for the energy-spectrum of both the quasi-particles and normal particles, the results above hold true, independent of the magnitude and particular shape of interaction.Some remarks are made on the hard-sphere interaction and on the interaction among the quasi-particles which seems necessary for the investigateions of the intermediate-density Bose system.
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.
An investigation is performed using normal-mode analysis on a quantum mechanical cell model whose general features are representative of 4He at low temperatures. Both random-phase and self-consistent field approximations are invoked. A Bogoliubov-type excitation energy spectrum is derived for the interacting Bose system, and the amplitudes of the system quasi-particles are obtained; these quantities are expressed in terms of zero-momentum occupation number. A relationship giving the temperature-dependent fractional occupation of the single-particle zero-momentum level is established, and special attention is devoted to the temperature extreme near absolute zero. An estimate is made of the zero-temperature sound speed corresponding to the special case of phonon excitation given by the model in low momentum. It is shown that at a critical temperature for which the zero-momentum occupation is depleted, the quasi-particles indeed become normal Bose particles. In first approximation, this critical temperature is akin in form to that obtained in treatments of ideal boson systems.