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.
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.
A finite temperature system of degenerate bosons is studied with the aid of the extended Bogoliubov transformation which has been proposed in a preceding part I. The energies of quasi-particles and the vertex functions describing their interaction are determined in an approximation which includes the lowest order effects of the interaction between non-zero momentum bosons. The quasi-particle energy depends on the distribution of thermal excitations, and is of phonon type in the long wavelength limit. The phonon nature of the spectrum is preserved by the interaction between quasi-particles, because the vertex functions vanish in the long wavelength limit. In the limit as the condensate is depleted completely, the proposed formalism simplifies to the Hartree-Fock theory.
The properties of Bose-Einstein-condensed systems of hard spheres at high density are studied by taking into account higher-order corrections beyond the Hartree-Fock-Bogoliubov approximation. Based on a diagrammatic technique of the perturbation expansion starting from the Bogoliubov approximation, we show that the effective interactions separate into two distinct parts: the screened and condensate-mediated interactions. The latter is described by the exchange of virtual quasiparticles between noncondensate bosons via the Bose-Einstein condensate and is attractive for small energy transfer. We calculate the condensation temperature, excitation spectrum, and condensate density in the whole region of the gas parameter. It is shown that the condensate temperature decreases compared to that of an ideal Bose gas due to hard-sphere interactions. We find that the condensate-mediated attraction between noncondensate bosons leads to an enhancement of the anomalous self-energy and, as a consequence, to the emergence of a roton minimum in the excitation spectrum and to the strong suppression of the condensate fraction in the region of high density.
We consider a dilute hard-sphere Bose gas in random external potentials at low temperatures, in D=3, using the technique of pseudopotentials and the Bogoliubov transformation. At absolute zero, the random potentials can deplete the Bose condensate, though not completely. On the other hand, they generate an amount of normal fluid equal to 4/3 the condensate depletion. This is a localization effect that can destroy superfluidity at absolute zero. General features of the superfluid density in the neighborhood of this transition point agree qualitatively with experimental results on helium in porous media.
The thermodynamics of a free Bose gas with effective temperature scale
T~ and hard-sphere Bose gas with the
T~ scale are studied.
T~ arises as the temperature experienced by a single particle in a quantum gas with 2-body harmonic oscillator interaction
Vosc, which at low temperatures is expected to simulate, almost correctly, the attractive part of the interatomic potential
VHe between
4He atoms. The repulsive part of
VHe is simulated by a hard-sphere (HS) potential. The thermodynamics of this system of HS bosons, with the
T~ temperature scale (HSET), is investigated, first, by the Bogoliubov-Huang method and next by a modified version of this method, which takes approximate account of those terms of the 2-body repulsion which are linear in the zero-momentum Bose operators
a0,a0∗ (originally rejected by Bogoliubov). Theoretical heat capacity
CV(T) exhibits good agreement, below 2.1 K, with the experimental heat capacity graph observed in
4He at saturated vapour pressure. The phase transition to the low-temperature phase, with a Bose-Einstein condensate, occurs in the HSET at $T_λ=$2.17 K, and is accompanied, in the modified HSET version, by a singularity of
CV(T). Other thermal properties of HSET, such as the momentum distribution function, the fraction of atoms in the momentum condensate and normal fluid density, agree qualitatively with those of
4He, but improve those of the free Bose gas.
This thesis is devoted to the study of the effect of disorder on low-dimensional weakly interacting Bose gases. In particular, the disorder triggers a quantum phase transition in one dimension at zero temperature that is investigated here through the study of the long-range behaviour of the one-body density matrix. An algebraic spatial decay of the coherence marks the quasicondensate, whereas, in the case of strong disorder, an exponential decay is recovered and it characterizes the insulating Bose-glass phase. This analysis is performed using an extended Bogoliubov theory to treat low dimensional Bose gases within a density-phase approach. A systematic numerical study allowed to draw the phase diagram of 1D weakly interacting bosons. The phase boundary obeys two different power laws between interaction and disorder strength depending on the regime of the gas where the transition occurs. These relations can be explained by means of scaling arguments valid in the white noise limit and in the Thomas-Fermi regime of the Bose gas. The phase transition to a quasicondensed phase comes along with the onset of superfluidity: the inspection of the superfluid fraction of the gas is consistent with these predictions for the boundary. The finite temperature case and the scenario in two dimensions are briefly discussed. The quantum phase transition is caused by low-energy phase fluctuations that destroy the quasi-long-range order characterizing the uniform system. Within the approach presented here, the phase fluctuations are identified as the low-lying Bogoliubov modes. Their properties have been investigated in detail to understand which changes trigger the phase transition and we found that the transition to the insulating phase is accompanied by a diverging density of states and a localization length, measured through the inverse participation ratio, that diverges as a power-law with power – 1 for vanishing energy. The fragmentation of the gas is also studied: this notion is very often associated with the onset of the insulating phase. The characterization of the density fragmentation is performed by analyzing the probability distribution of the density. A density profile is defined as fragmented when the probability distribution at vanishing density is finite or divergent and this happens for a gas in the Bose-glass phase. On the contrary, the superfluid phase is characterized by a zero limiting probability of having vanishing densities. This definition is derived analytically, and confirmed by a numerical study. This fragmentation criterion is particularly suited for detecting the phase transition in experiments: when a harmonic trap is included, the transition to the insulating phase can be extracted from the statistics of the local density distribution.