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.
We consider the many-body ground state and elementary excitations of the Matsubara-Matsuda cell model, which can be interpreted as a first approximation to Siegert's exact but untractable operator algebra for hard-core boson fields. The model has unphysical aspects, but allows treatment of some many-body effects of the interaction. It represents the hard core by restricting cell occupation numbers to \ensuremath{\le}1, which changes the algebra from Bose to Pauli type; kinetic energy and attraction appear as interactions between nearest-neighbor cell pseudospins, equivalent to a Heisenberg ferromagnet with anisotropy in pseudospin space. Our treatment, valid for scattering length ${f}_{0}\ensuremath{\ge}0$, splits the Hamiltonian into an isotropic "unperturbed" part
H0, describing a system with hard core plus attraction of strength making
f0=0, and
AH1, consisting of the attraction's deviation from this strength, the parameter $A\ensuremath{\ge}0$ giving the magnitude of this (repulsive) deviation. The exact ground state of
H0, for any density, is the state symmetric in all pseudospins, with the appropriate eigenvalue of the total pseudospin component
S(3) which measures
N. These states exhibit what corresponds to "incomplete Bose-Einstein condensation," the "excluded volume" effect of the hard-core constraint producing relative depletion ${\ensuremath{\xi}}_{0}$ of the condensate, proportional to $\ensuremath{\rho}$. Exact single excitations of
H0 are density fluctuations ${\ensuremath{\rho}}_{\mathrm{k}}$ with, however, free particle-like excitation spectrum, the ground-state energy being density-independent. Relaxing the restriction to eigenstates of
N permits definition, by rotation of the total pseudospin from the vacuum, of a quasiparticle vacuum and operators, for any mean density. These serve as starting points for treating the full Hamiltonian by the equations-of-motion method in the random-phase approximation. The excitation spectrum is now phonon-like for small
k, with $s\ensuremath{\sim}{(\ensuremath{\rho}A)}^{\frac{1}{2}}$,
f0 being expressible exactly in terms of
A,
NE0 can be written in terms of
f0, $\ensuremath{\rho}$, and ${\ensuremath{\xi}}_{0}$, self-consistent in the RPA to order
f025. In the low-density limit, ${\ensuremath{\xi}}_{0}\ensuremath{\rightarrow}0$, there results the well-known expansion in ($\ensuremath{\rho}{{f}_{0}}^{3}$), but there are higher density corrections including a term $\ensuremath{\sim}\ensuremath{\rho}{f}_{0}{(\ensuremath{\rho}{{f}_{0}}^{3})}^{\frac{1}{3}}{{\ensuremath{\xi}}_{0}}^{\frac{2}{3}}$, due to the strong interaction included in the unperturbed many-body ground state.
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.
Bosons interacting repulsively on a lattice with a flat lowest band energy dispersion may, at sufficiently small filling factors, enter into a Wigner-crystal-like phase. This phase is a consequence of the dispersionless nature of the system, which in turn implies the occurrence of single-particle localized eigenstates. We investigate one of these systems---the sawtooth lattice---filled with strongly repulsive bosons at filling factors infinitesimally above the critical point where the crystal phase is no longer the ground state. We find, in the hard-core limit, that the crystal retains its structure in all but one of its cells, where it is broken. The broken cell corresponds to an exotic kind of repulsively bound state, which becomes delocalized. We investigate the excitation spectrum of the system analytically and find that the bound state behaves as a single particle hopping on an effective lattice with reduced periodicity, and is therefore gapless. Thus, the addition of a single particle to a flat-band system at critical filling is found to be enough to make kinetic behavior manifest.
The ground state of a system of interacting Bose particles is examined by a pair-correlation variational method based on exact eigenstates of the total particle number, without assuming macroscopic occupation of the single-particle state k = 0 (``simple'' Bose condensation). Known results are recovered in the case of repulsive interactions of weak and intermediate strengths, where the ground state does in fact exhibit simple condensation. But it is shown that in the case of weak attractive interactions the ground state exhibits ``generalized'' condensation, in which a finite fraction of particles have momenta which are macroscopically negligible, but no one single-particle state is macroscopically occupied. A weak-coupling expansion is derived for the ground-state energy in this case, the leading term of which is three times as low as that of the state one would obtain in the case of simple condensation. By the use of a pseudopotential, this result is adapted to the case of a strongly interacting system with a small negative scattering length α and fixed low density ρ, giving E0/n=−(6π ℏ2 ρ | α | /m)[1−(128/45) π−1/2(ρ | α | 3)1/2+…]. The relationships between equilibrium density, ground-state energy, and sound velocity predicted by this expression are in order-of-magnitude agreement with those obtaining for liquid He4.
We study the ground-state phase diagram of a one-dimensional lattice model of interacting bosons at commensurate filling. When the ratio of interaction strengths to bandwidth is small, the bosons condense into a superfluid phase. For large on-site repulsion, there is a phase transition to a localized Mott phase where the gap \ensuremath{\Delta} is nonzero and the Fourier transform at zero momentum of the density-density correlation function S(q=0) diverges. For large intersite repulsion, S(q=\ensuremath{\pi}) diverges first. By examining the superfluid density ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$, in addition to the gap and S(q), we obtain a complete characterization of the phase diagram.