Pseudospin Model for Hard-Core Bosons with Attractive Interaction. Zero Temperature

Abstract

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 ${H}_{0}$, describing a system with hard core plus attraction of strength making ${f}_{0}=0$, and $A{H}_{1}$, 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 ${H}_{0}$, 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 ${H}_{0}$ 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 $s\ensuremath{\sim}{(\ensuremath{\rho}A)}^{\frac{1}{2}}$, ${f}_{0}$ being expressible exactly in terms of $A$, $\frac{{E}_{0}}{N}$ can be written in terms of ${f}_{0}$, $\ensuremath{\rho}$, and ${\ensuremath{\xi}}_{0}$, self-consistent in the RPA to order ${{f}_{0}}^{\frac{5}{2}}$. 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.