Rotating Bose gas with hard-core repulsion in a quasi-two-dimensional harmonic trap: Vortices in Bose-Einstein condensates

We consider a gas of $N(<~15)$ Bose particles with hard-core repulsion, contained in a quasi-two-dimensional harmonic trap and subjected to an overall angular velocity $\Omega$ about the z axis. Exact diagonalization of the $n\times n$ many-body Hamiltonian matrix in given subspaces of the total (quantized) angular momentum $L_z,$ with $n\sim {10}^5$ (e.g., for $L_z=N=15,\hspace{0.5em}n=240\mathrm{}782)$ was carried out using Davidson’s algorithm. The many-body variational ground-state wave function, as also the corresponding energy and the reduced one-particle density-matrix $\rho (\mathbf{r},{\mathbf{r}}^{\prime })={\sum }_{\mu }{\lambda }_{\mu }{\chi }_{\mu }^{*}\left(\mathbf{r}\right){\chi }_{\mu }\left({\mathbf{r}}^{\prime }\right)$ were determined. With the usual identification of $\Omega$ as the Lagrange multiplier associated with $L_z$ for a rotating system, the $L_z-\Omega$ phase diagram (or the stability line) was determined that gave a number of critical angular velocities ${\Omega }_{\mathrm{ci}},\hspace{0.5em}i=1,2,3,\dots ,$ at which the ground-state angular momentum and the associated condensate fraction, given by the largest eigenvalue of the reduced one-particle density matrix, undergo abrupt jumps. For a given N, a number of (total) angular momentum states were found to be stable at successively higher critical angular velocities ${\Omega }_{\mathrm{ci}},\hspace{0.5em}i=1,2,3,\dots .$ All the states in the regime $N>\mathrm{}{L}_z>0\mathrm{}$ are metastable. For $L_z>N,$ the $L_z$ values for the stable ground states generally increased with increasing critical angular velocities ${\Omega }_{\mathrm{ci}},$ and the condensate was strongly depleted. The critical ${\Omega }_{\mathrm{ci}}$ values, however, decreased with increasing interaction strength as well as the particle number, and were systematically greater than the nonvariational yrast-state values for the $L_z=N$ single vortex state. We have also observed that the condensate fraction for the single vortex state (as also for the higher vortex states) did not change significantly even as the two-body interaction strength was varied over several $\left(\sim 4\right)$ orders of magnitude in the moderately to the weakly interacting regime.