Vortices in rotating Bose gas interacting via finite range Gaussian potential in a quasi-two-dimensional harmonic trap

A system of harmonically trapped N=16 spin-$0$ bosons confined in quasi-$2$D symmetrical $x-y$ plane interacting via a finite range repulsive Gaussian potential is studied under an externally impressed rotation to an over all angular velocity $\Omega$ about the $z-$axis. The exact diagonalization (ED) of $n\times n$ many-body Hamiltonian matrix in a given subspace of quantized total angular momentum $0\le L_{z} \le 4N$ is performed using Davidson algorithm. For $N=16 \mbox{and}\ L_{z}=32$, the dimensionality of the Hilbert space turns out to be $n=384559$. The trap velocity $\Omega$ being the Langrange multiplier associated with the angular momentum $L_{z}$ for the rotating systems, the $L_{z}-\Omega$ phase diagram (or stability line) is drawn which determines the critical angular velocities, $\Omega_{\bf c_{i}}, i=1,2..$, at which, for a given angular momentum $L_{z}$, the system goes through a quantum phase transition. %condensate fraction, von-Neumann entropy exhibit abrupt %(quantum phase)jumps. Further with increase in interaction range $\sigma$, the quantum mechanical coherence extends over more and more particles in the system resulting in an enhanced stability of the $i^{th}$ vortical state with angular momentum $L_{z}\left(\Omega_{c_{i}}\right)$ leading to a delayed onset of the the next vortical state $L_{z}\left(\Omega_{c_{i+1}}\right)$ at a higher value of the next critical angular velocity $\Omega_{c_{i+1}}$. There is an increase in the critical angular velocity $\left(\Omega_{\textbf{c}_{i}}, i=1,2,3\cdots\right)$ and in the largest condensate fraction $\lambda_{1}$, calculated using single particle reduced density matrix(SPRDM) eigen-values with increase in the interaction range $\sigma$. We calculated the von-Neumann quantum entropy ($S_{1}$), degree of condensation ($C_{d}$) and the conditional probability density (CPDs).