Vortex patterns of a two-dimensional Bose-Einstein condensate at the almost critical rotation speed

Abstract

We study vortex patterns of a two-dimensional Bose-Einstein condensate rotating close to the centrifugal limit, treating the two signs of the contact interaction with the method each requires: for repulsion, a GPU-accelerated variational minimization with exact projection onto the Lowest Landau Level (LLL); for attraction, imaginary-time evolution of the full real-space Gross-Pitaevskii (GP) equation. For repulsive interactions, our approach reproduces Abrikosov vortex lattices, achieving quantitative alignment with Thomas-Fermi theory and recovering the Abrikosov constant $e^{\mathrm{Ab}}(1)\approx 1.1596$, in close analogy with the vortex ordering of type-II superconductors. In the attractive regime, the rotating ground state carries no vortex lattice at any rotation frequency: the cloud contracts and collapses as $G\to -G_{\ast }$, where $G_{\ast }=\Vert Q{\Vert }_{L^2}^2\simeq 11.70$, essentially independently of rotation. The only stationary vortex states we find are giant vortices, trapped vortex (Townes) solitons whose charge-$m$ sector collapses as $G\to -\Vert Q_m{\Vert }_{L^2}^2$ ($\Vert Q_m{\Vert }_{L^2}^2\simeq 11.70,48.3,89.8,132.4$ for $m=0,1,2,3$), fixed by an equivariant Gagliardo-Nirenberg inequality and confirmed numerically up to the critical region. These findings provide a numerical benchmark for vortex formation and collapse in rotating two-dimensional quantum gases.