Vortex-lattice formation in a spin-orbit coupled rotating spin-1 condensate

We study the vortex-lattice formation in a rotating {Rashba} spin-orbit (SO) coupled quasi-two-dimensional (quasi-2D) hyper-fine spin-1 spinor Bose-Einstein condensate (BEC) in the $x-y$ plane using a numerical solution of the underlying mean-field Gross-Pitaevskii equation. % The wave function for this system %has three components corresponding to the three projections of hyper-fine spin $F_z= +1,0,-1$. In this case, the non-rotating {Rashba} SO-coupled spinor BEC can have topological excitation in the form of vortices of different angular momenta in the three components, e.g. the $(0,+1,+2)$- and $(-1,0,+1)$-type states in ferromagnetic and anti-ferromagnetic spinor BEC: the numbers in the parenthesis denote the intrinsic angular momentum of the vortex states of the three components with the negative sign denoting an anti-vortex. The presence of these states with intrinsic vorticity breaks the symmetry between rotation with vorticity along the $z$ and $-z$ axes and thus generates a rich variety of vortex-lattice and anti-vortex-lattice states in a rotating quasi-2D spin-1 spinor ferromagnetic and anti-ferromagnetic BEC, not possible in a scalar BEC. {For weak SO coupling, } we find two types of symmetries of these states $-$ hexagonal and "square". The hexagonal (square) symmetry state has vortices arranged in closed concentric orbits with a maximum of $6, 12, 18...$ ($8,12,16...$) vortices in successive orbits. Of these two symmetries, the square vortex-lattice state is found to have the smaller energy.