Half-quantum vortex state and its excitations in a spin-orbit coupled spinor Bose-Einstein condensate

We investigate theoretically the condensate state and collective excitations of a spin-orbit coupled spinor Bose gas in two-dimensional harmonic traps. In the weakly interacting regime, when the inter-species interaction is larger than the intra-species interaction ($g_{\uparrow \downarrow} > g$), we find that the condensate state has a half-quantum-angular-momentum vortex configuration (half-vortex state) with spatial rotational symmetry and skyrmion-type spin texture. We investigate the stability of half-vortex state in the regime when $g$ is greater than a threshold $g_c$, and in the regime when $g_{\uparrow \downarrow} < g$, by solving the Bogoliubov equations for collective density oscillations. In addition, we also investigate the dynamical properties of the half-vortex state. We present the phase diagram as a function of interatomic interaction and spin-orbit coupling.