Fractional windings of the spinor condensates on a ring

We study the uniform solutions to the one-dimensional spinor Bose-Einstein condensates on a ring. These states explicitly display the associated motion of the super-current and the spin rotation, which give rise to fractional winding numbers according to the various compositions of the hyperfine states. It simultaneously yields a fractional factor to the global phase due to the gauge-spin symmetry. Our method can be applied to explore the fractional vortices by identifying the ring as the boundary of two-dimensional spinor condensates.