Excitation spectrum of a toroidal spin-1 Bose-Einstein condensate

We calculate analytically the Bogoliubov excitation spectrum of a toroidal spin-1 Bose-Einstein condensate that is subjected to a homogeneous magnetic field and contains vortices with arbitrary winding numbers in the $m_F=\pm 1$ components of the hyperfine spin. We show that a rotonlike spectrum can be obtained, or an initially stable condensate can be made unstable by adjusting the magnitude of the magnetic field or the trapping frequencies. The structure of the instabilities can be analyzed by measuring the particle densities of the spin components. We confirm the validity of the analytical calculations by numerical simulations.