Classifying Novel Phases of Spinor Atoms
  We consider many-body states of bosonic spinor atoms which, at the mean-field level, can be characterized by a single-particle wave function. Such states include BEC phases and insulating Mott states with one atom per site. We describe and apply a classification scheme that makes explicit spin symmetries of such states and enables one to naturally analyze their collective modes and topological excitations. Quite generally, the method allows classification of a spin F system as a polyhedron with 2F vertices. After discussing the general formalism we apply it to the many-body states of bosons with hyperfine spins two and three. For spin-two atoms we find the ferromagnetic state, a continuum of nematic states, and a state having the symmetry of the point group of the regular tetrahedron. For spin-three atoms we obtain similar ferromagnetic and nematic phases as well as states having symmetries of various types of polyhedra with six vertices: the hexagon, the pyramid with pentagonal base, the prism, and the octahedron. 
Ryan Barnett, Ari Turner, Eugene Demler