Classifying Novel Phases of Spinor Atoms

Abstract

We consider many-body states of bosonic spinor atoms which, at the mean-field level, can be characterized by a single-particle wave function for the Bose-Einstein condensation and Mott insulating states. We describe and apply a classification scheme that makes explicit the 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. We apply the method to the many-body states of bosons with 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. Ultracold atoms in either a single optical trap or in an optical lattice provide clean realizations of unique systems of spins which were previously studied only as toy mathematical problems (for a recent review, see [1]). Depending on which hyperfine state is populated, alkali atoms can have spin one or two. Different phases of spin-one alkali bosons have been experimentally realized [2 -5] and considered theoretically [6 -12]. Spin-two bosons have also been experimentally probed [3,13,14] and theoretically studied for the case of a single optical trap [15,16] as well as, very recently, an optical lattice [17,18]. Finally, the Stuttgart group has recently succeeded in obtaining a Bose-Einstein condensation of 52 Cr atoms [19] which are spin-three bosons. This was followed by theoretical work [20,21] showing the possible types of phases that can be realized for such a spin-three system.Classification schemes of single-particle states with nonzero spins are needed to describe both superfluid condensates and Mott insulating states of spinor bosonic atoms. However, such classification becomes increasingly more difficult for larger spins. For instance, classifying the state of a spin-half particle is straightforward; only knowledge of the expectation value of the spin operator hFi is needed. On the other hand, for a spin-one particle, knowledge of the expectation value of the nematic tensor familiar from the classical theory of liquid crystals [22]in addition to hFi is required. Proceeding along these lines, one finds that for larger spin such a classification scheme becomes quite cumbersome since one needs to consider order parameters that involve higher-order products of spin operators, and a physical interpretation is not immediate. In this Letter, we present an alternative classification scheme which will work well for large spin. This scheme allows the symmetries of a general spin F particle to be represented by a polyhedron with 2F vertices. To illustrate the method, we use it to discuss spin-two bosons which are naturally realized as a hyperfine state of alkali atoms. The previously discussed superfluid ferromagnetic, polar, and c...

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