We consider bosonic dipolar molecules in an optical lattice prepared in a mixture of different rotational states. The 1/r 3 interaction between molecules for this system is produced by exchanging a quantum of angular momentum between two molecules. We show that the Mott states of such systems have a large variety of quantum phases characterized by dipolar orderings including a state with ordering wave vector that can be changed by tilting the lattice. As the Mott insulating phase is melted, we also describe several exotic superfluid phases that will occur.

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...

In this work, we derive the equations of motion governing the dynamics of spin-F spinor condensates. We pursue a description based on standard physical variables ͑total density and superfluid velocity͒, alongside 2F "spin nodes:" unit vectors that describe the spin-F state and also exhibit the point-group symmetry of a spinor condensate's mean-field ground state. In the first part of our analysis, we derive the hydrodynamic equations of motion, which consist of a mass continuity equation, 2F Landau-Lifshitz equations for the spin nodes, and a modified Euler equation. In particular, we provide a generalization of the Mermin-Ho relation to spin one and find an analytic solution for the skyrmion texture in the incompressible regime of a spin-half condensate. In the second part, we study the linearized dynamics of spinor condensates. We provide a general method to linearize the equations of motion based on the symmetry of the mean-field ground state using the local stereographic projection of the spin nodes. We also provide a simple construction to extract the collective modes from symmetry considerations alone akin to the analysis of vibrational excitations of polyatomic molecules. Finally, we present a mapping between the spin-wave modes, and the wave functions of electrons in atoms, where the spherical symmetry is degraded by a crystal field. These results demonstrate the beautiful geometrical structure that underlies the dynamics of spinor condensates.

We perform analysis of the band structure, phonon dispersion, and electron-phonon interactions in three types of small-radius carbon nanotubes. We find that the ͑5,5͒ nanotube can be described well by the zonefolding method and the electron-phonon interaction is too small to support either a charge-density wave or superconductivity at realistic temperatures. For ultrasmall ͑5,0͒ and ͑6,0͒ nanotubes we find that the large curvature makes these tubes metallic with a large density of states at the Fermi energy and leads to unusual electron-phonon interactions, with the dominant coupling coming from the out-of-plane phonon modes. By combining the frozen-phonon approximation with the random phase approximation analysis of the giant Kohn anomaly in one dimension we find parameters of the effective Fröhlich Hamiltonian for the conduction electrons. Neglecting Coulomb interactions, we find that the ͑5,5͒ carbon nanotube ͑CNT͒ remains stable to instabilities of the Fermi surface down to very low temperatures while for the ͑5,0͒ and ͑6,0͒ CNTs a charge density wave instability will occur. When we include a realistic model of Coulomb interaction we find that the charge-density wave remains dominant in the ͑6,0͒ CNT with T CDW around 5 K while the charge-density wave instability is suppressed to very low temperatures in the ͑5,0͒ CNT, making superconductivity dominant with transition temperature around 1 K.