Motivated by recent numerical results, we study the phase diagram of the Kane-Mele-Hubbard (KHM) model, especially the nature of its quantum critical points. The phase diagram of the KaneMele-Hubbard model can be understood by breaking the SO(4) symmetry of our previous work down to U(1)spin × U(1) charge × PH symmetry. The vortices of the inplane Néel phase carry charge, and the proliferation of the charged magnetic vortex drives the transition between the inplane Néel phase and the QSH insulator phase; this transition belongs to the 3d XY universality class. The transition between the liquid phase and the inplane Néel phase is an anisotropic O(4) transition, which eventually becomes first order due to quantum fluctuation. The liquid-QSH transition is predicted to be first order based on a 1/N calculation.

Recent experiments on the anisotropic spin-1/2 triangular antiferromagnet Cs 2 CuBr 4 have revealed a remarkably rich phase diagram in applied magnetic fields, consisting of an unexpectedly large number of ordered phases. Motivated by this finding, we study the role of three ingredients-spatial anisotropy, Dzyaloshinskii-Moriya interactions, and quantum fluctuations-on the magnetization process of a triangular antiferromagnet, coming from the semiclassical limit. The richness of the problem stems from two key facts: (1) the classical isotropic model with a magnetic field exhibits a large accidental ground-state degeneracy and (2) these three ingredients compete with one another and split this degeneracy in opposing ways. Using a variety of complementary approaches, including extensive Monte Carlo numerics, spin-wave theory, and an analysis of Bose-Einstein condensation of magnons at high fields, we find that their interplay gives rise to a complex phase diagram consisting of numerous incommensurate and commensurate phases. Our results shed light on the observed phase diagram for Cs 2 CuBr 4 and suggest a number of future theoretical and experimental directions that will be useful for obtaining a complete understanding of this material's interesting phenomenology.

25 vol% relatively coarse opaque grain fragments and polycrystalline assemblages of kamacite, schreibersite, perryite, troilite (some grains with daubréelite exsolution lamellae), niningerite, oldhamite, and caswellsilverite; 2) ~30 vol% relatively coarse silicate grains including enstatite, albitic plagioclase, silica and diopside; and 3) an inferred fine nebular component (~45 vol%) comprised of submicrometer-size grains. Clastic matrix patches in ALH 81189 contain relatively coarse grains of opaques (~20 vol%; kamacite, schreibersite, perryite and troilite) and silicates (~30 vol%; enstatite, silica and forsterite) as well as an inferred fine nebular component (~50 vol%). The O-isotopic composition of clastic matrix in Y-691 is indistinguishable from that of olivine and pyroxene grains in adjacent chondrules; both sets of objects lie on the terrestrial mass-fractionation line on the standard three-isotope graph. Some patches of fine-grained matrix in Y-691 have distinguishable bulk concentrations of Na and K, inferred to be inherited from the solar nebula. Some patches in ALH 81189 differ in their bulk concentrations of Ca, Cr, Mn, and Ni. The average compositions of matrix material in Y-691 and ALH 81189 are similar but not identical-matrix in ALH 81189 is much richer in Mn (0.23 ± 0.05 versus 0.07 ± 0.02 wt%) and appreciably richer in Ni (0.36 ± 0.10 versus 0.18 ± 0.05 wt%) than matrix in Y-691. Each of the two whole-rocks exhibits a petrofabric, probably produced by shock processes on their parent asteroid.