We discuss phenomena arising from the combined influence of electron correlation and spin-orbit coupling, with an emphasis on emergent quantum phases and transitions in heavy transition metal compounds with 4d and 5d elements. A common theme is the influence of spin-orbital entanglement produced by spin-orbit coupling, which influences the electronic and magnetic structure. In the weak-to-intermediate correlation regime, we show how non-trivial band-like topology leads to a plethora of phases related to topological insulators. We expound these ideas using the example of pyrochlore iridates, showing how many novel phases such as the Weyl semi-metal, axion insulator, topological Mott insulator, and topological insulators may arise in this context. In the strong correlation regime, we argue that spin-orbital entanglement fully or partially removes orbital degeneracy, reducing or avoiding the normally ubiquitous Jahn-Teller effect. As we illustrate for the honeycomb lattice iridates and double perovskites, this leads to enhanced quantum fluctuations of the spin-orbital entangled states and the chance to promote exotic quantum spin liquid and multipolar ordered ground states. Connections to experiments, materials, and future directions are discussed.

We examine the role of spin-orbit coupling in the electronic structure of α-RuCl3, in which Ru ions in 4d 5 configuration form a honeycomb lattice. The measured optical spectra exhibit an optical gap of 220 meV and transitions within the t2g orbitals. The spectra can be described very well with firstprinciples electronic structure calculations obtained by taking into account both spin-orbit coupling and electron correlations. Furthermore, our x-ray absorption spectroscopy measurements at the Ru L edges exhibit distinct spectral features associated with the presence of substantial spin-orbit coupling, as well as an anomalously large branching ratio. We propose that α-RuCl3 is a spin-orbit assisted Mott insulator, and that the bond-dependent Kitaev interaction may be relevant for this compound.

The combination of electronic correlation and spin-orbit coupling is thought to precipitate a variety of highly unusual electronic phases in solids, including topological and quantum spin liquid states. We report a Raman scattering study that provides evidence for unconventional excitations in α-RuCl_{3}, a spin-orbit coupled Mott insulator on the honeycomb lattice. In particular, our measurements reveal unusual magnetic scattering, typified by a broad continuum. The temperature dependence of this continuum is evident over a large scale compared to the magnetic ordering temperature, suggestive of frustrated magnetic interactions. This is confirmed through an analysis of the phonon linewidths, which show a related anomaly due to spin-phonon coupling. These observations are in line with theoretical expectations for the Heisenberg-Kitaev model and suggest that α-RuCl_{3} may be close to a quantum spin liquid ground state.

We show that the Anomalous Hall Effect (AHE) observed in Colossal Magnetoresistance Manganites is a manifestation of Berry phase effects caused by carrier hopping in a non-trivial spin background. We determine the magnitude and temperature dependence of the Berry phase contribution to the AHE, finding that it increases rapidly in magnitude as the temperature is raised from zero through the magnetic transition temperature Tc, peaks at a temperature Tmax > Tc and decays as a power of T, in agreement with experimental data. We suggest that our theory may be relevant to the anomalous Hall effect in conventional ferromagnets as well. 75.20.Hr, 75.30.Hx, 75.30.Mb The Anomalous Hall Effect(AHE) is a fundamental but incompletely understood aspect of the physics of metallic ferromagnets [1,2]. The Hall effect is the development of a voltage which is transverse to an applied current; the constant of proportionality is the Hall resistivity ρ H . In non-magnetic materials, ρ H is proportional to the magnetic induction B and its sign is determined by the carrier charge. Many ferromagnets however exhibit an anomalous contribution to ρ H which is proportional to the magnetization M , thusThe definition of R s implies a sample with demagnetization factor N ∼ = 1 so that M represents the spin polarization in the material and the physical dipolar magnetic field caused by the ferromagnetically aligned spins cancels. The AHE thus involves a coupling of orbital motion of electrons to the spin polarization and must involve spin-orbit coupling.The conventional theoretical understanding of R s is based on a skew scattering mechanism which is a third order process involving interference between spin-orbit coupling (to first order ) and spin flip scattering ( to second order ) [1,3,4]. In conventional ferromagnets, this theory yields values of R s two orders of magnitude smaller than experimental data [3]. Also, some papers including Ref.[3] use a spin-orbit term involving coupling to the dipole fields produced by the spins which would apparently vanish for demagnetization factor N = 1.Recently, several groups measured the Hall resistivity ρ H of epitaxial films [5] and single crystals [6] of the 'colossal magnetoresistance' (CMR) material La 0.7 Ca 0.3 M nO 3 These materials involve carriers derived from Mn e g symmetry d-levels which may move through the lattice but are strongly ferromagnetically coupled to localized 'core spins' derived from Mn t 2g symmetry orbitals. The coupling is so strong that it may be taken to be infinite: a carrier on site i must have its spin parallel to the core spin on site i. The spin of the mobile carrier is thus quenched, but its amplitude to hop from site i to site j is modulated by a factor involving the relative spin states of core spins on the two sites. This physics is called 'double-exchange' [7].The Hall effect measurements found that in CMR materials ρ H was of the form of Eq.1 with R 0 hole-like and R s electron-like. R s becomes evident above 100K, increases sharply around T c , peaks at a te...

We construct a model for interacting electrons with strong spin orbit coupling in the pyrochlore iridates. We establish the importance of the direct hopping process between the Ir atoms and use the relative strength of the direct and indirect hopping as a generic tuning parameter to study the correlation effects across the iridates family. We predict novel quantum phase transitions between conventional and/or topologically non-trivial phases. At weak coupling, we find topological insulator and metallic phases. As one increases the interaction strength, various magnetic orders emerge. The novel topological Weyl semi-metal phase is found to be realized in these different orders, one of them being the all-in/all-out pattern. Our findings establish the possible magnetic ground states for the iridates and suggest the generic presence of the Weyl semi-metal phase in correlated magnetic insulators on the pyrochlore lattice. We discus the implications for existing and future experiments.

We report magnetic and thermodynamic properties of single crystal α-RuCl3, in which the Ru 3+ (4d 5 ) ion is in its low spin state and forms a honeycomb lattice. Two features are observed in both magnetic susceptibility and specific heat data; a sharp peak at 7 K and a broad hump near 10-15K. In addition, we observe a metamagnetic transition between 5 T and 10 T. Our neutron diffraction study of single crystal samples confirms that the low temperature peak in the specific heat is associated with a magnetic order with unit cell doubling along the honeycomb (100) direction, which is consistent with zigzag order, one of the types of magnetic order predicted within the framework of the Kitaev-Heisenberg model.

We report a combined experimental and theoretical investigation of the magnetic structure of the honeycomb lattice magnet Na2IrO3, a strong candidate for a realization of a gapless spin-liquid. Using resonant x-ray magnetic scattering at the Ir L3-edge, we find 3D long range antiferromagnetic order below TN =13.3 K. From the azimuthal dependence of the magnetic Bragg peak, the ordered moment is determined to be predominantly along the a-axis. Combining the experimental data with first principles calculations, we propose that the most likely spin structure is a novel "zig-zag" structure.

We study a model of fermions interacting with a gauge field and calculate gauge-invariant two-particle Green's functions or response functions. The leading singular contributions from the self-energy correction are found to be cancelled by those from the vertex correction for small $q$ and $\Omega$. As a result, the remaining contributions are not singular enough to change the leading order results of the random phase approximation. It is also shown that the gauge field propagator is not renormalized up to two-loop order. We examine the resulting gauge-invariant two-particle Green's functions for small $q$ and $\Omega$, but for all ratios of $\Omega / v_F q$ and we conclude that they can be described by Fermi liquid forms without a diverging effective mass.Comment: Plain Tex, 35 pages, 5 figures available upon request, Revised Version (Expanded discussion), To be published in Physical Review B 50, (1994) (December 15 issue