Topological Phases in Correlated Electron Systems (a short introduction)Topological phases are the states of matter that cannot be adiabatically transformed (without closing the bulk excitation gap) to a traditional simple phase that can be completely characterized by a local order parameter such as magnetization. In topological phases, there exist unusual emergent excitations in the bulk or at the surface/boundary. Sometimes these excitations carry only some part or a particular combination of the quantum numbers of the electrons. In contrast to traditional simple phases, the existence of these unusual excitations is protected by the topological properties of the many-body ground states, often characterized by a non-local topological invariant that can be computed from the quantum mechanical wavefunction. This phenomenon is often called “topological order”. A well known example is the quantum Hall state that has been observed in two-dimensional electron gas and more recently in Graphene. The scope of possible topological phases of matter has been greatly extended in the last several years via the discovery of topological insulators and candidate materials for spin liquid phases. In topological band insulators, the ordering of the bands near the Fermi level is inverted by strong spin-orbit coupling. At the interface with a trivial insulator, a protected surface state arises as the bulk bands unwind upon approaching the other side. The surface state allows an intriguing magneto-electric effect; that is, the electric polarization (magnetization) can be generated by magnetic (electric) field. A spin liquid is a correlated insulator with no broken lattice symmetry, where the low energy excitations are the so-called spinons that carry only the spin-1/2 quantum number, not the charge, of electrons. The protected surface states of topological insulators and the spectra of spinons in spin liquids are fingerprints of the topological properties of these phases.Emergent Phases in Correlated Materials with Strong-Spin Orbit Coupling In an attempt to understand the roles of electron interactions in topological phases, our group investigated the existence of topological insulators and spin liquids in 5d transition-metal oxides such as iridates. 5d transition-metal elements have strong spin-orbit coupling and intermediate strength of electron interactions (“Hubbard-U”) due to the relatively large extent of the outer-most-shell 5d orbitals. The strong spin-orbit coupling in iridates allows a complex orbital wavefunction and the relevant electronic state near the Fermi level is a spin-orbital-entangled pseudo-spin-1/2 doublet. In lattice systems, the corresponding band is half-filled and may open a charge gap upon entering a Mott insulator at large Hubbard-U. The intermediate strength of the Hubbard-U means that iridates are typically close to a metal-insulator transition. It is demonstrated that topological insulators and semi-metallic phases can emerge in pyrochlore iridates, A2Ir2O7. Here the Ir4+ ions are sitting on the pyrochlore lattice, a network of corner sharing tetrahedra, and the A-site can be occupied by a non-magnetic Yittrium or Lanthanides. In particular, it is shown that the ground state of such systems is sensitive to the relative strength of the direct hopping between Ir-sites via d-orbital overlap, and the indirect hopping through the oxygen sites. As a result, the ground state can be changed rather easily by applying pressure. Moreover, at intermediate strength of Hubbard-U, the so-called topological semi-metal phases, with different kinds of magnetic order, are identified in parts of the phase diagram. The topological semi-metal is characterized by linearly-dispersing Weyl-Dirac fermions in the bulk, and can be regarded as a three-dimensional analogue of Graphene. These findings also provide means to control the transitions between these phases. On the spin liquid front, our group for a putative spin liquid ground state of Na4Ir3O8, where Ir ions are sitting on the hyperkagome lattice, a network of corner sharing triangles. It was shown that the interplay between the strong spin-orbit coupling and the proximity to a metal-insulator transition stabilizes a spin liquid state with a spinon Fermi surface, which is consistent with the power-law-in-temperature behavior of the specific heat, constant spin-susceptibility, and insulating behavior. Once confirmed, this would be the first example of a three-dimensional spin liquid.Quantum Spin Liquid in Kitaev MaterialsQuantum spin liquids are highly entangled states of interacting spin moments, characterized by novel excitations that carry only the spin of the electron, but no charge. Such fractionalized excitations are known to occur in one-dimension, but possible examples in two and three-dimensional systems are still under investigation. There have been tremendous efforts to clearly identify a quantum spin liquid in various classes of materials. Recently it has been realized that 4d and 5d transition metal oxides with honeycomb or three dimensional hyperhoneycomb lattice structures may realize a novel form of quantum spin liquid. The spin-orbit-entangled local moment degrees of freedom in these systems offer a non-trivial bond-dependent interaction, often called the Kitaev interaction. Surprisingly such a model can be solved exactly and the ground state is a quantum spin liquid. Because of this, there have been intensive efforts to identify the Kiatev spin liquids in a number of iridium oxides and other related materials. We are studying the physics of two and three-dimensional versions of such materials. We examine possible generic theoretical models for these systems. We have shown that, while the Kitaev interaction is the dominant one, there exist additional interactions that drive the system to an unusual magnetically ordered state. Our findings are in good agreement with a variety of experiments and clarify how one may achieve the desired Kitaev limit via systematic engineering of additional interactions. Quantum Phase Transitions and Quantum Critical PhenomenaTheory of Quantum Magnetism and Quantum Spin LiquidsNon-equilibrium Quantum Systems |