| Spin-orbit assisted Mott insulators have emerged as a major interest in recent top-ics, such as 5d transition metal oxides that probably host novel magnetic phases and topological spin liquids. Among candidate materials, iridium oxides (iridates), e.g., Sr2IrO4, A2IrO3 (A=Na, Li), and more recently ?-RuCl3, have in particular caught a lot of attention because they are (1) ideal playgrounds to test the interplay between strong spin-orbit coupling (SOC) and Mott physics, and (2) candidate materials where the prominent Kitaev physics might be realized. The seminal spin-1/2 Kitaev model defined on the honeycomb lattice can be solved exactly and has a spin-liquid ground state. This model possesses Z2 topological order, and the original spins fractionalize into Majorana fermions and Z2 gauge-field excitations. One of the most important dif-ferences between the Kitaev spin liquid and the conventional RVB spin liquid is the spin-rotation symmetry, i.e., the latter has the maximal SU(2) spin symmetry but the former has only the minimal Q8 spin symmetry. More recently, people realize that the so-called Kitaev-Heisenberg model is one of the most important models which may capture the essential physics of the spin-orbit assisted Mott insulators.(1) First, using symmetry analysis, we give the detailed derivation of the Kane-Mele model, where we have used 4 kinds of symmetries:time-reversal symmetry, mirror symmetry,2- and 3-fold rotation symmetries. We find that these symmetries lead many matrix elements to be zero, thereby the simple expression of the Kane-Mele model Hamiltonian is obtained.Similar symmetry analysis can be also applied to the Kagome lattice with SOC. Most importantly, there exist the nearest-neighbor SOC hopping terms on the Kagome lattice. Because:The symmetries forbid the intra sublattice SOC hopping on the Kagome lattice. In contrast, for the Kane-Mele model on the honeycomb lattice, the symmetries forbid the inter sublattice SOC hopping.(2) We then study the possible ground states of the undoped and doped Kitaev-Heisenberg model on a triangular lattice. For the undoped system, a combination of the numerical exact diagonalization calculation and the four-sublattice transformation anal-ysis suggests one possible exotic phase and four magnetically ordered phases, including a collinear stripe pattern and a noncollinear spiral pattern in the global phase diagram. The exotic phase near the antiferromagnetic (AF) Kitaev point is further investigated using the Schwinger-fermion mean-field method, and we obtain an energetically fa-vorable Z2 chiral spin liquid with a Chern number ±2 as a promising candidate. At finite doping, we find that the AF Heisenberg coupling supports an s-wave or d+id-wave superconductivity (SC), while the AF and the ferromagnetic Kitaev interactions favor a d+id-wave SC and a time-reversal invariant topological p-wave SC, respec-tively. Our work is the first study on the ground-state phase diagram of the quantum Kitaev-Heisenberg model on a triangular lattice.(3) Finally, we study the half-filled Hubbard model on the triangular lattice with spin-dependent Kitaev-like hopping. Using the variational cluster approach, we iden-tify five phases in the phase diagram:a metallic phase, a non-coplanar chiral magnetic order, a 120° magnetic order, a nonmagnetic insulator (NMI), and an interacting Chern insulator (CI). In the absence of interactions, increasing the Kitaev-like hopping drives the system transiting from the metal phase to the CI phase. With the increasing of inter-action, the transition from CI to NMI is accompanied by the charge gap changing from the indirect band gap to the direct Mott gap. The interacting CI is topologically nontriv-ial in the sense that it has a nonzero Chern number. Using the slave-rotor approach, we find that the NMI phase might consist of a gapless Mott insulator and a fractionalized CI with spinon edge states. Our work highlights the rising field that interesting phases emerge from the interplay of band topology and Mott physics. |
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