Topological Phases in Perovskite Iridates with Strong Spin-Orbit Coupling

The study of topological phases of matter has attracted much attention recently, providing a promising route to understanding the classification of various phases in condensed matter physics. The Iridates, which are Iridium-based compounds, have already shown rich physical phenomena and can be a perfect platform to reveal topological phases in materials with strong spin-orbit coupling. This thesis focuses on possible exotic topological phases that stem from the interplay between strong spin-orbit coupling and non-trivial band structure in three-dimensional perovskite Iridates and other layered Iridate materials. Firstly, motivated from previous band structure results of SrIrO$_3$, we predict that three-dimensional perovskite Iridates can realize a new class of metals dubbed ``topological crystalline metals" with surface states protected by lattice symmetry and a pair of topological invariants. We show that these novel metallic phases can give rise to various non-trivial topological phases when different symmetries are broken, including a Weyl and nodal-line semimetal . Secondly, we provide a generic condition to identify a novel three-dimensional topological semi-metallic phase in materials with strong spin-orbit coupling. Such a semi-metallic phase is featured by bulk gapless four-fold degenerate nodal-line excitations. By using perovskite Iridates SrIrO$_3$ as an example, we give an explicit mathematical proof on the validity of this condition as well as verify the existence of such bulk nodal-line band degeneracy in the tight-binding model. Thirdly, we propose that a topological insulator can exist in a two-dimensional single layer Iridium oxide superlattice under certain lattice distortions; a tight-binding model is constructed using the basis of an effective J$_{\rm eff}$=1/2 pseudo-spins. A topological magnetic insulator with quantum anomalous Hall effect is found in the bilayer superlattice when a magnetic order or magnetic field is present in the system. Lastly, a pair of helical Majorana fermions emerges at the interface of a two-dimensional Weyl semimetal and a d-wave superconductor. We focused on the most promising and feasible realization: Iridium oxide layer without inversion symmetry, which is classified as Weyl semimetal, in proximity to d-wave high temperature Cuprates. In conclusion, we provide some future directions related to the topological phases in Iridates.Ph.D