Kitaev Materials

In transition-metal compounds with partially filled $4d$ and $5d$ shells spin-orbit entanglement, electronic correlations, and crystal-field effects conspire to give rise to a variety of novel forms of topological quantum matter. This includes Kitaev materials -- a family of spin-orbit assisted Mott insulators, in which local, spin-orbit entangled $j=1/2$ moments form that are subject to strong bond-directional interactions. On a conceptual level, Kitaev materials attract much interest for their unconventional forms of magnetism, such as spin liquid physics in two- and three-dimensional lattice geometries or the formation of non-trivial spin textures. Experimentally, a number of Kitaev materials have been synthesized, which includes the honeycomb materials Na$_2$IrO$_3$, $\alpha$-Li$_2$IrO$_3$, and RuCl$_3$, the triangular materials Ba$_3$Ir$_x$Ti$_{3-x}$O$_9$, as well as the three-dimensional hyper-honeycomb and stripy-honeycomb materials $\beta$-Li$_2$IrO$_3$ and $\gamma$-Li$_2$IrO$_3$. These lecture notes provide a short review of the current status of the theoretical and experimental exploration of these Kitaev materials.