Tuning the topological insulator states of artificial graphene

We develop a robust, non-perturbative approach to study the band structure of artificial graphene. Artificial graphene, as considered here, is generated by imposing a superlattice structure on top of a two dimensional hole gas in a semiconductor heterostructure, where the hole gas naturally possesses large spin-orbit coupling. Via tuning of the system parameters we demonstrate how best to exploit the spin-orbit coupling to generate time reversal symmetry-protected topological insulator phases. Our major conclusion is the identification of a second set of topological Dirac bands in the band structure (with spin Chern number $C=3$), which were not reliably obtainable in previous perturbative approaches to artificial graphene. Importantly, the second Dirac bands host more desirable features than the previously studied first set of Dirac bands (with $C=1$). Moreover, we find that upon tuning of the system parameters, we can drive the system to the highly desirable regime of the topological flat band. We discuss the possibilities this opens up for exotic, strongly correlated phases.