Engineering Topology in Massive Dirac Fermions

We engineer topology in massive Dirac fermions in transition-metal dichalcogenide (TMD) monolayer and bilayer materials using an artificial superlattice potential that can be created with contemporary experimental techniques such as with LaAlO$_3$/SrTiO$_3$ interface. We use symmetry analysis to analyze band inversions to provide the Chern number $mathcal C$ for the valence band as a function of tunable potential parameter space for a class of $C_4$ and $C_3$ symmetric potentials. We present a novel method to engineer Chern number $mathcal C=2$ for the valence band and show that the applied potential must have a scalar together with a non-scalar periodic part. We discover that externally applied potential allows the possibility of non-trivial topological transitions that cannot be achieved by naturally occurring moir'e patterns produced by a twist or lattice mismatch. We finally discuss the experimental implementation of our work which allows for the possibility to realize the quantum Spin Hall effect (QSHE), quantum Hall effect (QHE), and even exotic non-Abelian anyons in the fractional quantum Hall effect (FQHE).