Topological phases in a two-dimensional lattice: Magnetic field versus spin-orbit coupling

In this work, we explore the rich variety of two-dimensional topological phases that arise when considering the competing effects of spin-orbit couplings, Zeeman splitting, and uniform magnetic fields. We investigate minimal models, defined on a honeycomb lattice, which clarify the topological phases stemming from the intrinsic and Rashba spin-orbit couplings, and also from the Zeeman splitting. In this sense, our work provides an interesting path connecting the quantum Hall phases, generally produced by the uniform magnetic field, and the quantum spin Hall phases resulting from spin-dependent couplings. First, we analyze the properties of each coupling term individually and we point out their similarities and differences. Second, we investigate the subtle competitions that arise when these effects are combined. We finally explore the various possible experimental realizations of our model.