Zeroth Landau level crossings in two-dimensional topological insulators

Two-dimensional topological insulators (2D TIs) exhibit a characteristic Landau level spectrum in which a pair of zeroth Landau levels disperse oppositely as a result of band inversion. The two Landau levels cross at a critical magnetic field, marking the transition to a trivial phase. Experiments sometimes show an avoided crossing, signifying hybridization (mixing) of the zeroth Landau level wave functions. Here, we theoretically identify the essential symmetries that protect against zeroth Landau level mixing, using thin films of cadmium arsenide (${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$), a prototype 2D TI, as an example. We show that avoided crossings that result from zeroth Landau level mixing indicate the breaking of the four-fold rotational symmetry, inherent to bulk ${\mathrm{Cd}}_{3}{\mathrm{As}}_{2}$. We discuss potential reasons for the symmetry breaking, including the formation of novel correlated states, and the relevance of our results to other materials.