Magnetooptical determination of a topological index

When a Dirac fermion system acquires an energy-gap, it is said to have either trivial (positive energy-gap) or non-trivial (negative energy-gap) topology, depending on the parity ordering of its conduction and valence bands. The non-trivial regime is identified by the presence of topological surface or edge-states dispersing in the energy gap of the bulk and is attributed a non-zero topological index. In this work, we show that such topological indices can be determined experimentally via an accurate measurement of the effective velocity of bulk massive Dirac fermions. We demonstrate this approach analytically starting from the Bernevig-Hughes-Zhang Hamiltonian to show how the topological index depends on this velocity. We then experimentally extract the topological index in Pb 1-x Sn x Se and Pb1-x Sn x Te using infrared magnetooptical Landau level spectroscopy. This approach is argued to be universal to all material classes that can be described by a Bernevig-Hughes-Zhang-like model and that host a topological phase transition.