Measuring the Chern number with quantum oscillations

A peculiar feature of the majority of three-dimensional topological insulator surface states studied experimentally thus far, namely their particle-hole asymmetry, makes quantum oscillations (Shubnikov--de Haas and de Haas--van Alphen oscillations) in these materials particularly rich. I show that this peculiarity can be exploited to measure the Chern number and detect topological phase transitions in topological insulator surface states from the quantum spin Hall phase to the quantum anomalous Hall phase. I consider the behavior of quantum oscillations in topological insulator thin-film surface states in the presence of a topological exciton condensate or hybridization between the two surfaces. As a function of Zeeman field, the Chern number and phase transition from a quantum spin Hall to a quantum anomalous Hall phase can be measured using standard techniques. This effect relies necessarily on the particle-hole asymmetry, which is ubiquitous in currently known materials that exhibit topological insulator surface states.