Quantum anomalous Hall effect with field-tunable Chern number near Z$_2$ topological critical point

We study the practicability of achieving quantum anomalous Hall (QAH) effect with field-tunable Chern number in a magnetically doped, topologically trivial insulating thin film. Specifically in a candidate material, TlBi(S$_{1-\delta}$Se$_{\delta}$)$_2$, we demonstrate that the QAH phases with different Chern numbers can be achieved by means of tuning the exchange field strength or the sample thickness near the Z$_2$ topological critical point. Our physics scenario successfully reduces the necessary exchange coupling strength for a targeted Chern number. This QAH mechanism differs from the traditional QAH picture with a magnetic topological insulating thin film, where the ``surface'' states must involve and sometimes complicate the realization issue. Furthermore, we find that a given Chern number can also be tuned by a perpendicular electric field, which naturally occurs when a substrate is present.[1] High-Chern number QAH phase obtained from magnetically doped topological crystalline insulator thin films will also be discussed. \\ \\ REF: [1] Le Quy Duong, Hsin Lin, Wei-Feng Tsai, and Y. P. Feng, Phys. Rev. B 92, 115205 (2015).