  Quantum anomalous Hall (QAH) insulators are two-dimensional (2D) insulating
states exhibiting properties similar to those of quantum Hall states but
without external magnetic field. They have quantized Hall conductance
$\sigma^H=Ce^2/h$, where integer $C$ is called the Chern number, and represents
the number of gapless edge modes. Recent experiments demonstrated that chromium
doped thin-film (Bi,Sb)$_2$Te$_3$ is a QAH insulator with Chern number
$C=\pm1$. Here we theoretically predict that thin-film topological crystalline
insulators (TCI) can host various QAH phases, when doped by ferromagnetically
ordered dopants. Any Chern number between $\pm4$ can, in principle, be reached
as a result of the interplay between (a) the induced Zeeman field, depending on
the magnetic doping concentration, (b) the structural distortion, either
intrinsic or induced by a piezoelectric material through proximity effect and
(c) the thickness of the thin film. The tunable Chern numbers found in TCI
possess significant potential for ultra-low power information processing
applications.
