Possible quantization and half-quantization in the anomalous Hall effect caused by in-plane magnetic field

In this paper we propose that quantized and nearly half-quantized intrinsic anomalous Hall effect can be induced by in-plane external magnetic field through the Zeeman coupling in nonmagnetic two-dimensional (2D) systems with sizable spin-orbital but without twofold rotational symmetry. An analytical result is derived for 2D electron gas model ${C}_{3v}$ Further, based on $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ Hamiltonian from first principle calculations, find conductance observed ${\mathrm{Sb}}_{2}{\mathrm{Te}}_{3}$ thin film clean limit strong $B>20\phantom{\rule{4pt}{0ex}}\mathrm{T}$ low temperature $T<100\phantom{\rule{4pt}{0ex}}\mathrm{mK}$.