  We study QCD vacuum structure with the topological $\theta$ angle using a
recently proposed semiclassical approach on $\mathbb{R}^2 \times T^2$ with the
't Hooft and baryon magnetic fluxes. Under the assumption of adiabatic
continuity in this setup, the confining vacuum can be described by the dilute
gas of center vortices. With this semiclassical approach, we derive the 2d
effective description at small $T^2$ and successfully explain the reasonable
theta dependence of the QCD vacuum: In the one-flavor QCD at $\theta = \pi$,
the $CP$ symmetry is spontaneously broken for quark mass above a critical value
and restored for a subcritical mass, while the $CP$ symmetry is always
spontaneously broken in the multi-flavor QCD at $\theta = \pi$. From our
semiclassical description, we discuss implications to the $4$d chiral
Lagrangian and propose how the $\eta'$ meson should be incorporated in
consistent with known global structures: The periodicity of the $\eta'$ should
be extended from the naive one $2\pi$ to $2\pi N$. Additionally, we revisit the
phase diagram of $N_f = 1+1$ and $N_f = 1+1+1$ QCD on the up and down quark
mass plane, confirming and refining the existence of the $CP$-broken Dashen
phase.
