  This work examines non-perturbative dynamics of a $2$-dimensional QFT by
using discrete 't Hooft anomaly, semi-classics with circle compactification and
bosonization. We focus on charge-$q$ $N$-flavor Schwinger model, and also
Wess-Zumino-Witten model. We first apply the recent developments of discrete 't
Hooft anomaly matching to theories on $\mathbb{R}^2$ and its compactification
to $\mathbb{R} \times S^1_L$. We then compare the 't Hooft anomaly with
dynamics of the models by explicitly constructing eigenstates and calculating
physical quantities on the cylinder spacetime with periodic and flavor-twisted
boundary conditions. We find different boundary conditions realize different
anomalies. Especially under the twisted boundary conditions, there are $Nq$
vacua associated with discrete chiral symmetry breaking. Chiral condensates for
this case have fractional $\theta$ dependence $\mathrm{e}^{\mathrm{i}
\theta/Nq}$, which provides the $Nq$-branch structure with soft fermion mass.
We show that these behaviors at a small circumference cannot be explained by
usual instantons but should be understood by "quantum" instantons, which
saturate the BPS bound between classical action and quantum-induced effective
potential. The effects of the quantum-instantons match the exact results
obtained via bosonization within the region of applicability of semi-classics.
We also argue that large-$N$ limit of the Schwinger model with twisted boundary
conditions satisfy volume independence.
