This work examines non-perturbative dynamics of a $2$-dimensional QFT by<br />
using discrete &#039;t Hooft anomaly, semi-classics with circle compactification and<br />
bosonization. We focus on charge-$q$ $N$-flavor Schwinger model, and also<br />
Wess-Zumino-Witten model. We first apply the recent developments of discrete &#039;t<br />
Hooft anomaly matching to theories on $\mathbb{R}^2$ and its compactification<br />
to $\mathbb{R} \times S^1_L$. We then compare the &#039;t Hooft anomaly with<br />
dynamics of the models by explicitly constructing eigenstates and calculating<br />
physical quantities on the cylinder spacetime with periodic and flavor-twisted<br />
boundary conditions. We find different boundary conditions realize different<br />
anomalies. Especially under the twisted boundary conditions, there are $Nq$<br />
vacua associated with discrete chiral symmetry breaking. Chiral condensates for<br />
this case have fractional $\theta$ dependence $\mathrm{e}^{\mathrm{i}<br />
\theta/Nq}$, which provides the $Nq$-branch structure with soft fermion mass.<br />
We show that these behaviors at a small circumference cannot be explained by<br />
usual instantons but should be understood by &quot;quantum&quot; instantons, which<br />
saturate the BPS bound between classical action and quantum-induced effective<br />
potential. The effects of the quantum-instantons match the exact results<br />
obtained via bosonization within the region of applicability of semi-classics.<br />
We also argue that large-$N$ limit of the Schwinger model with twisted boundary<br />
conditions satisfy volume independence.