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This work examines non-perturbative dynamics of a $2$-dimensional QFT byusing discrete 't Hooft anomaly, semi-classics with circle compactification andbosonization. We focus on charge-$q$ $N$-flavor Schwinger model, and alsoWess-Zumino-Witten model. We first apply the recent developments of discrete 'tHooft anomaly matching to theories on $\mathbb{R}^2$ and its compactificationto $\mathbb{R} \times S^1_L$. We then compare the 't Hooft anomaly withdynamics of the models by explicitly constructing eigenstates and calculatingphysical quantities on the cylinder spacetime with periodic and flavor-twistedboundary conditions. We find different boundary conditions realize differentanomalies. Especially under the twisted boundary conditions, there are $Nq$vacua associated with discrete chiral symmetry breaking. Chiral condensates forthis 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 byusual instantons but should be understood by "quantum" instantons, whichsaturate the BPS bound between classical action and quantum-induced effectivepotential. The effects of the quantum-instantons match the exact resultsobtained via bosonization within the region of applicability of semi-classics.We also argue that large-$N$ limit of the Schwinger model with twisted boundaryconditions satisfy volume independence.

Fractional $\theta$ angle, 't Hooft anomaly, and quantum instantons in charge-$q$ multi-flavor Schwinger model