  We investigate the $\theta$-vacuum structure and the 't Hooft anomaly at
$\theta=\pi$ in a simple quantum mechanical system on $S^1$ to scrutinize the
applicability of the functional renormalization group (fRG) approach. Even
though the fRG is an exact formulation, a naive application of the fRG equation
would miss contributions from the $\theta$ term due to the differential nature
of the formulation. We first review this quantum mechanical system on $S^1$
that is solvable with both the path integral and the canonical quantization. We
discuss how to construct the quantum effective action including the $\theta$
dependence. Such an explicit calculation poses a subtle question of whether a
Legendre transform is well defined or not for general systems with the sign
problem. We then consider a deformed theory to relax the integral winding by
introducing a wine-bottle potential with the finite depth $\propto g$, so that
the original $S^1$ theory is recovered in the $g\to\infty$ limit. We
numerically solve the energy spectrum in the deformed theory as a function of
$g$ and $\theta$ in the canonical quantization. We test the efficacy of the
simplest local potential approximation (LPA) in the fRG approach and find that
the correct behavior of the ground state energy is well reproduced for small
$\theta$. When the energy level crossing is approached, the LPA flow breaks
down and fails in describing the ground state degeneracy expected from the 't
Hooft anomaly. We finally turn back to the original theory and discuss an
alternative formulation using the Villain lattice action. The analysis with the
Villain lattice at $\theta=\pi$ indicates that the nonlocality of the effective
action is crucial to capture the level crossing behavior of the ground states.
