  We examine the phase structure of the two-flavor Schwinger model as a
function of the $\theta$-angle and the two masses, $m_1$ and $m_2$. In
particular, we find interesting effects at $\theta=\pi$: along the
$SU(2)$-invariant line $m_1 = m_2 = m$, in the regime where $m$ is much smaller
than the charge $g$, the theory undergoes logarithmic RG flow of the
Berezinskii-Kosterlitz-Thouless type. As a result, in this regime there is a
non-perturbatively small mass gap $\sim e^{- A g^2/m^2}$. The $SU(2)$-invariant
line lies within a region of the phase diagram where the charge conjugation
symmetry is spontaneously broken and whose boundaries we determine numerically.
Our numerical results are obtained using the Hamiltonian lattice gauge
formulation that includes the mass shift $m_\text{lat} = m- g^2 a/4$ dictated
by the discrete chiral symmetry.
