  The square-lattice Ising antiferromagnet subjected to the imaginary magnetic
field $H=i \theta T /2 $ with the "topological" angle $\theta$ and temperature
$T$ was investigated by means of the transfer-matrix method. Here, as a probe
to detect the order-disorder phase transition, we adopt an extended version of
the fidelity susceptibility $\chi_F^{(\theta)}$, which makes sense even for
such a non-hermitian transfer matrix. As a preliminary survey, for an
intermediate value of $\theta$, we examined the finite-size-scaling behavior of
$\chi_F^{(\theta)}$, and found a pronounced signature for the criticality; note
that the magnetic susceptibility exhibits a weak (logarithmic) singularity at
the N\'eel temperature. Thereby, we turn to the analysis of the power-law
singularity of the phase boundary at $\theta=\pi$. With $\theta-\pi$ scaled
properly, the $\chi_F^{(\theta)}$ data are cast into the crossover scaling
formula, indicating that the phase boundary is shaped concavely. Such a feature
makes a marked contrast to that of the mean-field theory.
