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 Neel 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.
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