Imaginary-field-driven phase transition for the $2$D Ising antiferromagnet: A fidelity-susceptibility approach

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.