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Fidelity-susceptibility analysis of the honeycomb-lattice Ising antiferromagnet under the imaginary magnetic field

The honeycomb-lattice Ising antiferromagnet subjected to the imaginary magnetic field $H=iθT /2$ with the "topological" angle $θ$ and temperature $T$ was investigated numerically. In order to treat such a complex-valued statistical weight, we employed the transfer-matrix method. As a probe to detect the order-disorder phase transition, we resort to an extended version of the fidelity $F$, which makes sense even for such a non-hermitian transfer matrix. As a preliminary survey, for an intermediate value of $θ$, we investigated the phase transition via the fidelity susceptibility $χ_F^{(θ)}$. The fidelity susceptibility $χ_F^{(θ)}$ exhibits a notable signature for the criticality as compared to the ordinary quantifiers such as the magnetic susceptibility. Thereby, we analyze the end-point singularity of the order-disorder phase boundary at $θ=π$. We cast the $χ_F^{(θ)}$ data into the crossover-scaling formula with $δθ= π-θ$ scaled carefully. Our result for the crossover exponent $ϕ$ seems to differ from the mean-field and square-lattice values, suggesting that the lattice structure renders subtle influences as to the multi-criticality at $θ=π$.