Longitudinal-field fidelity susceptibility analysis of the $J_1$-$J_2$ transverse-field Ising model around $J_2/J_1 \approx 0.5$

The square-lattice $J_1$-$J_2$ transverse-field (TF) Ising model was investigated with the exact diagonalization (ED) method. In order to analyze the TF-driven phase transition, we applied the longitudinal-field fidelity susceptibility $χ^{(h)}_F$, which is readily evaluated via the ED scheme. Here, the longitudinal field couples with the absolute value of the magnetic moment $|M|$ rather than the raw $M$ so that the remedied fidelity susceptibility exhibits a peak around the critical point; note that the spontaneous magnetization does not appear for the finite-size systems. As a preliminary survey, the modified fidelity susceptibility $χ^{(h)}_F$ is applied to the analysis of criticality for $J_2=0$, where a number of preceding results are available. Thereby, properly scaling the distance from the multi-criticality, $η=0.5-J_2$, the $χ^{(h)}_F$ data were cast into the crossover-scaling formula, and the multi-critical exponent for $χ_F^{(h)}$ is estimated. The result is cross-checked by the numerically evaluated $β$-function behavior.