The two-dimensional easy-plane SU$(3)$ magnet subjected to the transverse field was investigated with the exact-diagonalization method. So far, as to the $XY$ model (namely, the easy-plane SU$(2)$ magnet), the transverse-field-driven order-disorder phase boundary has been investigated with the exact-diagonalization method, and it was claimed that the end-point singularity (multicriticality) at the $XX$-symmetric point does not accord with large-$N$-theorys prediction. Aiming to reconcile the discrepancy, we extend the internal symmetry to the easy-plane SU$(3)$ with the anisotropy parameter $eta$, which interpolates the isotropic ($eta=0$) and fully anisotropic ($eta=1$) cases smoothly. As a preliminary survey, setting $eta=1$, we analyze the order-disorder phase transition through resorting to the fidelity susceptibility $chi_F$, which exhibits a pronounced signature for the criticality. Thereby, with $eta$ scaled carefully, the $chi_F$ data are cast into the crossover-scaling formula so as to determine the crossover exponent $phi$, which seems to reflect the extension of the internal symmetry group to SU(3).
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