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The two-dimensional ($2$D) spin-$S=1$ $XY$ model was investigated numericallyas a realization of the $(2+1)$D superfluid-Mott-insulator (SF-MI) transition.The interaction parameters are extended so as to suppress corrections tofinite-size scaling. Thereby, the external field of a unit flux quantum($\Phi=2\pi$) is applied to the 2D cluster by incorporating the phase factor$e^{i\phi_{ij}}$ ($\phi_{ij}$: gauge angle between the $i$ and $j$ sites) intothe hopping amplitudes. Taking the advantage in that the exact-diagonalizationmethod allows us to treat such a complex-valued matrix element, we evaluatedthe excess energy cost $\Delta E(2\pi)$ due to the magnetic flux $\Phi=2\pi$explicitly in the SF ($XY$) phase. As a result, we found that the amplituderatio $\rho_s / \Delta E(2\pi)$ ($\rho_s$: spin stiffness) makes sense inproximity to the critical point, exhibiting a notable plateau in the SF-phaseside. The plateau height is estimated, and compared to the related studies.

Criticality of the excess energy cost due to the unit-flux-quantum external field for the $(2+1)$D superfluid-insulator transition