Duality-mediated critical amplitude ratios for the $(2+1)$-dimensional $S=1$ $XY$ model

The phase transition for the $(2+1)$-dimensional spin-$S=1$ $XY$ model was investigated numerically. Because of the boson-vortex duality, the spin stiffness ${\rho }_s$ in the ordered phase and the vortex-condensate stiffness ${\rho }_v$ in the disordered phase should have a close relationship. We employed the exact diagonalization method, which yields the excitation gap directly. As a result, we estimate the amplitude ratios ${\rho }_{s,v}\mathrm{/}\mathrm{\Delta }$ ($\mathrm{\Delta }$: Mott insulator gap) by means of the scaling analyses for the finite-size cluster with $N\le 22$ spins. The ratio ${\rho }_s\mathrm{/}{\rho }_v$ admits a quantitative measure of deviation from selfduality.