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

The two-dimensional (2D) spin-S = 1 XY model was investigated numerically as a realization of the (2 + 1)D superfluid-Mott-insulator (SF-MI) transition. The interaction parameters are extended so as to suppress corrections to finite-size scaling. Thereby, the external field of a unit flux quantum (φ = 2π) is applied to the 2D cluster by incorporating the phase factor eiφij (φ ij : gauge angle between the i and j sites) into the hopping amplitudes. Taking the advantage in that the exact-diagonalization method allows us to treat such a complex-valued matrix element, we evaluated the excess energy cost "E(2π) due to the magnetic flux φ = 2π explicitly in the SF (XY) phase. As a result, we found that the amplitude ratio ρ s/ "E(2π) (ρ s: spin stiffness) makes sense in proximity to the critical point, exhibiting a notable plateau in the SF-phase side. The plateau height is estimated, and compared to the related studies.