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

Abstract

The two-dimensional ($2$D) 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 ($\Phi =2\pi$) is applied to the 2D cluster by incorporating the phase factor $e^{i{\varphi }_{ij}}$ (${\varphi }_{ij}$: 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 $\Delta E(2\pi )$ due to the magnetic flux $\Phi =2\pi$ explicitly in the SF ($XY$) phase. As a result, we found that the amplitude ratio ${\rho }_s/\Delta E(2\pi )$ (${\rho }_s$: 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.

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Cited by in corpus (1)

• End-point singularity of the $XY$-paramagnetic phase boundary for the $(2+1)$D $S=1$ square-lattice $J_1$-$J_2$ $XY$ model with the single-ion anisotropy