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

TL;DR

This study numerically investigates the energy cost of a unit flux quantum in a 2D superfluid-insulator transition, revealing a plateau in the spin stiffness to energy cost ratio near the critical point.

Contribution

It introduces a detailed numerical analysis of the excess energy due to magnetic flux in a 2D superfluid-insulator transition, highlighting a plateau in the spin stiffness ratio.

Findings

Identified a plateau in the ratio of spin stiffness to excess energy near the critical point.

Quantified the amplitude of the plateau and compared it with previous studies.

Demonstrated the effectiveness of exact diagonalization in complex-valued matrix calculations.

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 $e^{i\phi_{ij}}$ ($\phi_{ij}$: 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.