Aspect-ratio dependence of the spin stiffness of a two-dimensional XY model

We calculate the superfluid stiffness of two-dimensional (2D) lattice hard-core bosons at half filling (equivalent to the $S=1/2$ $\mathrm{XY}$ model) using the squared winding number quantum Monte Carlo estimator. For ${L}_{x}\ifmmode\times\else\texttimes\fi{}{L}_{y}$ lattices with aspect ratio ${L}_{x}{/L}_{y}=R$ and ${L}_{x}{,L}_{y}\ensuremath{\rightarrow}\ensuremath{\infty},$ we confirm the recent prediction [N. Prokof'ev and B.V. Svistunov, Phys. Rev. B 61, 11 282 (2000)] that the finite-temperature stiffness parameters ${\ensuremath{\rho}}_{x}^{W}$ and ${\ensuremath{\rho}}_{y}^{W}$ determined from the winding number differ from each other and from the true superfluid density ${\ensuremath{\rho}}_{s}.$ Formally, ${\ensuremath{\rho}}_{y}^{W}\ensuremath{\rightarrow}{\ensuremath{\rho}}_{s}$ in the limit in which ${L}_{x}\ensuremath{\rightarrow}\ensuremath{\infty}$ first and then ${L}_{y}\ensuremath{\rightarrow}\ensuremath{\infty}.$ In practice we find that ${\ensuremath{\rho}}_{y}^{W}$ converges exponentially to ${\ensuremath{\rho}}_{s}$ for $Rg1.$ We also confirm that for 3D systems, ${\ensuremath{\rho}}_{x}^{W}={\ensuremath{\rho}}_{y}^{W}={\ensuremath{\rho}}_{z}^{W}={\ensuremath{\rho}}_{s}$ for any R. In addition, we determine the Kosterlitz-Thouless transition temperature to be ${T}_{\mathrm{KT}}/J=0.34303(8)$ for the 2D model.