positions. Fig.\[Fig:SF\] displays the behavior of $S(0)$ at fixed $V/t=1.6$ for different lattice sizes. At high temperatures, where spin correlations are short ranged, this quantity is independent of the size, $N$, of the system. However, when the ground state exhibits long range order, the sum over all sites in Eq. becomes dependent on $N$. More specifically, in Fig.\[Fig:SF\], curves with different sizes separate at $T/t \approx 0.05$, the temperature at which the correlation length $\xi$ starts to become comparable to the linear lattice size. The temperature, which is set by the effective exchange coupling $J_{\rm eff}$, where $\xi(T) \sim L$ decreases for larger $V/t$ (not shown). As a consequence, simulations for this parameter regime become very challenging for DQMC. The dependence of $J_{\rm eff}$ on $V$ has been studied in perturbation theory[@Potthoff15_1].

The order parameter is obtained by carrying out a finite-size scaling analysis of the spin structure factor. The saturated (large $\beta$) values of $S(0)$, for different lattice sizes are fit to a linear spin-wave scaling[@huse88], $$\begin{aligned}
\frac{S(0)}{N_{cell}} =
m^{2}_{tot} + \frac{a}{\sqrt{N_{cell}}},\end{aligned}$$ with $m^{2}_{tot}$ being the extrapolated global ferromagnetic order parameter; see Fig. \[Fig:Scaling\]. This result confirms the existence of long range ferromagnetism in the ground state, even for 