Variance-reduced estimator of the connected two-point function in the presence of a broken ${\mathbb{Z}}_2$-symmetry

The exchange or geometric cluster algorithm allows us to define a variance-reduced estimator of the connected two-point function in the presence of a broken ${\mathbb{Z}}_2$-symmetry. We present numerical tests for the improved Blume-Capel model on the simple-cubic lattice. We perform simulations for the critical isotherm, the low-temperature phase at vanishing external field, and, for comparison, also the high-temperature phase. For the connected two-point function, a substantial reduction of the variance can be obtained, allowing us to compute the correlation length $\xi$ with high precision. Based on these results, estimates for various universal amplitude ratios that characterize the universality class of the three-dimensional Ising model are computed.

Figure 1

We plot ${\xi }_{\text{exp}}/{\xi }_{2\text{nd}}$ along with its corrected counterpart $z_{\text{cor}}{\xi }_{\text{exp}}/{\xi }_{2\text{nd}}$ for the critical isotherm as a function of ${\xi }_{\text{exp}}^{-2}$. The dashed lines connecting the data points are there to guide the eye. Note that the symbols for the two smallest values of $h$ partially overlap. The solid horizontal line gives our estimate for the scaling limit. The dashed lines parallel to it indicate its error.