Two- and three-point functions at criticality: Monte Carlo simulations of the improved three-dimensional Blume-Capel model

We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to $L=1600$. As a check, also simulations of the spin-$\frac{1}{2}$ Ising model are performed. We find $f_{\sigma \sigma \epsilon }=1.051\left(1\right)$ and $f_{\epsilon \epsilon \epsilon }=1.533\left(5\right)$ for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of $N$-point functions. Finite size corrections vanish with $L^{-{\mathrm{\Delta }}_{\epsilon }}$, where $L$ is the linear size of the lattice and ${\mathrm{\Delta }}_{\epsilon }$ is the scaling dimension of the leading $Z_2$-even scalar $\epsilon$.

Figure 1

Two-dimensional sketch of the decomposition of the lattice into blocks and the remainder. The three points $x_1$, $x_2$, and $x_3$ are represented by red squares. The blocks around these three points are bordered by solid black lines. Lattice sites different from $x_1$, $x_2$, or $x_3$ are given by solid black circles.

Figure 2

We plot the effective dimension ${\mathrm{\Delta }}_{\sigma ,\mathrm{eff}}(x,s)$ of the field as defined by Eq. (35) for various linear lattice sizes $L$ and extrapolated estimates. The displacement between the points is in direction $r=a$. The dashed lines should only guide the eye. We give only results obtained from our simulations with stride $n_s=2$ to keep the figure readable. The horizontal solid line gives ${\mathrm{\Delta }}_{\sigma }=0.5181489$ obtained by the bootstrap method.

Figure 3

We plot the effective dimension ${\mathrm{\Delta }}_{\sigma ,\mathrm{eff}}(x,s)$ of the field as defined by Eq. (35) for the extrapolation obtained for $L=800$ and 1600. All results are obtained for the stride $n_s=2$. We give results for all three directions of the displacement that we have studied. The dashed lines should only guide the eye. The horizontal solid line gives ${\mathrm{\Delta }}_{\sigma }=0.5181489$ obtained by the bootstrap method.

Figure 4

We plot $g_{\sigma \sigma }{x}^{2{\mathrm{\Delta }}_{\sigma }}$ with ${\mathrm{\Delta }}_{\sigma }=0.5181489$ for the three directions studied. Most of the data are taken from the simulations with the stride $n_s=2$. A few data at small distances are taken from our preliminary simulations without variance reduction. The solid lines give the result of the fit with the Ansatz (36).

Figure 5

We plot $g_{\epsilon \epsilon }{x}^{2{\mathrm{\Delta }}_{\epsilon }}$ with ${\mathrm{\Delta }}_{\epsilon }=1.412625$ for the three directions studied. Analogous to Fig. 4.

Figure 6

We plot our estimates of $f_{\sigma \sigma \epsilon }$ for the direction $r=f$. We give results obtained for the linear lattice sizes $L=400$, 600, 800, 1200, and 1600 and the extrapolations for the pairs $(L,2L)=(400,800)$, $(600,1200)$, and $(800,1600)$. All data shown are based on our simulations with $n_s=2$. The straight line gives the bootstrap result.

Figure 7

We plot our estimates of $f_{\sigma \sigma \epsilon }$ as a function of the distance $x$. These estimates are obtained by the extrapolation of the pair of lattice sizes $(L,2L)=(800,1600)$. The solid lines represent the results of fits with the Ansatz (36).

Figure 8

We plot our estimates of $f_{\epsilon \epsilon \epsilon }$ as a function of the distance $x$. The solid lines give the results of fits with the Ansatz (36).