Interface tension in the improved Blume-Capel model

We study interfaces with periodic boundary conditions in the low-temperature phase of the improved Blume-Capel model on the simple cubic lattice. The interface free energy is defined by the difference of the free energy of a system with antiperiodic boundary conditions in one of the directions and that of a system with periodic boundary conditions in all directions. It is obtained by integration of differences of the corresponding internal energies over the inverse temperature. These differences can be computed efficiently by using a variance reduced estimator that is based on the exchange cluster algorithm. The interface tension is obtained from the interface free energy by using predictions based on effective interface models. By using our numerical results for the interface tension $\sigma$ and the correlation length $\xi$ obtained in previous work, we determine the universal amplitude ratios $R_{2\mathrm{nd},+}={\sigma }_0{f}_{2\mathrm{nd},+}^2=0.3863\left(6\right), R_{2\mathrm{nd},-}={\sigma }_0{f}_{2\mathrm{nd},-}^2=0.1028\left(1\right)$, and $R_{\exp ,-}={\sigma }_0{f}_{\exp ,-}^2=0.1077\left(3\right)$. Our results are consistent with those obtained previously for the three-dimensional Ising model, confirming the universality hypothesis.

Figure 1

We plot the size of the exchange cluster per area for $L=64$ and the two choices $L_0=64$ and 128 as a function of $\beta$.

Figure 2

We plot $r_{\mathrm{gain}}^{1/2}$, Eq. (46), for $L=64$ and $L_0=64$ and 128 as a function of $\beta$.

Figure 3

We plot minus the difference of the interface energy for $L=64$ between $L_0=64$ and 128. The solid red line simply indicates zero. For a discussion see the text.

Figure 4

We plot the difference of the interface free energy for $L=64$ between $L_0=64$ and 128. The solid red line simply indicates zero. For a discussion see the text.

Figure 5

We plot $\sigma {\xi }_{2\mathrm{nd}}^2$ as a function of ${\xi }_{2\mathrm{nd}}^{-\omega }$. The data for the Ising model are taken from Ref. [25]. The filled circle on the $y$ axis gives our estimate of $R_{2\mathrm{nd},-}$.