Spin models in three dimensions: Adaptive lattice spacing

Aiming at the study of critical phenomena in the presence of boundaries with a nontrivial shape we discuss how lattices with an adaptive lattice spacing can be implemented. Since the parameters of the Hamiltonian transform nontrivially under changes of the length scale, adapting the lattice spacing is much more difficult than in the case of the numerical solution of partial differential equations, where this method is common practice.

Here we shall focus on the universality class of the three-dimensional Ising model. Our starting point is the improved Blume-Capel model on the simple cubic lattice. In our approach, the system is composed of sectors with lattice spacing $a,2a,4a,....$ We work out how parts of the lattice with lattice spacing $a$ and $2a$, respectively, can be coupled in a consistent way. Here, we restrict ourself to the case where the boundary between the sectors is perpendicular to one of the lattice axes. Based on the theory of defect planes one expects that it is sufficient to tune the coupling between these two regions. To this end we perform a finite-size scaling study. However, first numerical results show that slowly decaying corrections remain. It turns out that these corrections can be removed by adjusting the strength of the couplings within the boundary layers. As a benchmark, we simulate films with strongly symmetry breaking boundary conditions. We determine the magnetization profile and the thermodynamic Casimir force. For our largest thickness $L_0=64.5$, we find that results obtained for the homogeneous system are nicely reproduced.

Figure 1

We give a two-dimensional sketch of the lattice with two lattice spacings. The couplings in the bulk of the fine and the coarse lattice are denoted by ${\beta }_1$ and ${\beta }_2$, respectively. The couplings within the layers at the boundary between the two parts of the lattice are denoted by ${\beta }_{1,b}$ and ${\beta }_{2,b}$, respectively. The corresponding links are given by dash-dotted lines in the sketch. The coupling between the fine and the coarse lattice is denoted by ${\beta }_{1,2}$. The corresponding links are indicated by blue dashed lines.

Figure 2

Profiles, Eq. (7), at $c_{1,b}=c_{2,b}=1$ and the lattice sizes $L=32$, 64, and 128. We plot $\tilde{q}=q\left(x_0\right)/\overline{q}$ as a function of $\tilde{x}=x_0/L$ and $\tilde{x}=(x_0-L)/L$ for the fine and the coarse lattice, respectively. The curves for the different lattice sizes can be identified by the number of data points. Furthermore, $\tilde{q}<1$ for the fine lattice and $\tilde{q}>1$ for the coarse one. The dashed lines should only guide the eye.

Figure 3

Profiles, Eq. (7), at $c_{1,b}=1.068,c_{2,b}=0.9234,\beta =0.387721735,{\beta }_{1,2}=0.185807$, and the lattice sizes $L=64$ and 128. We plot $\tilde{q}=q\left(x_0\right)/\overline{q}$ as a function of $\tilde{x}=x_0/L$ and $\tilde{x}=(x_0-L)/L$ for the fine and the coarse lattice, respectively. The curves for the different lattice sizes can be identified by the number of data points. Furthermore, $\tilde{q}<1$ for the fine lattice and $\tilde{q}>1$ for the coarse one. The dashed lines should only guide the eye.

Figure 4

We plot $\delta =\Delta E-\Delta E_{\text{hom}}$ for $(+,+)$ boundary conditions as a function of $\beta$. In addition we show as a solid red line, $0.1\Delta E_{\text{ex},\text{hom}}$. Results for the lattice size $(L_0,L)=(16.5,64)$ and $(L_0,L)=(64.5,256)$ are given in the upper and lower parts of the figure, respectively.

Figure 5

Same as Fig. 4 but for $(+,-)$ instead of $(+,+)$ boundary conditions.

Figure 6

We plot the difference $\delta m\left(x_0\right)$, Eq. (40), between the magnetization profiles of the composite and the homogeneous system for both $(+,+)$ and $(+,-)$ boundary conditions and the thickness $L_0=64$ at the critical temperature. We averaged $\delta m\left(x_0\right)$ over all values $0.3852\le \beta \le 0.38792$ we simulated at.

Figure 7

We plot the magnetization profile for both $(+,+)$ and $(+,-)$ boundary conditions and the thickness $L_0=64$ at the critical temperature. The data points denoted by “fine” are taken from simulations of the homogeneous system. The data points taken from the coarse part of the composite lattice are rescaled by a factor of $2^{-(1+\eta )/2}$. In the case of the fine lattice, the solid lines should only guide the eye.