Site-bond percolation in two-dimensional kagome lattices: Analytical approach and numerical simulations

The site-bond percolation problem in two-dimensional kagome lattices has been studied by means of theoretical modeling and numerical simulations. Motivated by considerations of cluster connectivity, two distinct schemes (denoted as $S\cap B$ and $S\cup B$) have been considered. In $S\cap B$ ($S\cup B$), two points are connected if a sequence of occupied sites and ($\mathit{or}$) bonds joins them. Analytical and simulation approaches, supplemented by analysis using finite-size scaling theory, were used to calculate the phase boundaries between the percolating and nonpercolating regions, thus determining the complete phase diagram of the system in the ($p_s,p_b$) space. In the case of the $S\cap B$ model, the obtained results are in excellent agreement with previous theoretical and numerical predictions. In the case of the $S\cup B$ model, the limiting curve separating percolating and nonpercolating regions is reported here.

Figure 1

Schematic diagram of a kagome lattice composed of four unit cells. Solid circles and thick segments represent sites and bonds in the lattice, respectively. Dashed open circles and dashed segments denote boundary sites and bonds, respectively.

Figure 2

Percolation functions in the ($p_s,p_b$) plane: (a) $h_1^{S\cap B}(p_s,p_b)$ [Eq. (10)], (b) $h_1^{S\cup B}(p_s,p_b)$ [Eq. (12)], (c) $h_2^{S\cap B}(p_s,p_b)$ [Eq. (14)], and (d) $h_2^{S\cup B}(p_s,p_b)$ [Eq. (16)].

Figure 3

Extrapolation of $p_{s,c}\left(L_c\right)$, towards the thermodynamic limit according to the theoretical prediction given by Eq. (17). The data correspond to the $S\cap B$ model with $p_b=0.60$ (squares), $p_b=0.70$ (circles), $p_b=0.80$ (triangles), and $p_b=0.90$ (diamonds). The plots were made using the exact critical exponent in two dimensions $\nu =4/3$.

Figure 4

Analytical and simulation $S\cap B$ phase diagrams for kagome lattices. The symbology is as follows: simulation results in Table 2 (solid circles); analytical data from scheme 1, first and second columns in Table 1 (open circles joined by lines); analytical data from scheme 2, fifth and sixth columns in Table 1 (open squares joined by lines); analytical data from Ref. [22], first and second columns in Table 4 (open triangles joined by lines); analytical data from Ref. [21], third and fourth columns in Table 4 (open diamonds joined by lines); and analytical data from Ref. [23], fifth and sixth columns in Table 4 (open stars joined by lines). In the inset, percolating and nonpercolating regions are indicated. The inset also includes numerical data obtained in the present paper (solid circles) and previous simulations from Ref. [23] (solid triangles).

Figure 5

Analytical and simulation $S\cup B$ phase diagrams for kagome lattices. The symbology is as follows: simulation results in Table 2 (solid squares); analytical data from scheme 1, third and fourth columns in Table 1 (open circles joined by lines); and analytical data from scheme 2, seventh and eighth columns in Table 1 (open squares joined by lines). As in Fig. 4, percolating and nonpercolating regions are indicated in the inset.

Figure 6

Gradient norm functions in the ($p_s,p_b$) plane: (a) $S_1^{S\cap B}$ [Eq. (A3)], (b) $S_1^{S\cup B}$ [Eq. (A4)], (c) $S_2^{S\cap B}$ [Eq. (A3)], and (d) $S_2^{S\cup B}$ [Eq. (A4)].