Critical Temperature(s) of Sierpiński Carpet(s)

Abstract

We present a key algorithmic improvement to the generalized combinatorial Feynman--Vdovichenko method for calculating the critical temperature of the Ising model on Sierpiński carpets $SC_k(a,b)$, originally introduced in {\tt arxiv:1505.02699}. By reformulating the method in terms of purely real-valued transfer matrices, we substantially reduce their dimension. This optimization, together with modern computational resources, enables us to reach generation $k=10$ for the canonical $SC_k(3,1)$ carpet. Extrapolation from these data yields the most accurate estimate to date of the critical temperature $T_c^{(3,1)} = 1.4782927(26)$. We further extend the analysis to additional members of the $SC_k(a,b)$ family and report their corresponding critical temperatures.

Figures (4)

• Figure 1: Construction pipeline for Sierpiński carpets $SC_k(a,b)$. As an example, the canonical case $(3,1)$ is highlighted together with its first finite iterations (middle row). The bottom panel illustrates the tiling of the plane using the selected approximation $SC_3(3,1)$. Convergence to the full carpet $SC(3,1)$ is obtained in the limit $k\to \infty$.
• Figure 2: Critical temperatures of the Sierpiński carpets $SC_k(a,b)$. $b$ decreases with $T_c$ while $k$ starts from two and goes up to the maximum value computed. Finally the size of the points is proportional to logarithm of the number of elements, highlighting the rapid growth in computational complexity.
• Figure 3: Critical eigenvalue $\lambda_c$ as a function of the fractal dimension $d_f$ for the Sierpiński carpets studied in this work. Dashed lines are guides to the eye highlighting trends in the carpet data. Two distinct branches can be identified: the upper branch smoothly connects to the two-dimensional Ising limit $(d_f,\lambda_c) = (2,2.4142)$, while the lower branch appears to extend toward the one-dimensional limit $(1,1)$.
• Figure 4: Critical eigenvalue $\lambda_c$ dependence on the scaled tilting $t/L$ for the $SC_k(3,1)$ carpet with $k=2,3,4,5$. The deviations from the average over tilting values are actually shown for each generation.