Phase Transition of the Ising Model on a 3-Dimensional Fractal Lattice

The critical behavior of the classical Ising model on a three-dimensional fractal lattice with Hausdorff dimension $d_H=\ln 32\mathrm{/}\ln 4=2.5$ is investigated using the higher-order tensor renormalization group (HOTRG) method. We determine the critical temperature $T_c\approx 2.65231$ and the critical exponents for magnetization $\beta \approx 0.059$ and field response $\delta \approx 35$. Unlike a previously studied 2D fractal with $d_H\approx 1.792$, the specific heat for this 3D fractal exhibits a divergent singularity at $T_c$. The results are compared with those for regular lattices and other fractal structures to elucidate the role of dimensionality in critical phenomena.