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

TL;DR

This study investigates the critical behavior of the Ising model on a 3D fractal lattice using HOTRG, revealing unique critical exponents and divergence in specific heat at the phase transition.

Contribution

It provides the first detailed analysis of the Ising model on a 3D fractal lattice, highlighting how fractal dimensionality influences critical phenomena.

Findings

Critical temperature T_c ≈ 2.65231

Critical exponents: β ≈ 0.059, δ ≈ 35

Specific heat diverges at T_c

Abstract

The critical behavior of the classical Ising model on a three-dimensional fractal lattice with Hausdorff dimension $d_H = \ln32 / \ln4 = 2.5$ 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.