Measurements of magnetization on the Sierpi\'{n}ski carpet

TL;DR

This study investigates the phase transition of the Ising model on a fractal Sierpiński carpet using tensor renormalization, revealing position-dependent critical behavior and accurately determining the global critical exponent.

Contribution

It introduces an adapted tensor renormalization approach to analyze phase transitions on fractal lattices and explores local versus global critical exponents.

Findings

Second-order phase transition at T_c ≈ 1.478

Local critical exponent β varies significantly by position

Global critical exponent β ≈ 0.135

Abstract

Phase transition of the classical Ising model on the Sierpi\'{n}ski carpet, which has the fractal dimension $\log_3^{~} 8 \approx 1.8927$, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature $T_{\rm c}^{~} \approx 1.478$. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent $\beta$ associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas $T_{\rm c}^{~}$ is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of $\beta \approx 0.135$.