Measurements of magnetization on the Sierpiński carpet

Abstract

Phase transition of the classical Ising model on the Sierpiński 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 $T_{\mathrm{c}}^{}\approx 1.478$. 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 $T_{\mathrm{c}}^{}$ 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 $\beta \approx 0.135$.

10 pages, 16 figures

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Cited by in corpus (2)

• Critical behavior of the Ising model on square-triangle tilings
• Critical Temperature(s) of Sierpinski Carpet(s)