Measurements of magnetization on the Sierpiński carpet

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 $T_{\rm c}^{~} = 1.4783(1)$. Position dependence of local functions is studied by means of impurity tensors, which are 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.