Measurements of magnetization on the Sierpi{n}ski carpet

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 $T_{rm c}^{~} = 1.4783(1)$. 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.