Tensor-network study of quantum phase transition on Sierpi\'nski fractal

The transverse-field Ising model on the Sierpi'nski fractal, which is characterized by the fractal dimension ${\log }_2^{}3\approx 1.585$, is studied by a tensor-network method, the Higher-Order Tensor Renormalization Group. We analyze the ground-state energy and the spontaneous magnetization in the thermodynamic limit. The system exhibits the second-order phase transition at the critical transverse field $h_{\mathrm{c}}^{}=1.865$. The critical exponents $\beta \approx 0.198$ and $\delta \approx 8.7$ are obtained. Complementary to the tensor-network method, we make use of the real-space renormalization group and improved mean-field approximations for comparison.