Tensor-network study of quantum phase transition on Sierpiński fractal

The transverse-field Ising model on the Sierpiński 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 \(h_{\rm c}^{~} = 1.865\). 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.