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

TL;DR

This paper investigates the quantum phase transition of the transverse-field Ising model on a Sierpiński fractal using tensor-network methods, revealing critical points, exponents, and comparing with other approaches.

Contribution

It applies tensor-network techniques to a fractal lattice, providing new insights into quantum phase transitions on non-integer dimensional structures.

Findings

Critical transverse field at h_c = 1.865

Critical exponents: β ≈ 0.198, δ ≈ 8.7

Comparison with real-space RG and mean-field results

Abstract

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 $h_{\rm c}^{~} = 1.865$. 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.