J1-J2 fractal studied by multi-recursion tensor-network method

TL;DR

This paper extends tensor-network algorithms to analyze thermodynamic properties of self-similar fractal spin lattices, introducing multiple local tensors and recursion relations, and applies it to the Ising model on a J1-J2 fractal.

Contribution

The authors develop a generalized HOTRG algorithm with multiple local tensors and recursion relations for studying fractal lattices, expanding the applicability of tensor networks to non-uniform, self-similar structures.

Findings

Successfully applied to the J1-J2 fractal lattice with Hausdorff dimension ~1.792.

Introduced ten independent local tensors with fifteen recursion relations.

Demonstrated applicability to models lacking translational invariance.

Abstract

We generalize a tensor-network algorithm to study thermodynamic properties of self-similar spin lattices constructed on a square-lattice frame with two types of couplings, $J_{1}^{}$ and $J_{2}^{}$, chosen to transform a regular square lattice ($J_{1}^{} = J_{2}^{}$) onto a fractal lattice if decreasing $J_{2}^{}$ to zero (the fractal fully reconstructs when $J_{2}^{} = 0$). We modified the Higher-Order Tensor Renormalization Group (HOTRG) algorithm for this purpose. Single-site measurements are performed by means of so-called impurity tensors. So far, only a single local tensor and uniform extension-contraction relations have been considered in HOTRG. We introduce ten independent local tensors, each being extended and contracted by fifteen different recursion relations. We applied the Ising model to the $J_{1}^{}-J_{2}^{}$ planar fractal whose Hausdorff dimension at $J_{2}^{} = 0$ is $d^{(H)} = \ln 12 / \ln 4 \approx 1.792$. The generalized tensor-network algorithm is applicable to a wide range of fractal patterns and is suitable for models without translational invariance.