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

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.

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Cited by in corpus (3)

• Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method
• Measurements of magnetization on the Sierpiński carpet
• Tensor-Network study of Ising model on infinite hyperbolic dodecahedral lattice