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Condensed Matter > Statistical Mechanics
Title: J1-J2 fractal studied by multi-recursion tensor-network method
(Submitted on 23 Jul 2021 (v1), last revised 14 Jan 2022 (this version, v2))
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.
Submission history
From: Andrej Gendiar [view email][v1] Fri, 23 Jul 2021 18:17:16 GMT (218kb,D)
[v2] Fri, 14 Jan 2022 17:26:05 GMT (239kb,D)
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