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arXiv:2110.03607 · Computer Physics Communications

A parallel computing method for the higher order tensor renormalization group

Takumi Yamashita, Tetsuya Sakurai

Published May 23, 2022• 5 citations
arXivHTMLPDF
DOI
10.1016/j.cpc.2022.108423

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Higher-order tensor renormalization group (HOTRG) is a coarse-graining algorithm for approximating the partition function in the field of elementary particle physics using a tensor network. Coarse-graining in HOTRG comprises an approximation step and a contraction step, and the contraction step is performed with tensor reorderings and matrix products. In this paper, we introduce a naive parallel implementation of HOTRG and propose optimal reordering procedures for a three-dimensional (3D) classical cubic lattice Ising model. Numerical experiments on the K computer show that the elapsed time of the proposed procedure is 6.88 times faster than the naive one for the reorderings.
We propose a different tensor renormalization group algorithm, anisotropic tensor renormalization group (ATRG), for lattice models in arbitrary dimensions. The proposed method shares the same versatility with the higher-order tensor renormalization group (HOTRG) algorithm, i.e., it preserves the lattice topology after the renormalization. In comparison with HOTRG, both the computation cost and the memory footprint of our method are drastically reduced, especially in higher dimensions, by renormalizing tensors in an anisotropic way after the singular value decomposition. We demonstrate the ability of ATRG for the square lattice and the simple cubic lattice Ising models. Although the accuracy of the present method degrades when compared with HOTRG of the same bond dimension, the accuracy with fixed computation time is improved greatly due to the drastic reduction of the computation cost.
We apply the higher-order tensor renormalization group to the four-dimensional ferromagnetic Ising model, which has been attracting interest in the context of the triviality of the scalar ${\ensuremath{\phi}}_{d=4}^{4}$ theory. We investigate the phase transition of this model with the higher-order tensor renormalization group enlarging the lattice size up to 10244102{4}^{4}10244 with parallel computation. The results for the internal energy and the magnetization are consistent with the weak first-order phase transition.