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arXiv:2005.01601 · Journal of the Physical Society of Japan

Higher-Order Tensor Renormalization Group Approach to Lattice Glass Model

Kota Yoshiyama, Koji Hukushima

Published Sep 15, 2020• 6 citations
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Abstract

In this study, the higher-order tensor renormalization group (HOTRG) method is applied to a lattice glass model that has local constraints on the occupation number of neighboring particles represented by many-body interactions. This model exhibits first- and second-order transitions depending on a certain model parameter. The results obtained by using the HOTRG method for the model were confirmed to be consistent with those obtained by a Markov-chain Monte Carlo (MCMC) method for systems of relatively small sizes. The transition points are accurately estimated by the HOTRG calculation for the systems of large sizes, which is challenging to perform using the MCMC method. These results demonstrate that the HOTRG method can be an efficient method for studying systems with many-body interactions.

DOI
10.7566/jpsj.89.104003

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In this study, the higher-order tensor renormalization group (HOTRG) method is applied to a lattice glass model that has local constraints on the occupation number of neighboring particles represented by many-body interactions. This model exhibits first- and second-order transitions depending on a certain model parameter. The results obtained by using the HOTRG method for the model were confirmed to be consistent with those obtained by a Markov-chain Monte Carlo (MCMC) method for systems of relatively small sizes. The transition points are accurately estimated by the HOTRG calculation for the systems of large sizes, which is challenging to perform using the MCMC method. These results demonstrate that the HOTRG method can be an efficient method for studying systems with many-body interactions.
Phase transitions of the J_{1}-J_{2} Ising model on a square lattice are studied using the higher-order tensor renormalization group (HOTRG) method. This system involves a competition between the ferromagnetic interaction J_{1} and antiferromagnetic interaction J_{2}, and in previous studies, weak first-order and second-order transitions were observed near the ratio g=J_{2}/|J_{1}|=1/2. It has also been suggested that the universality class of the second-order phase transition connected to the first-order transition line for g>1/2 belongs to the Ashkin-Teller class, which is characterized by a continuously varying critical exponent with g, as predicted by field-theoretical and other studies. Our results, based on the HOTRG calculations for significantly larger sizes, indicate that the region of the first-order transition is marginally narrower than that in previous studies. Furthermore, it is suggested that the region where the critical exponent changes does not necessarily coincide with the Ashkin-Teller region.
The higher-order tensor renormalization group (HOTRG) is a fundamental method to calculate the physical quantities by using a tensor network representation. This method is based on the singular value decomposition (SVD) to take the contraction of all indices in the network with an approximation. For the SVD, randomized singular value decomposition (R-SVD) is a powerful method to reduce computational costs of SVD. However, HOTRG with the randomized method is not established. We propose a randomized HOTRG method in a dimension ddd with the computational cost O(D3d)O(D^{3d})O(D3d) depending on the truncated bond dimension DDD. We also introduce the minimally-decomposed TRG (MDTRG) as the R-HOTRG on the tensor of order d+1d+1d+1 with O(D2d+1)O(D^{2d + 1})O(D2d+1) and a triad representation of the MDTRG (Triad-MDTRG) with O(Dd+3)O(D^{d+3})O(Dd+3). The results from these formulations are consistent with the HOTRG result with the same truncated bond dimension DDD.
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 reduce the computational cost of the bottleneck contraction in the higher-order tensor renormalization group (HOTRG) by using the singular value decomposition. To confirm the effectiveness of the proposed technique, we examine the HOTRG with or without the proposed technique by using the square lattice Ising model at criticality. Then, we find that the proposed method provides the consistent results with the original HOTRG while the computational cost and the elapsed time are reduced. Therefore, we conclude that the performance of the proposed method is better than that of the original one.
We investigate the 2- and 3-state ferromagnetic Potts models on the simple cubic lattice using the tensor renormalization group method with higher-order singular value decomposition (HOTRG). HOTRG works in the thermodynamic limit, where we use the Z q symmetry of the model, combined with a new measure for detecting the transition, to improve the accuracy of the critical point for the 2-state model by two orders of magnitude, obtaining T c = 4.51152469(1). The 3-state model is far more complex, and we improve the overall understanding of this case by calculating its thermodynamic quantities with high accuracy. Our results verify that the first-order nature of the phase transition and the HOTRG transition temperature benchmarks the most recent Monte Carlo result.
We investigate the phase transition of the four-dimensional Ising model with two types of tensor network scheme, one is the higher-order tensor renormalization group and the other is the anisotropic tensor renormalization group. The results for the internal energy and magnetization obtained by the former algorithm with the impure tensor method, enlarging the lattice volume up to 102441024^410244, are consistent with the weak first-order phase transition. For the later algorithm, our implementation successfully reduces the execution time thanks to the parallel computation and the results provided by ATRG seems comparable to those with HOTRG.