Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of the Physical Society of Japan

Residual Entropy of the Cubic Dimer Model: The Higher Order Tensor Renormalization Group Approach

Takahiro Otsuka, Yasuhiro Akutsu

Published Apr 25, 2022• 0 citations
Publisher

Abstract

The higher order tensor renormalization group (HOTRG) is a numerical method that can efficiently deal with high-dimensional statistical systems. We applied HOTRG to the cubic dimer model which has macroscopic ground-state degeneracy, to obtain the residual entropy with high accuracy.
DOI
10.7566/jpsj.91.055003

Related papers

Anisotropic tensor renormalization group

Daiki Adachi, Tsuyoshi Okubo, Synge Todo

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.

arXiv:1906.02007•Physical review. B./Physical review. B•Aug 24, 2020•83 citations

Higher-Order Tensor Renormalization Group Approach to Lattice Glass Model

Kota Yoshiyama, Koji Hukushima

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.

arXiv:2005.01601•Journal of the Physical Society of Japan•Sep 15, 2020•6 citations

Randomized higher-order tensor renormalization group

Katsumasa Nakayama

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 $d$ with the computational cost $O(D^{3d})$ depending on the truncated bond dimension $D$. We also introduce the minimally-decomposed TRG (MDTRG) as the R-HOTRG on the tensor of order $d+1$ with $O(D^{2d + 1})$ and a triad representation of the MDTRG (Triad-MDTRG) with $O(D^{d+3})$. The results from these formulations are consistent with the HOTRG result with the same truncated bond dimension $D$.

arXiv:2307.14191•Jul 26, 2023•2 citations

高精度テンソル繰り込み群法とその応用

Adachi, Daiki

High-accuracy tensor renormalization group algorithms and their applications (高精度テンソル繰り込み群法とその応用) 氏 名 足立 大樹We propose novel real-space renormalization group algorithms for classical and quantum lattice models.Conventionally, the tensor renormalization group (TRG) algorithm proposed by Levin and Nave in 2007 and the higher-order tensor renormalization group (HOTRG) algorithm proposed by Xie et al. in 2012 have been widely used as efficient numerical renormalization methods to study large-scale classical and quantum spin systems.The accuracy of the tensor network algorithms is generally controlled by the bond dimension χ, which is the cutoff of tensor indices.With a fixed bond dimension χ, TRG can renormalize the two-dimensional squarelattice tensor network within O ( χ 5 ) computational time.However, it is difficult to apply the TRG algorithm to models in three or higher dimensions.On the other hand, HOTRG can be applied, at least in principle, to higher-dimensional models, and it has a higher accuracy than TRG when comparing with the same bond dimension, while the computational cost of O ( χ 4d-1 ) for d-dimensional models is higher than TRG even in two dimensions and too high to execute simulations for higher-dimensional models in practice.New algorithms that can perform the real-space renormalization in general dimensions with a reasonable cost are strongly demanded.

•Mar 23, 2020•0 citations

Coarse-Graining Tensor Renormalization

Tao Xiang

Coarse-graining renormalization aims to reformulate a tensor network model with a coarse-grained one at a larger scale. It has attracted particular attention in recent years because it opens a new avenue to unveil the entanglement structure of a tensor network model under the scaling transformation. This chapter reviews and compares the tensor renormalization group (TRG) and other coarse-graining methods developed in the past two decades. The methods can be divided into two groups according to whether or not the renormalization effect of the environment tensors is incorporated in the optimization of local tensors. The local optimization methods include TRG, HOTRG (a variant of TRG based on the higher-order singular value decomposition), tensor network renormalization (TNR), and loop-TNR. The global optimization methods include the second renormalized TRG and HOTRG, referred to as SRG and HOSRG, respectively. Among all these coarse-graining methods, HOTRG and HOSRG are the only two that can be readily extended and efficiently applied to three-dimensional classical or two-dimensional quantum lattice models.

Cambridge University Press eBooks•Aug 31, 2023•0 citations

Optimization of Reordering Procedures in HOTRG for Distributed Parallel Computing

Haruka Yamada, Akira Imakura, Toshiyuki Imamura +1 more

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.

IEEE International Symposium on Parallel & Distributed Processing, Workshops and Phd Forum•May 1, 2018•2 citations
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.
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.
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.
High-accuracy tensor renormalization group algorithms and their applications (高精度テンソル繰り込み群法とその応用) 氏 名 足立 大樹We propose novel real-space renormalization group algorithms for classical and quantum lattice models.Conventionally, the tensor renormalization group (TRG) algorithm proposed by Levin and Nave in 2007 and the higher-order tensor renormalization group (HOTRG) algorithm proposed by Xie et al. in 2012 have been widely used as efficient numerical renormalization methods to study large-scale classical and quantum spin systems.The accuracy of the tensor network algorithms is generally controlled by the bond dimension χ, which is the cutoff of tensor indices.With a fixed bond dimension χ, TRG can renormalize the two-dimensional squarelattice tensor network within O ( χ 5 ) computational time.However, it is difficult to apply the TRG algorithm to models in three or higher dimensions.On the other hand, HOTRG can be applied, at least in principle, to higher-dimensional models, and it has a higher accuracy than TRG when comparing with the same bond dimension, while the computational cost of O ( χ 4d-1 ) for d-dimensional models is higher than TRG even in two dimensions and too high to execute simulations for higher-dimensional models in practice.New algorithms that can perform the real-space renormalization in general dimensions with a reasonable cost are strongly demanded.
Coarse-graining renormalization aims to reformulate a tensor network model with a coarse-grained one at a larger scale. It has attracted particular attention in recent years because it opens a new avenue to unveil the entanglement structure of a tensor network model under the scaling transformation. This chapter reviews and compares the tensor renormalization group (TRG) and other coarse-graining methods developed in the past two decades. The methods can be divided into two groups according to whether or not the renormalization effect of the environment tensors is incorporated in the optimization of local tensors. The local optimization methods include TRG, HOTRG (a variant of TRG based on the higher-order singular value decomposition), tensor network renormalization (TNR), and loop-TNR. The global optimization methods include the second renormalized TRG and HOTRG, referred to as SRG and HOSRG, respectively. Among all these coarse-graining methods, HOTRG and HOSRG are the only two that can be readily extended and efficiently applied to three-dimensional classical or two-dimensional quantum lattice models.
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.