Coarse graining tensor renormalization by the higher-order singular value decomposition

Z. Y. Xie, J. Chen, Mingpu Qin, Jian-Li Zhu, Li‐Ping Yang, Tao Xiang

Published Jan 5, 2012• 2 citations

Abstract

Beijing Computational Science Research Center, Beijing, 100084, China(Dated: January 10, 2012)We propose anovel coarse graining tensor renormalization group method based on the higher-ordersingular value decomposition (HOSVD). This method provides an accurate but low computationalcost technique for studying two- or three-dimensional (3D) lattice models. The method is demon-strated using the Ising model on the square and cubic lattices. By keeping up to 16 bond basisstates, we obtain by far the most accurate numerical renormalization group results for the 3D Isingmodel.

Related papers

Coarse-graining renormalization by higher-order singular value decomposition

Z. Y. Xie, Jing Chen, Mingpu Qin +3 more

We propose a novel coarse-graining tensor renormalization group method based on the higher-order singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two or three dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the three-dimensional Ising model. We have also applied the method to study the ground state as well as finite temperature properties for the two-dimensional quantum transverse Ising model and obtain the results which are consistent with published data.

arXiv:1201.1144Physical Review BJul 30, 2012392 citations

Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices

Hui‐Hai Zhao, Z. Y. Xie, Tao Xiang +1 more

We develop coarse-graining tensor renormalization group algorithms to compute physical properties of two-dimensional lattice models on finite periodic lattices. Two different coarse-graining strategies, one based on the tensor renormalization group and the other based on the higher-order tensor renormalization group, are introduced. In order to optimize the tensor network model globally, a sweeping scheme is proposed to account for the renormalization effect from the environment tensors under the framework of second renormalization group. We demonstrate the algorithms by the classical Ising model on the square lattice and the Kitaev model on the honeycomb lattice, and show that the finite-size algorithms achieve substantially more accurate results than the corresponding infinite-size ones.

arXiv:1510.03333Physical review. B./Physical review. BMar 9, 201640 citations

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:1906.02007Physical review. B./Physical review. BAug 24, 202083 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 eBooksAug 31, 20230 citations

Tensor renormalization group with randomized singular value decomposition

Satoshi Morita, Ryo Igarashi, Hui‐Hai Zhao +1 more

An algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition. Our algorithm is applicable to a broad range of two-dimensional classical models. In the case of a square lattice, its computational complexity and memory usage are proportional to the fifth and the third power of the bond dimension, respectively, whereas those of the conventional implementation are of the sixth and the fourth power. The oversampling parameter larger than the bond dimension is sufficient to reproduce the same result as full singular value decomposition even at the critical point of the two-dimensional Ising model.

arXiv:1712.01458Physical review. EMar 20, 201834 citations

Cost reduction of the bottleneck contraction in the higher-order tensor renormalization group

Hideaki Oba

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.

Jan 1, 20210 citations