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.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, 20200 citations

Generalization on renormalization group

อารีฟ บุญเสือ

Mar 25, 20260 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.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

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 $D$ .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:2307.14191Jul 26, 20232 citations

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