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arXiv:2401.06389 · International Symposium on Lattice Field Theory (LATTICE)

Application of the projective truncation and randomized singular value decomposition to a higher dimension

Katsumasa Nakayama

Published May 5, 2024• 2 citations
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Abstract

We study the tensor renormalization group (TRG) in the dimension larger than two as the Higher-order TRG (HOTRG) with the randomized SVD method. The randomized SVD and the detailed discussion on the low order tensor representation, we can calculate the HOTRG with the reduced computational cost. We also represent our method by using the cost function, and the details of the cost function for the isometry determine the precision, stability, and calculation time. In our study, we show calculation order improvement using randomized SVD. We also propose that the internal line respect for any TRG method improves the calculation without changing the order of the computational cost.

DOI
10.22323/1.453.0029

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We study the tensor renormalization group (TRG) in the dimension larger than two as the Higher-order TRG (HOTRG) with the randomized SVD method. The randomized SVD and the detailed discussion on the low order tensor representation, we can calculate the HOTRG with the reduced computational cost. We also represent our method by using the cost function, and the details of the cost function for the isometry determine the precision, stability, and calculation time. In our study, we show calculation order improvement using randomized SVD. We also propose that the internal line respect for any TRG method improves the calculation without changing the order of the computational cost.
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.
Tensor renormalization group (TRG) is a coarse-graining algorithm for approximating the partition function using a tensor network in the field of elementary particle physics.Although the computational cost of TRG can be reduced using a randomized singular value decomposition, its computation time is still large.In this paper, we propose a cost-efficient cutoff method for calculating TRG by truncating small tensor elements.Numerical experiments showed that the proposed method is faster than the conventional one without degrading accuracy.
We propose a hybrid stochastic method for the tensor renormalization group (TRG) approach. TRG is known as a powerful tool to study the many-body systems and quantum field theory on the lattice. It is based on a low-rank approximation of the tensor using the truncated singular value decomposition (SVD), whose computational cost increases as the bond dimension increases, so that efficient cost reduction techniques are highly demanded. We use noise vectors for the low-rank approximation with the truncated SVD, by which the truncation error is replaced with a statistical error due to noise, and an error estimation could be improved. We test this method in the classical Ising model and observe a better accuracy than TRG. We also discuss a cross contamination issue in a multiple use of the same noise vectors, and to remove this systematic error we consider position-dependent noise vectors.
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 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 projective truncation technique to the tensor renormalization group (TRG) algorithm in order to reduce the computational cost from $O({\ensuremath{\chi}}^{6})$ to $O({\ensuremath{\chi}}^{5})$, where $\ensuremath{\chi}$ is the bond dimension, and propose three kinds of algorithms for demonstration. On the other hand, the technique causes a systematic error due to the incompleteness of a projector composed of isometries and in addition requires iteration steps to determine the isometries. Nevertheless, we find that the accuracy of the free energy for the Ising model on a square lattice is recovered to the level of TRG with a few iteration steps even at the critical temperature for $\ensuremath{\chi}=32$, 48, and 64.