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High-accuracy tensor renormalization group algorithms and their applications

足立, 大樹 東京大学 DOI:10.15083/0002004672

2022.06.22

概要

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.

In this thesis, we propose two novel algorithms, the anisotropic tensor renormalization group (ATRG) and the bond-weighted tensor renormalization group (BTRG). ATRG is aimed to simulate lattice models in high dimensions efficiently, and BTRG is aimed to renormalize a two-dimensional square-lattice tensor network more accurately. ATRG can renormalize a ddimensional hypercubic-lattice tensor network with the computational cost of O(χ 2d+1) and the memory footprint of O(χ d+1), which are much smaller than O(χ 4d−1 ) computational time and O(χ 2d ) memory footprint, respectively, of the best existing algorithm, HOTRG. Although ATRG is less accurate than HOTRG when comparing with the same bond dimension, it is more efficient by orders of magnitude than HOTRG with fixed computation time, especially in higher dimensions. BTRG, on the other hand, is aimed to simulate the square-lattice tensor network more accurately. In BTRG, we consider the weights of the bonds in the tensor network. Despite the computational cost of BTRG [O(χ 5 )] being much lower than that of HOTRG [O(χ 7 )], we show that BTRG can yield more accurate results than HOTRG even in the vicinity of the critical point. In addition to these new algorithms, we introduce several techniques to improve the accuracy of ATRG, propose the combination of ATRG with BTRG, and discuss the performance of the algorithms. We also present the application of our new algorithms to classical and quantum lattice models to demonstrate the high accuracy of ATRG when applied to various simple and complex models.

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