JSIAM Letters

Cost-efficient cutoff method for tensor renormalization group with randomized singular value decomposition

Haruka Yamada, Akira Imakura, Tetsuya Sakurai

Published Jan 1, 2018• 0 citations

Abstract

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.

DOI: 10.14495/jsiaml.10.61

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JSIAM Letters

Cost-efficient cutoff method for tensor renormalization group with randomized singular value decomposition

Haruka Yamada, Akira Imakura, Tetsuya Sakurai

Published Jan 1, 2018• 0 citations

Publisher

Abstract

DOI: 10.14495/jsiaml.10.61

Related papers

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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

\mathcal {O}(d^2\log (L)\chi ^6)

, with the largest intermediate tensor size of

\mathcal {O}(\chi ^4)

, where d is the lattice dimension, L is the linear size of each lattice axis, and

\chi

is the cutoff of the number of lattice edge states. To validate our algorithm, we conducted numerical experiments on a Gaussian field model, for which the exact free energy is computable in any lattice dimensions. The results demonstrate that the tt-lattice method can calculate the free energy with reasonable accuracy even for a 10D lattice model with modern computational architectures.