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Progress of Theoretical and Experimental Physics

High-Dimensional Tensor Renormalization Group with Tensor Train Lattice

Hisanori Kiryu

Published Dec 26, 2025• 0 citations

Abstract

Abstract The tensor renormalization group (TRG) is a highly effective method for calculating the free energy of lattice models and has demonstrated exceptional accuracy for several low-dimensional systems. However, its application to higher-dimensional models is often hampered by a large memory footprint and high computational cost. In this paper, we propose a new algorithm, “tt-lattice”, which approximates the tensor at each lattice vertex using a tensor train (tt) decomposition. The algorithm is designed to maintain the tt-format throughout the TRG computation, which significantly reduces both memory usage and computational cost. For a uniform lattice model, the computational cost of tt-lattice scales as $\mathcal {O}(d^2\log (L)\chi ^6)$, 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.

DOI: 10.1093/ptep/ptaf193

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Abstract The tensor renormalization group (TRG) is a highly effective method for calculating the free energy of lattice models and has demonstrated exceptional accuracy for several low-dimensional systems. However, its application to higher-dimensional models is often hampered by a large memory footprint and high computational cost. In this paper, we propose a new algorithm, “tt-lattice”, which approximates the tensor at each lattice vertex using a tensor train (tt) decomposition. The algorithm is designed to maintain the tt-format throughout the TRG computation, which significantly reduces both memory usage and computational cost. For a uniform lattice model, the computational cost of tt-lattice scales as

\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.

The development of tensor renormalization group (TRG) algorithm in higher dimensions is an important and urgent task, as the TRG is expected to provide a way to overcome the sign problem in lattice quantum chromodynamics (QCD) calculations at finite density. One possible approach that enables faster computations in four-dimensional lattice theories is the anisotropic tensor renormalization group (ATRG). However, the computational cost remains substantial and requires significant computational resources. In this paper, we propose a novel algorithm, called the triad-ATRG, which is based on the ATRG and other improved TRG variants with triad network representation. This method achieves lower scaling with respect to the bond dimension, while minimizing the loss of accuracy in the free energy and other physical quantities. We also present parallel implementations of both the ATRG and triad-ATRG on multiple GPUs, which significantly improve performance compared to CPU-based calculations for the four-dimensional system.

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.

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We propose a modified form of tensor renormalization-group algorithm for evaluating partition functions of classical statistical mechanical models on two-dimensional lattices. This algorithm coarse-grains only the rows and columns of the lattice adjacent to a single core tensor at each step, such that the lattice size shrinks linearly with the number of coarse-graining steps as opposed to shrinking exponentially as in the usual tensor renormalization group (TRG). However, the cost of this approach only scales as $O({\ensuremath{\chi}}^{4})$ in terms of the bond dimension $\ensuremath{\chi}$, significantly cheaper than the $O({\ensuremath{\chi}}^{6})$ cost scaling of TRG, whereas numerical benchmarking indicates that both approaches have comparable accuracy for the same bond dimension $\ensuremath{\chi}$. It follows that the algorithm can allow for more accurate investigation of classical models on modestly sized lattices than is possible with standard TRG.

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.

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.