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arXiv:1906.09283 · Physical review. B./Physical review. B

Tensor renormalization group centered about a core tensor

Wangwei Lan, Glen Evenbly

Published Dec 11, 2019• 8 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.100.235118

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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(\ensuremathχ4)O({\ensuremath{\chi}}^{4})O(\ensuremathχ4) in terms of the bond dimension \ensuremathχ\ensuremath{\chi}\ensuremathχ, significantly cheaper than the O(\ensuremathχ6)O({\ensuremath{\chi}}^{6})O(\ensuremathχ6) cost scaling of TRG, whereas numerical benchmarking indicates that both approaches have comparable accuracy for the same bond dimension \ensuremathχ\ensuremath{\chi}\ensuremathχ. It follows that the algorithm can allow for more accurate investigation of classical models on modestly sized lattices than is possible with standard TRG.
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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 O(d2log⁡(L)χ6)\mathcal {O}(d^2\log (L)\chi ^6)O(d2log(L)χ6), with the largest intermediate tensor size of O(χ4)\mathcal {O}(\chi ^4)O(χ4), where d is the lattice dimension, L is the linear size of each lattice axis, and χ\chiχ 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.
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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 method to construct the initial tensor representation of partition functions and observables for the tensor renormalization group (TRG). The TRG is a numerical calculation technique that utilizes a tensor network representations of physical quantities to investigate physical properties without encountering the sign problem. To apply the TRG, it is essential to construct a locally connected tensor network suitable for recursive coarse-graining. We present a systematic approach for translating a general tensor representation of the partition function to this form. Furthermore, we show the dependence of TRG algorithms on the choice of the initial tensor network representation and propose an improvement of TRG algorithms in this respect