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Modern Physics Letters A

TENSOR RENORMALIZATION GROUP APPROACH TO A LATTICE BOSON MODEL

YUYA SHIMIZU

Published Feb 10, 2012• 39 citations
Publisher

Abstract

A method to apply the tensor renormalization group to a lattice boson model is proposed. It is based on the truncated singular value decomposition of a compact operator. We demonstrate it using the (1+1)-dimensional ϕ4 model. The evaluated critical points on the lattice are consistent with the Monte Carlo result.
DOI
10.1142/s0217732312500356

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An algorithm of the tensor renormalization group is proposed based on a randomized algorithm for singular value decomposition. Our algorithm is applicable to a broad range of two-dimensional classical models. In the case of a square lattice, its computational complexity and memory usage are proportional to the fifth and the third power of the bond dimension, respectively, whereas those of the conventional implementation are of the sixth and the fourth power. The oversampling parameter larger than the bond dimension is sufficient to reproduce the same result as full singular value decomposition even at the critical point of the two-dimensional Ising model.
We propose a novel coarse-graining tensor renormalization group method based on the higher-order singular value decomposition. This method provides an accurate but low computational cost technique for studying both classical and quantum lattice models in two or three dimensions. We have demonstrated this method using the Ising model on the square and cubic lattices. By keeping up to 16 bond basis states, we obtain by far the most accurate numerical renormalization group results for the three-dimensional Ising model. We have also applied the method to study the ground state as well as finite temperature properties for the two-dimensional quantum transverse Ising model and obtain the results which are consistent with published data.
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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.