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The tensor network (TN) is a powerful tool to calculate and understand strongly correlated many-body problems. The partition function in classical statistical systems can easily be represented as a tensor network. Combining the real-space renormalization group and TNs, the tensor renormalization group method (TRG) provides an efficient contraction scheme for TNs. Information compression based on the singular-value decomposition avoids exponential divergence of a computational cost. In this talk, we will present our recent new ideas in the tensor renormalization group. The first one is a calculation method for higher-order moments of physical quantities such as magnetization [1]. The systematic summation technique provides the recursion formula for coarse-graining tensors including multiple impurities. Our method easily approaches a huge system size which the Monte Carlo method cannot treat. The second one is an efficient algorithm for global optimization technique which improves the accuracy of TRG [2]. By replacing the environment tensor with the corner transfer matrices, we succeed to reduce the order of the computational cost.

[1]    Satoshi Morita and Naoki Kawashima, Comput. Phys. Comm. 236, 65–71 (2019).
[2]    Satoshi Morita and Naoki Kawashima, Phys. Rev. B 103, 045131 (2021)

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