We apply the projective truncation technique to the tensor renormalization group (TRG) algorithm in order to reduce the computational cost from O(χ 6 ) to O(χ 5 ), where χ 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 χ = 32, 48, and 64.

We propose a second renormalization group (SRG) in the triad representation of tensor networks. The SRG method improves two parts of the triad tensor renormalization group, which are the decomposition of intermediate tensors and the preparation of isometries, taking the influence of environment tensors into account. Every fundamental tensor including environment tensor is given as a rank-3 tensor, and the computational cost of the proposed algorithm scales with $$ \mathcal{O} $$ O (χ5) employing the randomized SVD where χ is the bond dimension of tensors. We test this method in the classical Ising model on the two dimensional square lattice, and find that numerical results are obtained in good accuracy for a fixed computational time.

The bottleneck part of an anisotropic tensor renormalization group (ATRG) is a bond-swapping part that consists of a contraction of two tensors and a partial singular value decomposition of a matrix, and their computational costs are $O(\chi^{2d+1})$, where $\chi$ is the maximum bond dimension and $d$ is the dimensionality of the system. We propose an alternative method for the bond-swapping part and it scales with $O(\chi^{\max(d+3,7)})$, though the total cost of ATRG with the method remains $O(\chi^{2d+1})$. Moreover, the memory cost of the whole algorithm can be reduced from $O(\chi^{2d})$ to $O(\chi^{\max(d+1,6)})$. We examine ATRG with or without the proposed method in the 4D Ising model and find that the free energy density of the proposed algorithm is consistent with that of the original ATRG while the elapsed time is significantly reduced. We also compare the proposed algorithm with a higher-order tensor renormalization group (HOTRG) and find that the value of the free energy density of the proposed algorithm is lower than that of HOTRG in the fixed elapsed time.

The bottleneck part of anisotropic tensor renormalization group (ATRG) is a swapping bonds part which consists of a contraction of two tensors and a partial singular value decomposition of a matrix, and their computational costs are O(χ 2d+1 ), where χ is the maximum bond dimension and d is the dimensionality of a system. We propose an alternative method for the swapping bonds part and it scales with O(χ max(d+3,7) ), though the total cost of ATRG with the method remains O(χ 2d+1 ). Moreover, the memory cost of the whole algorithm can be reduced from O(χ 2d ) to O(χ max(d+1,6) ). We examine ATRG with or without the proposed method in the four-dimensional Ising model and find that the free energy density of the proposed algorithm is consistent with that of the original ATRG while the elapsed time is significantly reduced. We also compare the proposed algorithm with higher-order tensor renromalization group (HOTRG) and find that the value of the free energy density of the proposed algorithm is lower than that of HOTRG in the fixed elapsed time.