Cost Reduction of Swapping Bonds Part in Anisotropic Tensor Renormalization Group

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 $O(\chi^{2d+1})$, where $\chi$ 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 $O(\chi^{\max(d+3,7)})$, 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 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.