Cost reduction of the bond-swapping part in an anisotropic tensor renormalization group

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.