Renormalization group on a triad network

TL;DR

This paper introduces a new renormalization scheme for tensor networks composed of third order tensors, which efficiently coarse-grains the network with manageable computational costs, and demonstrates its effectiveness on the 3D Ising model.

Contribution

The authors develop a novel renormalization method for third order tensor networks with optimized computational cost, applicable in higher dimensions.

Findings

Successfully applied to 3D Ising model

Achieves large bond dimensions with reasonable errors

Reduces computational complexity for tensor coarse-graining

Abstract

We propose a new renormalization scheme of tensor networks made only of third order tensors. The isometry used for coarse-graining the network can be prepared at an $O(D^6)$ computational cost in any $d$ dimension ($d \ge 2$), where $D$ is the truncated bond dimension of tensors. Although it is reduced to $O(D^5)$ if a randomized singular value decomposition is employed, the total cost is $O(D^{d+3})$ because the contraction part for creating a renormalized tensor with isometries has $D^{d+3}$ multiplications. We test our method in three dimensional Ising model and find that the numerical results are obtained for large $D$s with reasonable errors.