Triad second renormalization group

TL;DR

This paper introduces a second renormalization group method for tensor networks that accounts for environment effects, improving accuracy in classical Ising model simulations with manageable computational costs.

Contribution

It presents a novel SRG approach in the triad tensor network representation that enhances tensor decomposition and isometry preparation by considering environment tensors.

Findings

Achieves good accuracy in 2D Ising model simulations

Computational cost scales as O(χ^5) with randomized SVD

Method improves tensor network renormalization accuracy

Abstract

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 ${\cal O}(\chi^5)$ employing the randomized SVD where $\chi$ 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.