Applying the Triad network representation to four-dimensional ATRG method

TL;DR

This paper introduces a triad tensor network representation for four-dimensional ATRG, significantly reducing computational costs while maintaining accuracy, especially when implemented on GPUs.

Contribution

The paper develops a novel triad representation for 4D ATRG, improving computational efficiency and enabling parallel GPU implementation without sacrificing accuracy.

Findings

Reduced computational cost in 4D ATRG calculations

Maintained convergence accuracy of free energy in 4D Ising model

Enhanced performance with GPU parallelization

Abstract

Anisotropic Tensor Renormalization Group (ATRG) is a powerful algorithm for four-dimensional tensor network calculations. However, the larger bond dimensions are known to be difficult to achieve in practice due to the higher computational cost. Adopting the methods of the minimally decomposed TRG and its triad prescriptions, we construct a triad representation of the four-dimensional ATRG by decomposing the unit-cell tensor. We observe that this combining approach can significantly improve the computational cost even with maintaining the convergence accuracy of the free energy in the four-dimensional Ising model. In addition, we also show that a further improvement can be achieved in terms of the computational cost when our proposed approach is implemented in parallel on GPUs.