Triad representation for the anisotropic tensor renormalization group in four dimensions

TL;DR

This paper introduces the triad-ATRG algorithm, an improved tensor renormalization group method for four-dimensional lattice theories, reducing computational cost and enabling efficient GPU parallelization.

Contribution

It proposes the triad-ATRG algorithm with lower scaling and minimal accuracy loss, along with GPU parallel implementations for 4D tensor calculations.

Findings

Lower computational scaling with bond dimension

Maintains accuracy in physical quantities

GPU implementations significantly improve performance

Abstract

The development of tensor renormalization group (TRG) algorithm in higher dimensions is an important and urgent task, as the TRG is expected to provide a way to overcome the sign problem in lattice quantum chromodynamics (QCD) calculations at finite density. One possible approach that enables faster computations in four-dimensional lattice theories is the anisotropic tensor renormalization group (ATRG). However, the computational cost remains substantial and requires significant computational resources. In this paper, we propose a novel algorithm, called the triad-ATRG, which is based on the ATRG and other improved TRG variants with triad network representation. This method achieves lower scaling with respect to the bond dimension, while minimizing the loss of accuracy in the free energy and other physical quantities. We also present parallel implementations of both the ATRG and triad-ATRG on multiple GPUs, which significantly improve performance compared to CPU-based calculations for the four-dimensional system.