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  • Open Access

Triad representation for the anisotropic tensor renormalization group in four dimensions

Yuto Sugimoto* and Shoichi Sasaki†

  • *Contact author: sugimoto@nucl.phys.tohoku.ac.jp
  • †Contact author: ssasaki@nucl.phys.tohoku.ac.jp
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Phys. Rev. D 112, 094514 – Published 25 November, 2025

DOI: https://doi.org/10.1103/c3qc-tn48

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.

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Phys. Rev. D 112, 094514– Published 25 November, 2025
  • Received 4 August 2025
  • Accepted 22 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

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