Triad representation for the anisotropic tensor renormalization group in four dimensions

TL;DR

This work introduces triad-ATRG, a novel tensor renormalization group algorithm for four-dimensional systems that combines anisotropic TRG with a triad representation to lower the coarse-graining cost from $O(D^{9})$ (ATRG) toward $O(D^{7})$, while maintaining accuracy comparable to ATRG in calculating the free energy and other observables. An internal-line overspanning (ILO) variant further enhances accuracy and reduces memory demands, particularly in the bond-swapping step. The authors implement both ATRG and triad-ATRG on multi-GPU platforms, achieving significant performance gains and demonstrating favorable scaling with bond dimension $D$ on large 4D lattices, exemplified by the four-dimensional Ising model. The results indicate that the triad-ATRG framework, especially with ILO, offers a practical and scalable route to applying TRG methods to four-dimensional lattice theories, with potential relevance to finite-density QCD calculations. Overall, the paper establishes a promising pathway for high-dimensional TRG applications by reducing computational cost, preserving accuracy, and leveraging GPU parallelism.

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.

Figures (13)

• Figure 1: Illustration of the HOTRG method on a two-dimensional square lattice. The procedure is as follows: (a) Apply a low-rank projector constructed from the HOSVD of $\Gamma$. (b) Perform tensor contractions around the yellow patches. (c) Rotate the system and repeat the procedure.
• Figure 2: The decomposition of the tensor $\Gamma$ in the ATRG. The left figure shows $\Gamma$ in four dimension, and the center figure illustrates the Eq. \ref{['eq:ABCD']}, and right figure illustrates the Eq. \ref{['eq:GammaATRG']}
• Figure 3: Schematic illustration of the triad representation of $\Gamma$ in the three-dimensional case. The unit-cell tensor $\Gamma$ (left) is expressed in the triad representation (right). The bond dimension of the dotted lines is overspanned to $rD$, while that of the solid lines is set to $D$.
• Figure 4: The triad representation of $\Gamma$. The left figure shows $\Gamma$ in four dimension, and the right figure illustrates the triad representation. Note that the bond dimension of magenta dotted lines are overspanned from $D$ to $rD$.
• Figure 5: Schematic overview of the triad-ATRG algorithm: (a) Decomposition of the fundamental tensor. (b) Bond-swapping step. (c) Construction of the triad representation. (d) Derivation of the squeezer. (e) Contraction (coarse-graining). (f) Moving onto the next step.