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arXiv:2005.04645 · Journal of High Energy Physics

Tensor renormalization group approach to four-dimensional complex ϕ4 theory at finite density

Shinichiro Akiyama, Daisuke Kadoh, Yoshinobu Kuramashi, Takumi Yamashita, Yusuke Yoshimura

Published Sep 1, 2020• 36 citations
arXivHTMLPDF

Abstract

Abstract Tensor network is an attractive approach to the field theory with negative sign problem. The complex ϕ4 theory at finite density is a test bed for numerical algorithms to verify their effectiveness. The model shows a characteristic feature called the Silver Blaze phenomenon associated with the sign problem in the large volume limit at low temperature. We analyze the four-dimensional model employing the anisotropic tensor renormalization group algorithm with a parallel computation. We find a clear signal of the Silver Blaze phenomenon on a large volume of V = 10244, which implies that the tensor network approach is effective even for four-dimensional field theory beyond two dimensions.

DOI
10.1007/jhep09(2020)177

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The tensor renormalization group attracts great attention as a new numerical method that is free of the sign problem. In addition to this striking feature, it also has an attractive aspect as a coarse-graining of space-time; the computational cost scales logarithmically with the space-time volume. This fact allows us to aggressively approach the thermodynamic limit. While taking this advantage, we study the critical coupling of the two dimensional ϕ4\phi^{4}ϕ4 theory on large and fine lattices. We present the numerical results along with the extrapolation procedure to the continuum limit and compare them with the previous ones by Monte Carlo simulations.
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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.