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.
We report recent progress on the application of the tensor renormalization group (TRG) to quantum field theories pursued by the Tsukuba group. We explain how to treat the scalar, fermion, and gauge theories with the TRG method presenting the results for the phase transitions in the (3+1)-dimensional ((3+1)
d) complex
ϕ4 theory at finite density, (1+1)
d pure U(1) lattice gauge theory with a
θ term, (3+1)
d Nambu--Jona-Lasinio model at finite density and (1+1)
d and (2+1)
d Hubbard models at an arbitrary chemical potential. It is demonstrated that the TRG method is free from the sign problem in practical calculations and applicable to the four-dimensional models.
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.