Tensor renormalization group approach to four-dimensional complex \u03d54 theory at finite density

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

doi:10.1007/jhep09(2020)177

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.

Highlights

JHEP09(2020)[177] using the 4d Ising model, where a parallel computation with 2D processes is used to reduce the cost per process of tensor contractions from O(D9) to O(D8) in four dimensions [20]
Testing a few types of tensor network representation including it, we found that the current one described here provided the most convergent result of them all in 4d complex scalar field theory
We have slightly modified the original algorithm of ATRG [3]; we avoid taking the square root for the singular values obtained by the randomized SVD (RSVD)

Summary

Method

2.1 Tensor network representation The complex scalar field theory at finite chemical potential in 4d euclidean space is given by. We employ the polar coordinate φn = rneiπsn (rn ≥ 0 and sn ∈ [−1, 1)) to express the partition function Z as a tensor network. The partition function is represented by a tensor network as. Testing a few types of tensor network representation including it, we found that the current one described here provided the most convergent result of them all in 4d complex scalar field theory. With the choice of p ≥ 4D and q ≥ 2D, we have confirmed that the numerical results obtained by the ATRG do not depend on these parameters.[2] In addition, we have slightly modified the original algorithm of ATRG [3]; we avoid taking the square root for the singular values obtained by the RSVD. We have checked the improvement of the accuracy with this modification, benchmarking with the 2d Ising model

Numerical results

Summary