Efficient calculation of three-dimensional tensor networks

TL;DR

This paper introduces an efficient algorithm for calculating physical quantities in three-dimensional tensor networks, improving computational efficiency for classical and quantum lattice models.

Contribution

The paper presents a novel method using projected entangled simplex states to compute dominant eigenvalues in 3D tensor networks, enhancing efficiency over traditional approaches.

Findings

Accurately computed internal energy for 3D Ising model.

Calculated spontaneous magnetization consistent with literature.

Demonstrated improved efficiency in tensor network calculations.

Abstract

We have proposed an efficient algorithm to calculate physical quantities in the translational invariant three-dimensional tensor networks, which is particularly relevant to the study of the three-dimensional classical statistical models and the (2+1)-dimensional quantum lattice models. In the context of a classical model, we determine the partition function by solving the dominant eigenvalue problem of the transfer matrix, whose left and right dominant eigenvectors are represented by two projected entangled simplex states. These two projected entangled simplex states are not Hermitian conjugate to each other but are appropriately arranged so that their inner product can be computed much more efficiently than in the usual prescription. For the three-dimensional Ising model, the calculated internal energy and spontaneous magnetization agree with the published results in the literature. The possible improvement and extension to other models are also discussed.