Efficient calculation of three-dimensional tensor networks

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.