Renormalização de redes de tensores : uma abordagem analítica

Abstract

The renormalization group in the network representation of tensors has proved to be a powerful theoretical tool for the analysis of strongly interacting physical systems. The technique is based on the representation of the partition function of the system under study, by a network of tensors, that is, at each site of the network we associate a translationally invariant tensor. The tensor codes, which we call the "legs", correspond to the links between the tensor network sites. Thus, the calculation of the partition function reduces to the contraction of a net of tensors, that is, a sum of the indices common to any two tensors of the lattice, Here we apply the technique to the Ising model de fi ned in a square lattice, in which case each tensor has four indices and each index or leg can we have to assume two values: we divide the grid into blocks of four tensors, so that each tensor is only part of one of these blocks. results in another network, with the same geometry as the original network, but the number of tensors is reduced to a quarter. In principle, the procedure can be repeated until only one The contraction of the legs of this last tensor would give the exact partition function. However, the number of states of the tensor legs resulting from the contraction increases exponentially, limiting the practical use of such procedure. Thus, we need to "renormalize" the tensors to limit the number of states, the tensors at each stage of the process. The renormalization used here is based on the decomposition into singular high order values ​​of the tensors (a generalization of the decomposition of a matrix into singular values). As a result, we present an analytical calculation of the partition function of the Ising model in the square network, maintaining only two states for the renormalized tensor. We were able to obtain the critical temperature with good precision, considering the radical approximation made. In addition, we numerically apply the procedure for cutting dimensions as high as maintaining up to thirty states in the renormalized tensor. The thermodynamics of the model was obtained with good agreement with the exact known results.