Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order

We study the renormalization group flow of the Lagrangian for statistical and quantum systems by representing their path integral in terms of a tensor network. Using a tensor-entanglement-filtering renormalization approach that removes local entanglement and produces a coarse-grained lattice, we show that the resulting renormalization flow of the tensors in the tensor network has a nice fixed-point structure. The isolated fixed-point tensors ${T}_{\text{inv}}$ plus the symmetry group ${G}_{\text{sym}}$ of the tensors (i.e., the symmetry group of the Lagrangian) characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as…