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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 (TEFR) approach that removes local entanglement and coarse grain the 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 $T_{inv}$ plus the symmetry group $G_{sym}$ 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 illustrated by 2D statistical Ising model, 2D statistical loop gas model, and 1+1D quantum spin-1/2 and spin-1 models. In particular, using such a $(G_{sym}, T_{inv}) $ characterization, we show that the Haldane phase for a spin-1 chain is a phase protected by the time-reversal, parity, and translation symmetries. Thus the Haldane phase is a symmetry protected topological phase. The $(G_{sym}, T_{inv})$ characterization is more general than the characterizations based on the boundary spins and string order parameters. The tensor renormalization approach also allows us to study continuous phase transitions between symmetry breaking phases and/or topological phases. The scaling dimensions and the central charges for the critical points that describe those continuous phase transitions can be calculated from the fixed-point tensors at those critical points.