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

entanglement-filtering renormalization approach and symmetry-protected topological order. Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. 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 inv plus the symmetry group G sym of the tensors ͑i.e., the symmetry group of the Lagrangian͒ characterize various phases of the system. Such a characterization can describe both the symmetry breaking phases and topological phases, as illustrated by two-dimensional ͑2D͒ 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.