Symmetry protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry G. They can all be smoothly connected to the same trivial product state if we break the symmetry. The Haldane phase of spin-1 chain is the first example of SPT phases which is protected by SO(3) spin rotation symmetry. The topological insulator is another example of SPT phases which is protected by U (1) and time reversal symmetries. In this paper, we show that interacting bosonic SPT phases can be systematically described by group cohomology theory: distinct d-dimensional bosonic SPT phases with on-site symmetry G (which may contain antiunitary time reversal symmetry) can be labeled by the elements in H 1+d [G, UT (1)] -the Borel (1 + d)-group-cohomology classes of G over the G-module UT (1). Our theory, which leads to explicit ground state wave functions and commuting projector Hamiltonians, is based on a new type of topological term that generalizes the topological θ-term in continuous non-linear σ-models to lattice non-linear σ-models. The boundary excitations of the non-trivial SPT phases are described by lattice non-linear σ-models with a non-local Lagrangian term that generalizes the Wess-ZuminoWitten term for continuous non-linear σ-models. As a result, the symmetry G must be realized as a non-on-site symmetry for the low energy boundary excitations, and those boundary states must be gapless or degenerate. As an application of our result, we can use] to obtain interacting bosonic topological insulators (protected by time reversal Z T 2 and boson number conservation), which contain one non-trivial phases in 1D or 2D, and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time reversal symmetry only), in term of, which contain one non-trivial phase in odd spatial dimensions and none for even. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use H 1+d [U (1) × Z T 2 , UT (1)] to construct the SPT phases of integer spin systems with time reversal and U (1) spin rotation symmetry, which contain three non-trivial SPT phases in 1D, none in 2D, and seven in 3D. Even more generally, we find that the different bosonic symmetry breaking short-range-entangled phases are labeled by the following three mathematical objects: GH , GΨ, H 1+d [GΨ, UT (1)] , where GH is the symmetry group of the Hamiltonian and GΨ the symmetry group of the ground states.

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 fixedpoint 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.

Quantum many-body systems divide into a variety of phases with very different physical properties. The question of what kind of phases exist and how to identify them seems hard especially for strongly interacting systems. Here we make an attempt to answer this question for gapped interacting quantum spin systems whose ground states are short-range correlated. Based on the local unitary equivalence relation between short-range correlated states in the same phase, we classify possible quantum phases for 1D matrix product states, which represent well the class of 1D gapped ground states. We find that in the absence of any symmetry all states are equivalent to trivial product states, which means that there is no topological order in 1D. However, if certain symmetry is required, many phases exist with different symmetry protected topological orders. The symmetric local unitary equivalence relation also allows us to obtain some simple results for quantum phases in higher dimensions when some symmetries are present.

We construct a 2D quantum spin model that realizes an Ising paramagnet with gapless edge modes protected by Ising symmetry. This model provides an example of a "symmetry-protected topological phase." We describe a simple physical construction that distinguishes this system from a conventional paramagnet: we couple the system to a Z2 gauge field and then show that the π-flux excitations have different braiding statistics from that of a usual paramagnet. In addition, we show that these braiding statistics directly imply the existence of protected edge modes. Finally, we analyze a particular microscopic model for the edge and derive a field theoretic description of the low energy excitations. We believe that the braiding statistics approach outlined in this paper can be generalized to a large class of symmetry-protected topological phases.