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.

Symmetry protected topological (SPT) states are bulk gapped states with gapless edge excitations protected by certain symmetries. The SPT phases in free fermion systems, like topological insulators, can be classified by the K-theory. However, it is not known what SPT phases exist in general interacting systems. In this paper, we present a systematic way to construct SPT phases in interacting bosonic systems, which allows us to identify many new SPT phases, including three bosonic versions of topological insulators in three dimension and one in two dimension protected by particle number conservation and time reversal symmetry. Just as group theory allows us to construct 230 crystal structures in 3D, we find that group cohomology theory allows us to construct different interacting bosonic SPT phases in any dimensions and for any symmetry groups. In particular, we are going to show how topological terms in the path integral description of the system can be constructed from nontrivial group cohomology classes, giving rise to exactly soluble Hamiltonians, explicit ground state wave functions and symmetry protected gapless edge excitations. We used to believe that different phases of matter are different because they have different symmetries.1-3 Recently, we see a deep connection between quantum phases and quantum entanglement 4-6 which allows us to go beyond this framework. First it was realized that even in systems without any symmetry there can be distinct quantum phases -topological phases 7,8 due to different patterns of long-range entanglement in the states.6 For systems with symmetries, difference in long-range entanglement and in symmetry still lead to distinct phases. Moreover, even short-range entangled states with the same symmetry can belong to different phases. These symmetric short-range entangled states are said to contain a new kind of order -symmetry protected topological (SPT) order.9 The SPT phases have symmetry protected gapless edge modes despite the bulk gap, which clearly indicates the topological nature of this order. On the other hand, the gapless edge modes disappear when the symmetry of the system is broken, indicating that this is a different type of topological order than that found in fractional quantum Hall systems 10,11 whose edge modes cannot be removed with any local perturbation. 12Also, SPT orders have no factional statistics or fractional charges, while intrinsic topological orders from long range entanglement can have them. The discovery of SPT order hence greatly expands our original understanding of possible phases in many-body systems.One central issue is to understand what SPT phases exist and much progress has been made in this regard. The first system with SPT order was discovered decades ago in spin-1 Haldane chains. The Haldane chain with antiferromagnetic interactions was shown to have a gapped bulk 13,14 and degenerate modes at the ends of the chain 15-17 which are protected by spin rotation or time reversal symmetry of the system.9,18 This model has been generalized, l...

Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry protected topological orders exist. In this paper, we present a model in a 2D interacting spin system with nontrivial on-site Z2 symmetry protected topological order. The order is nontrivial because we can prove that the 1D system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the Z2 group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if Z2 symmetry is not broken. We prove the latter by developing the tool of matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and revealing the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with on-site Z2 symmetry protected topological order.