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.

We report the theoretical discovery of a class of 2D tight-binding models containing nearly flatbands with nonzero Chern numbers. In contrast with previous studies, where nonlocal hoppings are usually required, the Hamiltonians of our models only require short-range hopping and have the potential to be realized in cold atomic gases. Because of the similarity with 2D continuum Landau levels, these topologically nontrivial nearly flatbands may lead to the realization of fractional anomalous quantum Hall states and fractional topological insulators in real materials. Among the models we discover, the most interesting and practical one is a square-lattice three-band model which has only nearest-neighbor hopping. To understand better the physics underlying the topological flatband aspects, we also present the studies of a minimal two-band model on the checkerboard lattice.

Two gapped quantum ground states in the same phase are connected by an adiabatic evolution which gives rise to a local unitary transformation that maps between the states. On the other hand, gapped ground states remain within the same phase under local unitary transformations. Therefore, local unitary transformations define an equivalence relation and the equivalence classes are the universality classes that define the different phases for gapped quantum systems. Since local unitary transformations can remove local entanglement, the above equivalence/universality classes correspond to pattern of long range entanglement, which is the essence of topological order. The local unitary transformation also allows us to define a wave function renormalization scheme, under which a wave function can flow to a simpler one within the same equivalence/universality class. Using such a setup, we find conditions on the possible fixed-point wave functions where the local unitary transformations have finite dimensions. The solutions of the conditions allow us to classify this type of topological orders, which generalize the string-net classification of topological orders. We also describe an algorithm of wave function renormalization induced by local unitary transformations. The algorithm allows us to calculate the flow of tensor-product wave functions which are not at the fixed points. This will allow us to calculate topological orders as well as symmetry breaking orders in a generic tensor-product state.

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...

Quantum phases with different orders exist with or without breaking the symmetry of the system. Recently, a classification of gapped quantum phases which do not break time reversal, parity or on-site unitary symmetry has been given for 1D spin systems in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. It was found that, such symmetry protected topological (SPT) phases are labeled by the projective representations of the symmetry group which can be viewed as a symmetry fractionalization. In this paper, we extend the classification of 1D gapped phases by considering SPT phases with combined time reversal, parity, and/or on-site unitary symmetries and also the possibility of symmetry breaking. We clarify how symmetry fractionalizes with combined symmetries and also how symmetry fractionalization coexists with symmetry breaking. In this way, we obtain a complete classification of gapped quantum phases in 1D spin systems. We find that in general, symmetry fractionalization, symmetry breaking and long range entanglement(present in 2 or higher dimensions) represent three main mechanisms to generate a very rich set of gapped quantum phases. As an application of our classification, we study the possible SPT phases in 1D fermionic systems, which can be mapped to spin systems by Jordan-Wigner transformation.