In this thesis, we study the topological phases of quantum spin systems. One project is to investigate a class of anti-ferromagnetic SU(N) Heisenberg models, describing Mott insulators of fermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubic lattice. Our large-N analysis maps a rich phase diagram. One particularly striking state we found spontaneously breaks lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state. In the other projects, we study the phenomenon of symmetry fractionalization on anyons as a tool to characterize two-dimensional symmetry enriched topological phases. In particular, we focus on how crystalline symmetries may fractionalize in gapped Z2 spin liquids. If the system has the symmetry of the square lattice, then there are 2080 symmetry fractionalization patterns possible. With exactly solvable models, we realize 487 of these in strictly two-dimensional systems. In addition, we succeed to understand why the remaining patterns cannot be found in the family of models we construct. Some can only appear on the surface of three-dimensional systems with non-trivial point group symmetry protected topological (pgSPT) order, whose boundary degrees of freedom transform non-locally under the symmetries. We construct a simple toy model to show this anomalous crystalline symmetry fractionalization phenomenon associated with a reflection. Moreover, our approach establishes the connection between the pgSPT phases and the topological phases with on-site symmetries in lower dimensions. This insight is very useful for classification of pgSPT orders in general.
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 are 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
H1+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 $\ensuremath{\theta}$ term in continuous nonlinear $\ensuremath{\sigma}$ models to lattice nonlinear $\ensuremath{\sigma}$ models. The boundary excitations of the nontrivial SPT phases are described by lattice nonlinear $\ensuremath{\sigma}$ models with a nonlocal Lagrangian term that generalizes the Wess-Zumino-Witten term for continuous nonlinear $\ensuremath{\sigma}$ 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 ${\mathcal{H}}^{1+d}[U(1)\ensuremath{\rtimes}{Z}_{2}^{T},{U}_{T}(1)]$ to obtain interacting bosonic topological insulators (protected by time reversal
Z2T and boson number conservation), which contain one nontrivial phase in one-dimensional (1D) or 2D and three in 3D. We also obtain interacting bosonic topological superconductors (protected by time-reversal symmetry only), in term of
H1+d[Z2T,UT(1)], which contain one nontrivial phase in odd spatial dimensions and none for even dimensions. Our result is much more general than the above two examples, since it is for any symmetry group. For example, we can use ${\mathcal{H}}^{1+d}[U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{T},{U}_{T}(1)]$ to construct the SPT phases of integer spin systems with time-reversal and
U(1) spin rotation symmetry, which contain three nontrivial 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: $({G}_{H},{G}_{\ensuremath{\Psi}},{\mathcal{H}}^{1+d}[{G}_{\ensuremath{\Psi}},{U}_{T}(1)])$, where
GH is the symmetry group of the Hamiltonian and ${G}_{\ensuremath{\Psi}}$ the symmetry group of the ground states.
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 two-dimensional (2D) interacting spin system with nontrivial onsite
Z2 symmetry-protected topological order. The order is nontrivial because we can prove that the one-dimensional (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 a matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and reveal the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with onsite
Z2 symmetry-protected topological order.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry
G, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have
T2=1 time-reversal
Z2T and fermion-number-parity
Z2f symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the
K-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge
c=1 gapless edge excitations, if the symmetry is not broken.
In Liu et al. [Phys. Rev. B 84, 075135 (2011)] we studied eight gapped symmetric quantum phases in
S=1 spin chains/ladders which respect a discrete spin rotation ${D}_{2}\ensuremath{\subset}SO(3)$ and time reversal
T symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of one-dimensional (1D) integer spin systems with
D2+T symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in
S=1 spin chain or spin ladder models. The gapped symmetric phases protected by subgroups of
D2+T are also studied. Again, all these phases can be distinguished by physically measuring their end ``spins.''
Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.