Microscopic Realization of 2-Dimensional Bosonic Topological Insulators

It is well known that a Bosonic Mott insulator can be realized by condensing vortices of a bo- son condensate. Usually, a vortex becomes an anti-vortex (and vice-versa) under time reversal symmetry, and the condensation of vortices results in a trivial Mott insulator. However, if each vortex or anti-vortex interacts with a spin trapped at its core, the time reversal transformation of the composite vortex operator will contain an extra minus sign. It turns out that such a composite vortex condensed state is a bosonic topological insulator (BTI) with gapless boundary excitations protected by $U(1)\rtimes Z_2^T$ symmetry. We point out that in BTI, an external $\pi$ flux monodromy defect carries a Kramers doublet. We propose lattice model Hamiltonians to realize the BTI phase, which might be implemented in cold atom systems or spin-1 solid state systems.Comment: 5 pages + supplementary materia

Symmetry protected topological orders and the group cohomology of their symmetry group

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 phase which is protected by SO(3) spin rotation symmetry. The topological insulator is another exam- ple of SPT phase which is protected by U(1) and time reversal symmetries. It has been shown that free fermion SPT phases can be systematically described by the K-theory. 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 anti-unitary time reversal symmetry) can be labeled by the elements in H^{1+d}[G, U_T(1)] - the Borel (1 + d)-group-cohomology classes of G over the G-module U_T(1). The boundary excitations of the non-trivial SPT phases are gapless or degenerate. 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_{\Psi}, H^{1+d}[G_{\Psi}, U_T(1)], where G_H is the symmetry group of the Hamiltonian and G_{\Psi} the symmetry group of the ground states.Comment: 55 pages, 42 figures, RevTeX4-1, included some new reference

Study on E- Business Platform of Electric Enterprise Group Based on the Perspective of System Synergy

the paper showcase the synergistic effect framework and apply it to a study on electricity energy groups of internet bidding platform on B2B supply business. We enumerate electricity grid enterprise group’s purchase system. Since 2006, by setting up the multi-layer purchase system based on internet. Chinese energy enterprise group began to adopt internet supply platform and take full advantage of synergistic effect and scale effects, decrease the trade cost of whole supply chain. From the perspective of system synergy, the paper firstly analyzed the model building on e-procurement platform, and then offered a set of evaluation index on synergistic effect, lastly through empirical analysis, point out there is not only positive synergistic effect but also negative synergistic effect on the e-business platform of the purchase system

New gapped quantum phases for S=1 spin chain with D2h symmetry

We study different quantum phases in integer spin systems with on-site D2h=D2xZ2 and translation symmetry. We find four distinct non-trivial phases in S=1 spin chains despite they all have the same symmetry. All the four phases have gapped bulk excitations, doubly-degenerate end states and the doubly-degenerate entanglement spectrum. These non-trivial phases are examples of symmetry protected topological (SPT) phases introduced by Gu and Wen. One of the SPT phase correspond to the Haldane phase and the other three are new. These four SPT phases can be distinguished experimentally by their different response of the end states to weak external magnetic fields. According to Chen-Gu-Wen classification, the D2h symmetric spin chain can have totally 64 SPT phases that do not break any symmetry. Here we constructed seven nontrivial phases from the seven classes of nontrivial projective representations of D2h group. Four of these are found in S=1 spin chains and studied in this paper, and the other three may be realized in S=1 spin ladders or S=2 models.Comment: 15+ pages,5 figures, 9 table

Symmetry protected Spin Quantum Hall phases in 2-Dimensions

Symmetry protected topological (SPT) states are short-range entangled states with symmetry. Nontrivial SPT states have symmetry protected gapless edge excitations. In 2-dimension (2D), there are infinite number of nontrivial SPT phases with SU(2) or SO(3) symmetry. These phases can be described by SU(2)/SO(3) nonlinear-sigma models with a quantized topological \theta-term. At open boundary, the \theta-term becomes the Wess-Zumino-Witten term and consequently the boundary excitations are decoupled gapless left movers and right movers. Only the left movers (if \theta>0) carry the SU(2)/SO(3) quantum numbers. As a result, the SU(2) SPT phases have a half-integer quantized spin Hall conductance and the SO(3) SPT phases an even-integer quantized spin Hall conductance. Both the SU(2)/SO(3) SPT phases are symmetric under their U(1) subgroup and can be viewed as U(1) SPT phases with even-integer quantized Hall conductance.Comment: 5 pages + appendi