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

Correlated metallic state in honeycomb lattice: Orthogonal Dirac semimetal

A novel gapped metallic state coined orthogonal Dirac semimetal is proposed in the honeycomb lattice in terms of $Z_{2}$ slave-spin representation of Hubbard model. This state corresponds to the disordered phase of slave-spin and has the same thermaldynamical and transport properties as usual Dirac semimetal but its singe-particle excitation is gapped and has nontrivial topological order due to the $Z_{2}$ gauge structure. The quantum phase transition from this orthogonal Dirac semimetal to usual Dirac semimetal is described by a mean-field decoupling with complementary fluctuation analysis and its criticality falls into the universality class of 2+1D Ising model while a large anomalous dimension for the physical electron is found at quantum critical point (QCP), which could be considered as a fingerprint of our fractionalized theory when compared to other non-fractionalized approaches. As byproducts, a path integral formalism for the $Z_{2}$ slave-spin representation of Hubbard model is constructed and possible relations to other approaches and the sublattice pairing states, which has been argued to be a promising candidate for gapped spin liquid state found in the numerical simulation, are briefly discussed. Additionally, when spin-orbit coupling is considered, the instability of orthogonal Dirac semimetal to the fractionalized quantum spin Hall insulator (fractionalized topological insulator) is also expected. We hope the present work may be helpful for future studies in $Z_{2}$ slave-spin theory and related non-Fermi liquid phases in honeycomb lattice.Comment: 12 pages,no figures, more discussions added. arXiv admin note: text overlap with arXiv:1203.063

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