Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect

We discuss physical properties of `integer' topological phases of bosons in D=3+1 dimensions, protected by internal symmetries like time reversal and/or charge conservation. These phases invoke interactions in a fundamental way but do not possess topological order and are bosonic analogs of free fermion topological insulators and superconductors. While a formal cohomology based classification of such states was recently discovered, their physical properties remain mysterious. Here we develop a field theoretic description of several of these states and show that they possess unusual surface states, which if gapped, must either break the underlying symmetry, or develop topological order. In the latter case, symmetries are implemented in a way that is forbidden in a strictly two dimensional theory. While this is the usual fate of the surface states, exotic gapless states can also be realized. For example, tuning parameters can naturally lead to a deconfined quantum critical point or, in other situations, a fully symmetric vortex metal phase. We discuss cases where the topological phases are characterized by quantized magnetoelectric response \theta, which, somewhat surprisingly, is an odd multiple of 2\pi. Two different surface theories are shown to capture these phenomena - the first is a nonlinear sigma model with a topological term. The second invokes vortices on the surface that transform under a projective representation of the symmetry group. A bulk field theory consistent with these properties is identified, which is a multicomponent `BF' theory supplemented, crucially, with a topological term. A possible topological phase characterized by the thermal analog of the magnetoelectric effect is also discussed.Comment: 25 pages+ 3 pages Appendices, 3 figures. Introduction rewritten for clarity, minor technical changes and additional details of surface topological order adde

Spin Liquid States on the Triangular and Kagome Lattices: A Projective Symmetry Group Analysis of Schwinger Boson States

A symmetry based analysis (Projective Symmetry Group) is used to study spin liquid phases on the triangular and Kagom\'e lattices in the Schwinger boson framework. A maximum of eight distinct $Z_2$ spin liquid states are found for each lattice, which preserve all symmetries. Out of these only a few have nonvanishing nearest neighbor amplitudes which are studied in greater detail. On the triangular lattice, only two such states are present - the first (zero-flux state) is the well known state introduced by Sachdev, which on condensation of spinons leads to the 120 degree ordered state. The other solution which we call the $\pi$-flux state has not previously been discussed. Spinon condensation leads to an ordering wavevector at the Brillouin zone edge centers, in contrast to the 120 degree state. While the zero-flux state is more stable with just nearest-neighbor exchange, we find that the introduction of either next-neighbor antiferromagnetic exchange or four spin ring-exchange (of the sign obtained from a Hubbard model) tends to favor the $\pi$-flux state. On the Kagom\'e lattice four solutions are obtained - two have been previously discussed by Sachdev, which on spinon condensation give rise to the $q=0$ and $\sqrt{3}\times\sqrt{3}$ spin ordered states. In addition we find two new states with significantly larger values of the quantum parameter at which magnetic ordering occurs. For one of them this even exceeds unity, $\kappa_c\approx 2.0$ in a nearest neighbor model, indicating that if stabilized, could remain spin disordered for physical values of the spin. This state is also stabilized by ring exchange interactions with signs as derived from the Hubbard model.Comment: revised, 21 pages, 19 figures, RevTex 4, corrected references, added 4 references, accepted by Phys.Rev.