Ashvin Vishwanath, T. Senthil
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. Recently, the mathematical classification of such states was discussed in terms of cohomology theory . However, 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. 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 theories of surface states are shown to capture these phenomena - the first is a nonlinear sigma model with a topological term. The second invokes vortices on the surface with fractional quantum numbers, 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. Bulk sigma model field theories of these phases are also provided. Topological phases that lie beyond the cohomology classification, characterized by the thermal analog of the quantized magnetoelectric effect, are also discussed.
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http://arxiv.org/abs/1209.3058
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