Correlated topological insulators and the fractional magnetoelectric effect

Topological insulators are characterized by the presence of gapless surface modes protected by time-reversal symmetry. In three space dimensions the magnetoelectric response is described in terms of a bulk $\theta$ term for the electromagnetic field. Here we construct theoretical examples of such phases that cannot be smoothly connected to any band insulator. Such correlated topological insulators admit the possibility of fractional magnetoelectric response described by fractional $\theta /\pi$. We show that fractional $\theta /\pi$ is only possible in a gapped time-reversal-invariant system of bosons or fermions if the system also has deconfined fractional excitations and associated degenerate ground states on topologically nontrivial spaces. We illustrate this result with a concrete example of a time-reversal-symmetric topological insulator of correlated bosons with $\theta =\frac{\pi }{4}$. Extensions to electronic fractional topological insulators are briefly described.