Bosonic topological insulator in three dimensions and the statistical Witten effect

It is well known that one signature of the three-dimensional electron topological insulator is the Witten effect: if the system is coupled to a compact electromagnetic gauge field, a monopole in the bulk acquires a half-odd-integer polarization charge. In the present work, we propose a corresponding phenomenon for the topological insulator of bosons in three dimensions protected by particle number conservation and time-reversal symmetry. We claim that although a monopole inside a topological insulator of bosons can remain electrically neutral, its statistics are transmuted from bosonic to fermionic. We demonstrate that this ``statistical Witten effect'' directly implies that if the surface of the topological insulator is neither gapless, nor spontaneously breaks the symmetry, it necessarily supports an intrinsic two-dimensional topological order. Moreover, the surface properties can not be fully realized in a purely two-dimensional system. We also confirm that the surface phases of the bosonic topological insulator proposed by Vishwanath and Senthil Phys. Rev. X 3, 011016 (2013) provide a consistent termination of a bulk exhibiting the statistical Witten effect. In a forthcoming paper, we will provide an explicit field-theoretic, lattice-regularized construction of the three-dimensional topological insulator of bosons, employing a parton decomposition and subsequent condensation of parton-monopole composites.