# 3D bosonic topological insulator and its exotic electromagnetic response

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Recently, many new types of bosonic symmetry-protected topological phases, including bosonic topological insulators, were predicted using group cohomology theory.  The bosonic topological insulators have  both  U(1) symmetry (particle number conservation) and time-reversal symmetry, described by symmetry group $U(1)\rtimes Z_2^T$.  In this paper, we propose a projective construction of three-dimensional correlated gapped bosonic state with $U(1)\rtimes Z_2^T$ symmetry.  The gapped bosonic insulator is formed by eight kinds of charge-1 bosons. We show that, in our bosonic state, an {\it electromagnetic} monopole with a unit magnetic charge is fermionic while an {\it electromagnetic} dyon with a unit magnetic charge and a unit electric charge is bosonic.  This indicates that the constructed bosonic state is a non-trivial bosonic topological insulator, since in a trivial bosonic Mott insulator, the monopole is bosonic while the dyon is fermionic.  We also constructed a three-dimensional correlated gapless bosonic insulator with $U(1)\rtimes Z_2^T$ symmetry, that has two emergent gapless $U(1)$ gauge fields, and excitations with fractional gauge charges for both the emergent and electromagnetic gauge fields.  Both  bosonic insulators can have protected conducting surface states. The gapless boundary excitations of the gapless bosonic insulator can even be fermionic.