Magnetoelectric response in bosonic insulators

In the absence of inversion and time-reversal symmetry, insulators exhibit a non-quantized magneto-electric response. For fermionic band insulators, it is well-known how to compute the corresponding linear response tensor from the band structure and Berry connection. We adapt this analysis to the case of gapped bosonic systems with a U(1) charge, and find a formula for the magneto-electric response in terms of the Berry connection on the bosonic modes. Such responses could be observed in Mott insulators coupled to synthetic gauge fields. Our formulae also permit the computation of the effective axion couplings in quantum spin ice coupled to appropriate antiferromagnetic orders.