Zone-center dynamical matrix in magnetoelectrics

In ordinary dielectrics, the dynamical matrix at the zone center in general is a nonanalytic function of degree zero in the wave vector q. Its expression (for a crystal of arbitrary symmetry) is well known and is routinely implemented in first-principles calculations. The nonanalytic behavior occurs in polar crystals and owes to the coupling of the macroscopic electric field E to the lattice. In magnetoelectric crystals both electric and magnetic fields, E and H, are coupled to the lattice, formally on equal footing. We provide the general expression for the zone-center dynamical matrix in a magnetoelectric, where the E and H couplings are accounted for in a symmetric way. As in the ordinary case, the dynamical matrix is a nonanalytic function of degree zero in q, and is exact in the harmonic approximation. For the sake of completeness, we address other issues, and in particular, we solve a problem, which might arise in first-principles implementations, where-differently than here-the basic fields are E and B (not H).