Nonlinear orbital magnetization

We calculate the nonlinear electric polarization induced by a magnetic field in crystals (metals and insulators). Two second order, we find general expressions suitable for ab-initio numerical studies of materials nonlinear magnetoelectric response. In the limit of a homogeneous magnetic field, we recover the known ground state orbital magnetization, and the orbital magnetoelectric response including the Chern-Simons (CS) term. In the nonlinear response, there are higher order analogs of the CS term. In general, these terms are small compared with a large background. Applications to recent experiments is also discussed.