On the energy of a Chern-Simons-Higgs vortex lattice

We study the interaction of Chern-Simons-Higgs vortices with an external magnetic field. Away from self-duality, we give a detailed picture and compute critical fields using a I-convergence approach involving a Jacobian compactness result. Closer to the self-dual scaling, where the Gamma-convergence approach fails, we analyze energy minimizers with supercritical magnetic fields and show that their energy scales asymptotically with the energy of a vortex lattice for most parameter ranges.