Modulated phases of a one-dimensional sharp interface model in a magnetic field

We investigate the ground states of one-dimensional continuum models having short-range ferromagnetic-type interactions and a wide class of competing longer-range antiferromagnetic-type interactions. The model is defined in terms of an energy functional, which can be thought of as the Hamiltonian of a coarse-grained microscopic system or as a mesoscopic free-energy functional describing various materials. We prove that the ground state is simple periodic whatever the prescribed total magnetization might be. Previous studies of this model of frustrated systems assumed this simple periodicity but, as in many examples in condensed-matter physics, it is neither obvious nor always true that ground states do not have a more complicated, or even chaotic structure.