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We explore the equilibrium properties of a two-dimensional Ising spin modelwith short-range exchange and long-range dipolar interactions as a function ofthe applied magnetic field H. The model is studied through extensive MonteCarlo simulations that show the existence of many modulated phases with longrange orientational order for a wide range of fields. These phases arecharacterized by different wave vectors that change discontinuously with themagnetic field. We provide numerical evidence supporting the existence of firstorder transitions between these phases. At higher fields our results suggest aKosterliz-Thouless scenario for the transition from a bubble to a ferromagneticphase.

H-T phase diagram of the bi-dimensional Ising model with exchange and dipolar interactions