We extend the Blume-Emery-Griffiths (BEG) model to a two-component BEG model in order to study 2D systems with two order parameters, such as magnetic superconductors or two-component Bose-Einstein condensates. The model is investigated using Monte Carlo simulations, and the temperature-concentration phase diagram is determined in the presence and absence of an external magnetic field. This model exhibits a rich phase diagram, including a second-order transition to a phase where superconductivity and magnetism coexist. Results are compared with experiments on Cerium-based heavy-fermion superconductors. To study cold atom mixtures, we also simulate the BEG and two-component BEG models with a trapping potential. In the BEG model with a trap, there is no longer a first order transition to a true phase-separated regime, but a crossover to a kind of phase-separated region. The relation with imbalanced fermi-mixtures is discussed. We present the phase diagram of the two-component BEG model with a trap, which can describe boson-boson mixtures of cold atoms. Although there are no experimental results yet for the latter, we hope that our predictions could help to stimulate future experiments in this direction.
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