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Microscopically derived Ginzburg-Landau theory for magnetic order in the iron pnictides

We examine the competition of the observed stripe spin density wave (SDW) with other commensurate and incommensurate SDW phases in a two-band model of the pnictides. Starting from this microscopic model, we rigorously derive an expansion of the free energy in terms of the different order parameters at the mean-field level. We show that three distinct commensurate SDW states are possible and study their appearance as a function of the doping and the electronic structure. We show that the stripe phase is generally present, but its extent in the phase diagram depends strongly upon the number of hole Fermi pockets that are nested with the electron Fermi pockets. Electron pockets competing for the same portion of a hole pocket play a crucial role. We discuss the relevance of our results for the antiferromagnetism of the pnictides.