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Phase diagrams and Thomas-Fermi estimates for spin-orbit coupled Bose-Einstein Condensates under rotation

We provide complete phase diagrams describing the ground state of a trapped spinor BEC under the combined effects of rotation and a Rashba spin-orbit coupling. The interplay between the different parameters (magnitude of rotation, strength of the spin-orbit coupling and interaction) leads to a rich ground state physics that we classify. We explain some features analytically in the Thomas-Fermi approximation, writing the problem in terms of the total density, total phase and spin. In particular, we analyze the giant skyrmion, and find that it is of degree 1 in the strong segregation case. In some regions of the phase diagrams, we relate the patterns to a ferromagnetic energy.