Control, competition and coexistence of effective magnetic orders by interactions in Bose-Einstein condensates with high-Q cavities

Ultracold atomic systems confined in optical cavities have been demonstrated as a laboratory for the control of quantum matter properties and analog quantum simulation. Often neglected, but soon amenable to manipulation in a new generation of experiments, we show that atomic many-body interactions allow additional control in the cavity driven self-organization of effective spinor Bose-Einstein condensates (BEC). We theoretically show that a rich landscape of magnetic ordering configurations emerges. This can be controlled by modifying the geometry of the light-fields in the system with the interplay of two-body interactions and the cavity induced interactions. This leads to competition scenarios and phase separated dynamics. Our results show that it is possible to tailor on demand configurations possibly useful for analog quantum simulation of magnetic materials with highly controllable parameters in a single experimentally realistic setup.