Magnetothermodynamics of BPS baby skyrmions

The magnetothermodynamics of skyrmion type matter described by the gauged BPS baby Skyrme model at zero temperature is investigated. We prove that the BPS property of the model is preserved also for boundary conditions corresponding to an asymptotically constant magnetic field. The BPS bound and the corresponding BPS equations saturating the bound are found. Further, we show that one may introduce pressure in the gauged model by a redefinition of the superpotential. Interestingly, this is related to non-extremal type solutions in the so-called fake supersymmetry method. Finally, we compute the equation of state of magnetized BSP baby skyrmions inserted into an external constant magnetic field $H$ and under external pressure $P$, i.e., $V=V(P,H)$, where $V$ is the "volume" (area) occupied by the skyrmions. We show that the BPS baby skyrmions form a ferromagnetic medium.