A Complete Study of the Ground State Phase Diagrams of Spin-1 Bose–Einstein Condensates in a Magnetic Field Via Continuation Methods

We present a complete investigation of the ground state patterns and phase diagrams of the spin-1 Bose–Einstein condensates (BEC) confined in a harmonic or box potential under the influence of a homogeneous magnetic field. A pseudo-arclength continuation method with parameter switching technique is developed to study the BEC systems numerically. The continuation process is performed on the parameter space consisting of the spin–independent interaction, spin–exchange interaction and the quadratic Zeeman (QZ) energy parameters. In the first stage of the parameter switching process, we fix the QZ energy term to be zero and vary the interaction parameters from zero to the desired physical values. Next, we fix the interaction parameters and vary the QZ energy parameter in both positive and negative regions. Two types of phase transitions are found, as we vary the QZ parameter. The first type is a transition from a two-component state to a three-component (3C) state. The second type is a symmetry breaking in the 3C state. Then, a phase separation of the spin components follows. In the semi-classical regime, we find that these two phase transition curves are gradually merged.