Common Features in Multi-species Bose–Einstein Condensates and the Origin of the Inherent Critical Phenomena

A general approach is proposed to solve the coupled Gross–Pitaevskii equations (CGP) for K-species Bose–Einstein condensates under the Thomas–Fermi approximation. We aim at finding out the common features of these condensates different in K. In particular, two types of phase transitions, full-state transition and partial-state transition, are found. In the former, all species are involved in the transition, while in the latter only a few specified species are significantly involved. This leads to the criticality and the hidden criticality (namely, the criticality found in a condensate with fewer species recovers in a condensate with more species). The former originates from the singularity of the whole matrix of the CGP, while the latter originates from the singularity of a specified sub-matrix (which is related to a few specified species). In general, for any many-body system, the singularity inherent in the equations that govern the system is crucial to its critical behavior.