Generic Phase Diagram for Bose-Einstein Condensation of Weakly Interacting Symmetric Bosonic Mixtures

Weakly interacting Bose gas represents {\it strongly} correlated classical field within a domain (determined by the gas parameter ) of its Bose-Einstein condensation (BEC) temperature $T=T_c$. Thus, $N$-component {\it weakly} interacting mixtures representing some symmetry can potentially exhibit rich phase diagram (PD). In particular, it can feature {\it quasi-molecular} phases preceding actual formation of the ODLRO in the vicinity of $T_c$. However, realization of a specific part of the PD depends on details of interactions. As examples, we consider mixtures characterized by O(2)$\times$O(2) symmetry ($N=2$) and spin $S=1$ with the symmetry reduced to U(1)$\times$U(1) ($N=3$). Monte Carlo simulations of these systems find a {\it single} line of the respective two- and three-component BEC transitions which has tricritical point separating II and I order transitions. No {\it quasi-molecular} phases have been found despite that na\"ive mean field (with one loop correction) predicts it. We discuss how such phases can emerge above the actual $N$-component BEC transition. One suggestion relies on Feschbach resonance detuned into negative inter-specie scattering length even when the gas parameter remains small. We acknowledge support from NSF grants PHY 0653135, 0653183 and CUNY grant 80209-0914.