  A binary mixtures of Bose-Einstein condensate structures exhibit an
incredible richness in terms of holding different kinds of phases. Depending on
the ratio of the inter- and intra-atomic interactions, the transition from
mixed to separated phase, which is also known as the miscibility-immiscibility
transition, has been reported in different setups and by different groups.
Here, we describe such type of quantum phase transition in an effective
Hamiltonian approach, by applying Holstein-Primakoff transformation in the
limit of large number of particles. We demonstrate that non-trivial geometric
phase near the critical coupling is present, which confirms the connection
between Berry phase and quantum phase transition. We also show that, by using
the spin form of Hillery & Zubairy criterion, a two mode entanglement
accompanies this transition in the limit of large, but not infinite number of
particles.
