Quantum phase transition of two-mode Bose-Einstein condensates with an entanglement order parameter

The ground state entanglement of the two-mode Bose-Einstein condensate is investigated through a quantum phase transition approach. The entanglement measure is taken as the order parameter and this is a non-local order parameter, which is different from the conventional order parameter of the Mott insulator-superfluid phase transitions. For this non-local order parameter, scaling behavior corresponding to a continuous phase transition is obtained and a power-law divergence near the critical region follows it. This scaling behavior of quantum entanglement is analyzed by the finite-size scaling and the critical exponents are obtained as $\nu = 1.01$ and $\gamma = 0.86$. A close connection between quantum fluctuations and the phase transition of entanglement is also obtained.