  We investigate two solvable models for Bose-Einstein condensates and extract
physical information by studying the structure of the solutions of their Bethe
ansatz equations. A careful observation of these solutions for the ground state
of both models, as we vary some parameters of the Hamiltonian, suggests a
connection between the behavior of the roots of the Bethe ansatz equations and
the physical behavior of the models. Then, by the use of standard techniques
for approaching quantum phase transition - gap, entanglement and fidelity - we
find that the change in the scenery in the roots of the Bethe ansatz equations
is directly related to a quantum phase transition, thus providing an
alternative method for its detection.
