  We adopt a three-level bosonic model to investigate the quantum phase
transition in an ultracold atom-molecule conversion system which includes one
atomic mode and two molecular modes. Through thoroughly exploring the
properties of energy level structure, fidelity, and adiabatical geometric
phase, we confirm that the system exists a second-order phase transition from
an atommolecule mixture phase to a pure molecule phase. We give the explicit
expression of the critical point and obtain two scaling laws to characterize
this transition. In particular we find that both the critical exponents and the
behaviors of ground-state geometric phase change obviously in contrast to a
similar two-level model. Our analytical calculations show that the ground-state
geometric phase jumps from zero to ?pi/3 at the critical point. This
discontinuous behavior has been checked by numerical simulations and it can be
used to identify the phase transition in the system.
