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 struc
ture, 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.
A two-dimensional Frenkel-Kontorova model is set up. Its application to the tribology is considered. The materials and the commensurability between two layers strongly affect the static friction force. It is found that the static friction force is la
rger between two layer of same materials than that for different materials. For two-dimensional case the averaged static friction force is larger for the uncommensurate case than that for the commensurate case, which is completely different from one-dimensional case. The directions of the propagation of the center of mass and the external driving force are usually different except at some special symmetric directions. The possibility to obtain superlubricity is suggested.
