Commutability between the Semiclassical and Adiabatic Limits

We study the adiabatic limit and the semiclassical limit with a second-quantized two-mode model of a many-boson interacting system. When its mean-field interaction is small, these two limits are commutable. However, when the interaction is strong and over a critical value, the two limits become incommutable. This change of commutability is associated with a topological change in the structure of the energy bands. These results reveal that nonlinear mean-field theories, such as Gross-Pitaevskii equations for Bose-Einstein condensates, can be invalid in the adiabatic limit