  We study the semiclassical limit and the adiabatic limit with a
second-quantized two-mode model, which describes 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.
