  Self-intersecting energy band structures in momentum space can be induced by
nonlinearity at the mean-field level, with the so-called nonlinear Dirac cones
as one intriguing consequence. Using the Qi-Wu-Zhang model plus power law
nonlinearity, we systematically study in this paper the Aharonov-Bohm (AB)
phase associated with an adiabatic process in the momentum space, with two
adiabatic paths circling around one nonlinear Dirac cone. Interestingly, for
and only for Kerr nonlinearity, the AB phase experiences a jump of $\pi$ at the
critical nonlinearity at which the Dirac cone appears or disappears, whereas
for all other powers of nonlinearity the AB phase always changes continuously
with the nonlinear strength. Our results may be useful for experimental
measurement of power-law nonlinearity and shall motivate further fundamental
interest in aspects of geometric phase and adiabatic following in nonlinear
systems.
