  One milestone in quantum physics is Berry's seminal work
[Proc.~R.~Soc.~Lond.~A \textbf{392}, 45 (1984)], in which a quantal phase
factor known as geometric phase was discovered to solely depend on the
evolution path in state space. Here, we unveil that even an infinitesimal
deviation of the initial state from the eigenstate of the initial Hamiltonian
can yield a significant change of the geometric phase accompanying an adiabatic
evolution. This leads to the surprising observation that two geometric phases
can dramatically differ from each other even if their evolution paths are
sufficiently close in a pointwise manner.
