  We demonstrate theoretically that the topology of energy bands and Fermi
surface in bilayer graphene undergoes a very sensitive transition when
extremely tiny lateral interlayer shift occurs in arbitrary directions. The
phenomenon originates from a generation of effective non-Abelian vector
potential in Dirac Hamiltonian by the sliding motions. The characteristics of
the transition such as pair annihilations of massless Dirac fermions are
dictated by the sliding direction owing to a unique interplay between the
effective non-Abelian gauge fields and Berry's phases associated with massless
electrons. The transition manifests itself in various measurable quantities
such as anomalous density of states, minimal conductivity, and distinct Landau
level spectrum.
