  We derive a geometrical expression of the excess entropy production for
quasi-static transitions between nonequilibrium steady states of Markovian jump
processes, which can be exactly applied to nonlinear and nonequilibrium
situations. The obtained expression is geometrical; the excess entropy
production depends only on a trajectory in the parameter space, analogous to
the Berry phase in quantum mechanics. Our results imply that vector potentials
are needed to construct thermodynamics of nonequilibrium steady states.
