TWO-BRANCH ENERGY OF MAGNETIC SURFACES AND CONNECTION WITH ANHOLONOMY
For general boundary conditions, we show that the energy H of a classical Heisenberg ferromagnetic spin system on a curved surface satisfies the inequality H≥|Γ|. Here Γ is a certain geometric phase or anholonomy associated with the spin vector field. This is a generalization of the well known Bogomol'nyi inequality H≥4πQ (Q=integer). For a wide variety of curved surfaces, seeking solutions with certain symmetries, we find that the variational equation δH=0 can be reduced to a sine-Gordon equation with a geometry-dependent parameter. Soliton lattice solutions of this equation are analyzed. It is shown that both H and Γ develop two branches each, all of which merge at the one-soliton limit. This branching is interpreted as a topological transition of the spin textures that occurs at the one-soliton point.