Infinite-order cumulant expansion for spins

The cumulant expansion for spins developed previously is rearranged and extended up to an infinite order in perturbation. A pair of spin-deviation lines is considered as a propagator and, as an additional propagator appears and interacts with the original one, that part of the diagram is defined as a self-energy correction. All cumulant corrections are decomposed and added to the corresponding self-energy corrections, so that self-energy corrections and their cumulant corrections are automatically calculated together exactly. The contribution of a self-energy correction to a diagram depends on the geometry of the incoming and outgoing spin-deviation lines of the self-energy diagram, yielding a new type of perturbation expansion. Numerical results suggest that the nature of the spin-spin correlation in the spin-1/2 case is distinct from that in other cases and that spins behave like fermions rather than bosons in this limit.