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Mapping a many-body state on a loop in parameter space is a simple way tocharacterize a quantum state. The connections of such a geometricalrepresentation to the concepts of Chern number and Majorana zero mode areinvestigated based on a generalized quantum spin system with short andlong-range interactions. We show that the topological invariants, the Chernnumbers of corresponding Bloch band is equivalent to the winding number in theauxiliary plane, which can be utilized to characterize the phase diagram. Weintroduce the concept of Majorana charge, the magnitude of which is defined bythe distribution of Majorana fermion probability in zero-mode states, and thesign is defined by the type of Majorana fermion. By direct calculations of theMajorana modes we analytically and numerically verify that the Majorana chargeis equal to Chern numbers and winding numbers.

Majorana charges, winding numbers and Chern numbers in quantum Ising models