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