  With the commutation relations of the spin operators, we first write out the
equations of motion of the spin susceptibility and related correlation
functions that have a hierarchical structure, then under the "soft cut-off"
approximation, we give a set of equations of motion of spin susceptibilities
for a spin S=1/2 antiferromagnetic Heisenberg model, that is independent of
whether or not the system has a long range order in the low energy/temperature
limit. Applying for a chain, a square lattice and a honeycomb lattice,
respectively, we obtain the upper and the lowest boundaries of the low-lying
excitations by solving this set of equations. For a chain, the upper and the
lowest boundaries of the low-lying excitations are the same as that of the
exact ones obtained by the Bethe ansatz, where the elementary excitations are
the spinon pairs. For a square lattice, the spin wave excitation (magnons)
resides in the region close to the lowest boundary of the low-lying
excitations, and the multispinon excitations take place in the high energy
region close to the upper boundary of the low-lying excitations. For a
honeycomb lattice, we have one kind of "mode" of the low-lying excitation. The
present results obey the Lieb-Schultz-Mattis theorem, and they are also
consistent with recent neutron scattering observations and numerical
simulations for a square lattice.
