  We investigate the excited states of the quasi-one-dimensional quantum
antiferromagnets on hexagonal lattices, including the longitudinal modes based
on the magnon-density waves. A model Hamiltonian with a uniaxial single-ion
anisotropy is first studied by a spin-wave theory based on the one-boson
method; the ground state thus obtained is employed for the study of the
longitudinal modes. The full energy spectra of both the transverse modes (i.e.,
magnons) and the longitudinal modes are obtained as functions of the
nearest-neighbor coupling and the anisotropy constants. We have found two
longitudinal modes due to the non-collinear nature of the triangular
antiferromagnetic order, similar to that of the phenomenological field theory
approach by Affleck. The excitation energy gaps due to the anisotropy and the
energy gaps of the longitudinal modes without anisotropy are then investigated.
We then compare our results for the longitudinal energy gaps at the magnetic
wavevectors with the experimental results for several antiferromagnetic
compounds with both integer and non-integer spin quantum numbers, and we find
good agreement after the higher-order contributions are included in our
calculations.
