  Quantum many-body edge and extended magnon excitations from the 1/3 --
plateau of the anisotropic Heisenberg model on an open AB$_2$ chain in a
magnetic field $h$ are unveiled using the density matrix renormalization group
and exact diagonalization. By tuning both the anisotropy and $h$ in the rich
phase diagram, the edge states penetrate in the bulk, whose gap closes in a
symmetry-protected topological Kosterlitz-Thouless transition. Also, we witness
the squeezed chain effect, the breaking of the edge states degeneracy, and a
topological change of the excitations from gapped magnons with quadratic
long-wavelength dispersion to a linear spinon dispersion in the Luttinger
liquid gapless phase as the anisotropy $\lambda$ approaches the critical point
from the $\lambda>0$ side of the phase diagram.
