  We employ tensor network renormalization to explore the dynamical spectra of
the easy-axis triangular-lattice antiferromagnet (TLAF) in a magnetic field.
Our analysis identifies two distinct low-energy magnon excitations: a gapless
Goldstone mode and a gapped mode. At zero field, the spectra display two nearly
degenerate roton modes near the M point. With the increase of the magnetic
field within the Y-shape superfluid phase, these modes diverge, with the roton
excitation vanishing from the Goldstone mode branch, suggesting that the roton
dip in this mode may just result from the energy-level repulsion imposed by the
roton excitation in the gapped mode. Moreover, the in-plane spectral function
shows substantial weight in high energies in the same spin excitation channel
where the low-energy roton excitation appears. However, these roton excitations
are absent in the V-shape supersolid phase.
