Dynamical Spectra of Spin Supersolid States in Triangular Antiferromagnets

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.