Quantum Spin Supersolid as a precursory Dirac Spin Liquid in a Triangular Lattice Antiferromagnet

Based on the recent experiments on the triangular lattice antiferromagnet Na${}_2$BaCo(PO${}_4$)${}_2$, we propose the easy-axis XXZ spin-1/2 model on the triangular lattice, that exhibits a quantum spin supersolid, to be a precursory Dirac spin liquid. Despite the presence of a three-sublattice magnetic order as a spin supersolid, we suggest that this system is close to a Dirac spin liquid by exploring its spectroscopic response. The physical consequence is examined from the spectroscopic response, and we establish the continuous spectra near the M point in addition to the K point excitation from the spinon continuum on top of the three-sublattice order. Moreover, the satellite peaks were predicted at the mid-points connecting the $\mathrm{\Gamma }$ and K points. This proposal offers a plausible understanding of the recent inelastic neutron scattering measurement in Na${}_2$BaCo(PO${}_4$)${}_2$ and could inspire further research in relevant models and materials, such as K${}_2$Co(SeO${}_3$)${}_2$ and Rb${}_2$Co(SeO${}_3$)${}_2$, and even more anisotropic magnets like PrMgAl${}_{11}$O${}_{19}$.