Exploring Reionization-Era Quasars III: Discovery of 16 Quasars at $6.4\lesssim z\lesssim 6.9$ with DESI Legacy Imaging Surveys and UKIRT Hemisphere Survey and Quasar Luminosity Function at $z\sim 6.7$

This is the third paper in a series aims at finding reionzation-era quasars with the combination of DESI Legacy imaging Surveys (DELS) and near-infrared imaging surveys, such as the UKIRT Hemisphere Survey (UHS), as well as the Wide-field Infrared Survey Explore ($WISE$) mid-infrared survey. In this paper, we describe the updated quasar candidate selection procedure, report the discovery of 16 quasars at $6.4\lesssim z\lesssim 6.9$ from area of $\sim$13,020 deg${}^2$, and present the quasar luminosity function (QLF) at $z\sim 6.7$. The measured QLF follows $\mathrm{\Phi }(L_{1450})\propto L_{1450}^{-2.35}$ in the magnitude range $27.6<M_{1450}<-25.5$. We determine the quasar comoving spatial density at $⟨z⟩$=6.7 and $M_{1450}<-26.0$ to be $\rm 0.39\pm0.11 Gpc^{-3}$ and find that the exponential density evolution parameter to be $k=-0.78\pm 0.18$ from $z\sim 6$ to $z\sim 6.7$, corresponding to a rapid decline by a factor of $\sim 6$ per unit redshift towards earlier epoch, a rate significantly faster than that at $z\sim 3-5$. The cosmic time between $z\sim 6$ and $z\sim 6.7$ is only 121 Myrs. The quasar density declined by a factor of more than three within such short time requires that SMBHs must grow rapidly or they are less radiatively efficient at higher redshifts. We measured quasar comoving emissivity at $z\sim 6.7$ which indicate that high redshift quasars are highly unlikely to make a significant contribution to hydrogen reionization. The broad absorption line (BAL) quasar fraction at $z\gtrsim 6.5$ is measured to be $\gtrsim$22%. In addition, we also report the discovery of additional five quasars at $z\sim 6$ in the appendix.