Direct spectroscopic evidence for phase competition between the pseudogap and superconductivity in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$

In the high-temperature ($T_c$) cuprate superconductors, increasing evidence suggests that the pseudogap, existing below the pseudogap temperature $T$*, has a distinct broken electronic symmetry from that of superconductivity. Particularly, recent scattering experiments on the underdoped cuprates have suggested that a charge ordering competes with superconductivity. However, no direct link of this physics and the important low-energy excitations has been identified. Here we report an antagonistic singularity at $T_c$ in the spectral weight of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ as a compelling evidence for phase competition, which persists up to a high hole concentration $p$ ~ 0.22. Comparison with a theoretical calculation confirms that the singularity is a signature of competition between the order parameters for the pseudogap and superconductivity. The observation of the spectroscopic singularity at finite temperatures over a wide doping range provides new insights into the nature of the competitive interplay between the two intertwined phases and the complex phase diagram near the pseudogap critical point.