Doping and Magnetic Field Dependence of In-Plane Tunneling into ${{\mathrm{YBa}}_2{\mathrm{Cu}}_{3}O}_{7-\mathit{x}}$: Possible Evidence for the Existence of a Quantum Critical Point

We present tunneling measurements into $(1,1,0)$ ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-x}$ films at various doping levels around the optimum. We find that, above a certain doping level near optimum doping, a spontaneous zero bias conductance peak splitting, $\delta$, appears. It increases with doping. It also increases with magnetic field applied along the $c$ axis, for both underdoped and overdoped films. The low field susceptibility $\chi =\frac{d\delta }{\mathrm{dH}}{|}_{H\to 0}$ is maximum, possibly diverging when the spontaneous value of $\delta$ goes to zero. These results suggest a transition from a pure $d_{x^2{-y}^2}$ to a ${d+id}_{\mathrm{xy}}$ or $d+is$ order parameter.