Superconducting quantum criticality and the anomalous scaling: A nonlinear relativistic equation

By using the Landau-Ginzburg-Wilson paradigm, we show that, near a quantum critical point (QCP), Cooper pairs at zero temperature would obey a nonlinear relativistic equation, where the imaginary time emerges as a novel dimension. This relativistic equation is applicable to certain superconductors at zero temperature for which the Faber-Pippard coherence length formula holds at and above the upper critical dimension. Here, we further show that the relativistic equation leads to a testable prediction in the vicinity of the QCP $T_c=0$, with $T_c$ being the transition temperature. That is, for 2D overdoped (clean) superconducting films, when the parameter $T_c/(c_0v_F)$ is lower than a characteristic scale, the Lorentz symmetry of relativistic equation arouses an anomalous scaling $\xi_0 \propto T_c^{-1.34}$, where $\xi_0$ denotes the zero-temperature coherence length, $v_F$ denotes the Fermi velocity, and $c_0$ denotes the Faber-Pippard coefficient. However, when the parameter $T_c/(c_0v_F)$ is large enough, the Lorentz symmetry may be broken so that the Faber-Pippard scaling $\xi_0 \propto T_c^{-1}$ is restored.