Microwave Spectroscopy Evidence of Superconducting Pairing in the Magnetic-Field-Induced Metallic State of InO$_x$ Films at Zero Temperature

We investigate the field tuned quantum phase transition in a 2D low-disorder amorphous InO$_x$ film in the frequency range of 0.05 to 16 GHz employing microwave spectroscopy. In the zero temperature limit, the AC data are consistent with a scenario where this transition is from a superconductor to a metal instead of a direct transition to an insulator. The intervening metallic phase is unusual with a small but finite resistance that is much smaller than the normal state sheet resistance at the lowest measured temperatures. Moreover, it exhibits a superconducting response on short length and time scales while global superconductivity is destroyed. We present evidence that the true quantum critical point of this 2D superconductor metal transition is located at a field $B_{sm}$ far below the conventionally defined critical field $B_{cross}$ where different isotherms of magnetoresistance cross each other. The superfluid stiffness in the low frequency limit and the superconducting fluctuation frequency from opposite sides of the transition both vanish at B $approx B_{sm}$. The lack of evidence for finite-frequency superfluid stiffness surviving $B_{cross}$ signifies that $B_{cross}$ is a crossover above which superconducting fluctuations make a vanishing contribution to DC and AC measurements.