Universal scaling of tunable Yu-Shiba-Rusinov states across the quantum phase transition

Quantum magnetic impurities give rise to a wealth of phenomena attracting tremendous research interest in recent years. On a normal metal, magnetic impurities generate the correlation-driven Kondo effect. On a superconductor, bound states emerge inside the superconducting gap called the Yu-Shiba-Rusinov (YSR) states. Theoretically, quantum impurity problems have been successfully tackled by numerical renormalization group (NRG) theory, where the Kondo and YSR physics are shown to be unified and the normalized YSR energy scales universally with the Kondo temperature divided by the superconducting gap. However, experimentally the Kondo temperature is usually extracted from phenomenological approaches, which gives rise to significant uncertainties and cannot account for magnetic fields properly. Using scanning tunneling microscopy at 10 mK, we apply a magnetic field to several YSR impurities on a vanadium tip to reveal the Kondo effect and employ the microscopic single impurity Anderson model with NRG to fit the Kondo spectra in magnetic fields accurately and extract the corresponding Kondo temperature unambiguously. Some YSR states move across the quantum phase transition (QPT) due to the changes in atomic forces during tip approach, yielding a continuous universal scaling with quantitative precision for quantum spin-1/2 impurities.

Automated summary

Quantum magnetic impurities have attracted tremendous research interest due to the wealth of phenomena they give rise to. The Kondo effect is generated on a normal metal by magnetic impurities, while bound states called Yu-Shiba-Rusinov (YSR) states emerge on a superconductor. Theoretical studies have successfully unified the Kondo and YSR physics using numerical renormalization group (NRG) theory, showing that the normalized YSR energy scales universally with the Kondo temperature divided by the superconducting gap. However, experimental extraction of the Kondo temperature usually involves significant uncertainties and cannot properly account for magnetic fields. In this study, scanning tunneling microscopy and NRG are employed to accurately fit the Kondo spectra in magnetic fields and unambiguously extract the corresponding Kondo temperature.