Quantum Geometric Origin of Yu-Shiba-Rusinov States

Abstract

In this work, we revisit the classic Yu-Shiba-Rusinov (YSR) problem of magnetic impurities in $s$-wave superconductors and uncover a quantum geometric origin of the YSR bound-state structure. The key geometric quantity is the momentum-space average of Bloch band projector over low-energy electrons, whose matrix rank, positive eigenvalues, and corresponding eigenvectors directly control the number, energies, and symmetry representations of the YSR states, respectively. As a result, a superconductor with nontrivial normal-state quantum geometry can host more bound states than its trivial counterpart, even if they share the same electronic dispersion. We apply our theory to monolayer $1H$-NbSe$_2$ and find that the geometric texture of its single superconducting band enforces three symmetry-distinct YSR channels for a magnetic point impurity at Nb or Se sites. Our work opens a route to probing normal-state wavefunction geometry through local YSR spectroscopy.