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The interplay of magnetic impurity and vortex in a topological superconductoris of fundamental interest with major implications for implementing quantumcomputation. There are multiple degrees of freedom interacting with theimpurity in the system, including the Majorana zero mode (MZM), the Caroli deGennes Matricon (CdGM) states, and the electron bulk states that form Cooperpairs, which makes the impurity pinned vortex state elusive to date intopological superconductors. Here, we present an accurate solution of theproblem, based on a generalized mapping scheme and the density-matrixrenormalization group (DMRG) method. We identify in-gap states that aredistinct from the established Yu-Shiba-Rusinov (YSR) states. The newly foundin-gap physics is driven by three prominent mechanisms: (i) the coupling ofimpurity and bulk states leading to competition between Kondo screening andCooper pairing, resulting in a singlet-doublet quantum phase transition, (ii)the coupling of impurity and CdGM states introducing an effective Zeemansplitting to the doublet state, and (iii) the coupling of impurity and MZMturning the singlet-doublet transition into a crossover. These mechanismscooperatively produce a unique spin-resolved local density of states. Despitethe MZM, a robust nearly zero-energy peak is generated, with comparable heightbut opposite spin polarization as that of the MZM. These results signify novelemergent physics and offer insights to elucidate intriguing experimentalphenomena.

Unique quantum impurity states driven by a vortex in topological superconductors