Caroli-de Gennes-Matricon (CdGM) states are localized states with a discrete energy spectrum bound to the core of vortices in superconductors. In topological superconductors, CdGM states are predicted to coexist with zero energy, chargeless states widely known as Majorana zero modes (MZMs). Due to their energy difference, current experiments rely on scanning tunneling spectroscopy methods to distinguish between them. This work shows that electrostatic inhomogeneities can push trivial CdGM states arbitrarily close to zero energy in nontopological systems where no MZM is present. Furthermore, the BCS charge of CdGM states is suppressed under the same mechanism. Through exploration of the impurity parameter space, we establish that these two phenomena generally happen in consonance. Our results show that energy and charge shifts in CdGM may be enough to imitate the spectroscopic signatures of MZMs even in cases where the estimated CdGM level spacing (in the absence of impurities) is much larger than the typical experimental level broadening.

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Research into the realization of Majorana quasiparticles has been driven by the relevance of quantum computing and topological materials. Initially, efforts were focused on nanowire realization of Majoranas, but experimental limitations and skepticism led to investigations into two-dimensional setups such as superconducting vortices [1]- [7]. In this study, we explore the potential and constraints of superconducting vortices as hosts of topological quasiparticles. We examine how scalar impurities can cause false-positive signatures of Majoranas in fully trivial superconductivity. Our findings show that scalar impurities can modify the Caroli-de Gennes-Matricon (CdGM) spectrum, resulting in spectral properties similar to Majorana Zero Modes (MZMs). Local scalar perturbations lower the energy of CdGM states and suppress their total Bardeen-Cooper-Schrieffer (BCS) charge, leading to impurity-driven zero energy states that cannot be distinguished from topological MZMs using local spectroscopic techniques. Additionally, extrinsic broadening of the particle-hole symmetric levels results in vanishing non-local transport signals, making it appear that the BCS charge is zero. These findings have implications for experimental works that rely on finite energy CdGM states [8]. We also investigate the potential of hybrid systems of semiconductors and trivial superconductors as substitutes for intrinsic topological superconductivity, which may provide a more robust way of retaining topological character [9], [10]. Whereas the p−wave order parameter's complex phase places limitations on the number of vortices that can be introduced, the scalar s−wave order parameter lacks this constraint. This discrepancy is significant because, despite having an equivalent total number of modes, the edge mode is not suitable for braiding. Furthermore, we remark that unequivocally topological zero energy mode in the topological region is not guaranteed for four vortices and is dependent on a suitably large scattering space, even in the approximation of minimal radii.