Localized states near the Abrikosov vortex core in type-II superconductors within zero-range potential model

We propose to treat the lowest bound states near the Abrikosov vortex core in type-II superconductors on the basis of the self-adjoint extension of the Hamiltonian of Aharonov–Bohm type with the localized magnetic flux. It is shown that the Hamiltonian for the excitations near the vortex core can be treated in terms of the generalized zero-range potential method when the magnetic field penetration depth $\delta$ is much greater than the coherence length $\xi$ i.e. in the limit $\varkappa=\delta/\xi\gg1$ In addition, it is shown that in this limit it is the singular behavior of $d\Delta/dr|_{r=0}$ and not the details of the order parameter $\Delta(\mathbf r)$ profile that is important. In support of the proposed model, we reproduce the spectrum of the Caroli–de Gennes–Matricon states and provide direct comparison with the numerical calculations of Hayashi, N. et al. [Phys. Rev. Lett. 80, p. 2921 (1998)]. In contrast to the empirical formula for the energy of the ground state in Hayashi, N. we use no fitting parameter. The parameters for the boundary conditions are determined in a self-consistent manner with Caroli–de Gennes–Matricon formula.