Quantized bound states around a vortex in anisotropic superconductors

TL;DR

This paper develops a variational theory based on the Andreev approximation to analyze low-energy bound states around vortices in anisotropic superconductors, extending previous models and providing detailed wave functions and spatial profiles.

Contribution

It generalizes existing theories to arbitrary Fermi surfaces, enabling calculation of bound state wave functions and revealing new features due to pairing anisotropy.

Findings

Proves equidistant energy spectrum inside vortices for a broad class of superconductors.

Obtains wave functions of bound states beyond quasiclassical Eilenberger results.

Describes spatial profiles of local density of states influenced by anisotropy.

Abstract

The bound states around a vortex in anisotropic superconductors is a longstanding yet important issue. In this work, we develop a variational theory on the basis of the Andreev approximation to obtain the energy levels and wave functions of the low-energy quantized bound states in superconductors with anisotropic pairing on arbitrary Fermi surface. In the case of circular Fermi surface, the effective Schr\"odinger equation yielding the bound state energies gets back to the theory proposed by Volovik and Kopnin many years ago. Our generalization here enables us to prove the equidistant energy spectrum inside a vortex in a broader class of superconductors. More importantly, we are now able to obtain the wave functions of these bound states by projecting the quasiclassical wave function on the eigenmodes of the effective Schr\"odinger equation, going beyond the quasiclassical Eilenberger results, which, as we find, are sensitive to the scattering rate. For the case of isotropic Fermi surface, the spatial profile of the low-energy local density of states is dominated near the vortex center and elongates along the gap antinode directions, in addition to the ubiquitous Friedel oscillation arising from the quantum inteference neglected in the Eilenberger theory. Moreover, as a consequence of the pairing anisotropy, the quantized wave functions develop a peculiar distribution of winding number, which reduces stepwise towards the vortex center. Our work provides a flexible way to study the vortex bound states in the future.