Index-theoretic route to the subgap Andreev bands and topological response in Josephson junctions

TL;DR

This paper links subgap Andreev bound states in Josephson junctions to an index theorem, revealing topological effects and differences between superconductor types, with implications for Josephson current behavior and topological protection.

Contribution

It provides an exact solution for subgap states using the index theorem in supersymmetric quantum mechanics and analyzes their topological and dispersive properties across different superconductor types.

Findings

Subgap states depend only on the asymptotic pair-potential properties.

Non-chiral p-wave junctions exhibit a topologically protected 4π Josephson effect.

Chiral and s-wave junctions lack such topological protection.

Abstract

We demonstrate that the subgap Andreev bound states in a transparent Josephson junction, comprising of either chiral or non-chiral superconductors, can be viewed as a consequence of the index theorem in supersymmetric quantum mechanics. We provide an exact solution for these states starting from the Bogoliubov-de Gennes (BdG) equations describing quasiparticles in such junctions. We demonstrate that the dispersion of these subgap states depends only on the asymptotic properties of the pair-potential and not on its local spatial variation. Our study reveals the crucial distinction between junctions of non-chiral $p$-wave superconductors and those of $s$-wave or chiral superconductors by analyzing the wavefunction of their subgap bound states. We find a stable topological response leading to the well-known $4\pi$ periodic Josephson effect protected against weak disorder potential for the non-chiral $p$-wave junctions; no such protection is found for junctions of $s$-wave or chiral superconductors. We supplement our analytic results with numerical computation of the Josephson currents in such junctions using exact numerical Green functions and starting from a lattice model of an itinerant altermagnet which is expected to host triplet $p$-wave superconductivity with equal-spin-pairing. We also discuss the implications of our results for Josephson junctions away from the transparent limit.