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

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.