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

TL;DR

This work develops an index-theoretic, SUSY QM framework to understand subgap Andreev bound states in transparent Josephson junctions, showing that a sign-changing superpotential across the junction guarantees bound states whose dispersion follows $E=\pm\Delta_0\cos(\varphi/2)$. The authors demonstrate a topological protection of the $4\pi$ Josephson effect for non-chiral $p$-wave junctions, arising from the structure of the ABS wavefunctions which forbids coupling by scalar impurities or weak barriers, while $s$-wave and chiral junctions lack this protection and exhibit conventional $2\pi$ periodicity. They corroborate the theory with numerical simulations on a lattice model of 2D altermagnets with equal-spin $p$-wave pairing, using Green's functions to show robust $4\pi$ CPRs against weak disorder and across barrier strengths, consistent with the index-theoretic picture. The results suggest a broad, platform-agnostic route to 4$\pi$ Josephson physics beyond Majorana-based approaches, with experimental relevance for altermagnetic and related topological superconductors. The work also opens avenues for perturbative studies and extended junction geometries guided by the index theorem.

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 $4π$ 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.

Figures (7)

• Figure 1: Schematic figure for the setup for a Josephson junction showing two superconductors (labeled as S) separated by a normal insulating region (labeled as N). The insulator is modeled by a potential $U_0 \gg t,\Delta_0$. The direction normal to the junction is taken to be $\hat{x}$ in all analysis throughout the work.
• Figure 2: Plot of the Fermi surface of an altermagnets obtained from Eq. \ref{['disp']} for chemical potential $\mu / t = 0$ and $\eta=t_R /t=0.5$ (left panel) and $0.9$ (right panel). The Fermi lines flatten out as $\eta$ approaches unity. The spin labels, shown in the figures, indicates anisotropic spin-momentum locking.
• Figure 3: (a) - (d) Andreev bound state spectrum of the Josephson junction between two effective 1D superconductors with fixed $k_{y} = 0$ and weak-link hopping $t_J$. The number of sites of each superconductor is $N=120$ along the $x$-direction, $\Delta_0 / t = 0.1$ and $\mu / t = 0$. We consider two limiting cases - (a), (b) AB limit for which $t_J/t = 0.01$ and (c), (d) KO limit for which $t_J / t = 1.0$. The corresponding $t_R / t$ values are shown in the figures. The subgap energy spectrum versus $\varphi$ shows a gap of $2\delta$ at $\varphi = \pi$, where $\delta$ in the energy splitting between Majorana edge modes in each superconductor. This gap increases with $t_R$ in the AB limit but is almost independent of it in the KO limit for $N=120$. (e) Plot of $\alpha \equiv \mathrm{log}_{10}(\delta / t)$ as a function of $N$, confirming exponential decay of $\delta$ with $N$. (f) Plot of $\alpha$ with $1 + t_R / t$ showing linear scaling of coherence length $\xi \sim \alpha$ with $t_R$. For both plots (e) and (f) $t_J/t=0.01$.
• Figure 4: The current-phase relationship for the spin-$\uparrow$ current in the Josephson junction between two 2D altermagnet superconductors for junction hopping $t_J / t = 1.0$, shown for a few values of $t_R / t$. The number of sites of each superconductor is $N=120$ along $x$-direction, $W = 10$ along $y$-direction. For all plots, $\Delta_0 / t = 0.1$ and $\mu / t = 0$. The Josephson current remains $4\pi$-periodic with a sharp jump at $\varphi=\pi$ for all values of $t_R / t$.
• Figure 5: Current-phase relationship for the spin-$\uparrow$ current in the Josephson junction between two 2D altermagnet superconductors for junction hopping $t_J / t = 0.01$, shown for a few values of $t_R / t$. The number of sites of each superconductor is $N=120$ along $x$-direction, $W = 10$ along $y$-direction. For all plots, $\Delta_0 / t = 0.1$ and $\mu / t = 0$. With decrease in $t_R / t$, the Josephson current $J$ (in units of $t$) changes from almost $2\pi$-periodic to $4\pi$-periodic, along with a discontinuity at $\varphi=\pi$ where the fermion parity switches sign.