Fig. 2

Theory of spin–orbit defect and magnetic exchange. a Map of the local density of states (LDOS) at zero energy in the model of Eq. (1) solved in a circular system with a single spin–orbit vortex at the center of a disk (dashed line), which has constant magnetic exchange V_z = 5Δ_S, and V_z = 0 outside, while all system parameters are (ξ/a, R/a, l_V/a, l_F/a, l_SO/a) = (400; 300; 80; 45; 3.3) (see Methods for definitions and technical details), where superconducting coherence length ξ is larger than disk radius R, and system size is L = 8000. This LDOS is contributed by exactly two Majorana zero-mode wavefunctions. b Angularly averaged amplitude of the wavefunction at zero energy (dashed lines are guide to eye). The superconducting coherence length ξ/a = 20 is smaller than disk radius R/a = 80 (vertical line) in a system of size L = 300. See Methods for definitions, parameter values and technical details of the two-dimensional tight-binding calculation of model Eq. (1). c Energy-dependent LDOS at the center of the island with a thermal broadening of k_BT/Δ_S = 0.1. d Energy-dependent LDOS at the edge of the magnetic disk (light blue curve) and inside the magnetic disk at distance 0.75R from the center (dark blue curve). The inset show a zoom on the zero-bias density of states, revealing a well-defined zero-bias peak. e Spectrum of excitations in superconductor of pairing energy Δ_S for spin–orbit vortex—anti-vortex pair in a plane of size L = 450 with periodic boundary conditions and with constant magnetic exchange V_z (see Methods). The lowest energies are plotted for each V_z, with lowest two in red. Above a critical magnetic exchange two zero-energy states are localized at the two spin–orbit vortices