Magnetic tilting and emergent Majorana spin connection in topological superconductors

Abstract

Due to the charge neutral and localized nature of surface Majorana modes, detection schemes usually rely on local spectroscopy or interference through the Josephson effect. Here, we theoretically study the magnetic response of a two-dimensional cone of Majorana fermions localized at the surface of class DIII Topological Superconductors. For a field parallel to the surface the Zeeman term vanishes and the orbital term induces a Doppler shift of the Andreev levels resulting in a tilting of the surface Majorana cone. For fields larger than a critical threshold field H * the system undergoes a transition from type I to type II Dirac-Majorana cone. In a spherical geometry the surface curvature leads to the emergence of the Majorana spin connection in the tilting term via an interplay between orbital and Zeeman, that generates a finite non-trivial coupling between negative and positive energy states. Majorana modes are thus expected to show a finite response to the applied field, that acquires a universal character in finite geometries and opens the way to detection of Majorana modes via time-dependent magnetic fields.

Signatures of surface Majorana modes in the magnetic response of topological superconductors

“…Defining m * = µ/2v 2 , we can write the field dependent velocity as v H = eHλ/m * , in analogy with the results of Ref. [36]. Introducing the coherence length ξ = v/∆ and equating the two velocities v H = v ∆ we find the threshold tilting field (restoring )…”

“…The spectrum of the surface Majorana Hamiltonian has two branches, k,± = ±v ∆ + v H k y . At H = H * one of the two branches, k,− , becomes flat along the line k x = 0 and a structural change of the dispersion takes place, that is characterised by a formally diverging number of states at zero energy 36 . The current operator is obtained by the usual relationĵ i = ∂H BdG /∂A i .…”

“…Recently, the authors have shown that for class DIII topological superconductors a coupling to a vector potential can arise at finite momentum and finite energy 36 . In a planar geometry a magnetic field laying on the surface of the system produces a supercurrent, that Doppler shifts the Majorana modes and results in a tilting of the cone.…”

“…The tilting direction in momentum space is orthogonal to the applied field and parallel to the supercurrent. When Majorana modes are confined on a curved surface an additional coupling of geometric origin arises, that involves the Majorana spin connection 36 . The latter generates finite matrix elements between empty and occupied states of the surface Majorana cone and a response is expected at finite frequency.…”

Majorana multipole response: General theory and application to wallpaper groups

“…As an example, we discuss the behavior of tunneling conductance under a magnetic field or with a magnet attached, where we assume that only the Zeeman magnetic field affects the MKP. We note that the orbital magnetization is also useful to measure the topological surface states on TSCs [84][85][86][87][88][89] . Tunnel conductance detects the MKP as a zero bias conductance peak 9 .…”