Magnetic tilting and emergent Majorana spin connection in topological superconductors

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 a 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 terms that generates a finite nontrivial coupling between negative- and positive-energy states. Majorana modes are thus expected to show a finite response to the applied field, which acquires a universal character in finite geometries and opens the way to detection of Majorana modes via time-dependent magnetic fields.

Figure 1

Evolution with the field of the spectrum on a cylindrical geometry for the case $k_z=0$ versus angular momentum ${\ell }_z=j_z-1/2$. The applied field is $H=0$ in (a), $H=H^{*}$ in (b), and $H>H^{*}$ in (c), and it exponentially decays towards the center of the cylinder on a scale $\lambda =0.06R$. PH symmetry requires that ${\epsilon }_{{\ell }_z}^n\left(H\right)=-{\epsilon }_{-{\ell }_z-1}^n\left(H\right)$.

Figure 2

Sketch of the surface Andreev states of the BW states in a spherical geometry. (a) Surface Majorana cone without magnetic field and (b) the effect of the matrix elements (13) on the spectrum. Large values of the external field have been chosen to highlight the effect.

Figure 3

Screened full-induction field $\mathbf{B}$ associated with the Meissner effect for $R=10\lambda$.