Pairing symmetry and vortex zero mode for superconducting Dirac fermions

We study vortex zero-energy bound states in presence of pairing between low-energy Dirac fermions on the surface of a topological insulator. The pairing symmetries considered include the $s$-wave, $p$-wave, and, in particular, the mixed-parity symmetry, which arises in absence of the inversion symmetry on the surface. The zero mode is analyzed within the generalized Jackiw-Rossi-Dirac Hamiltonian that contains a momentum-dependent mass term, and includes the effects of the electromagnetic gauge field and the Zeeman coupling as well. At a finite chemical potential, as long as the spectrum without the vortex is fully gapped, the presence of a single Fermi surface with a definite helicity always leads to one Majorana zero mode, in which both electron’s spin projections participate. In particular, the critical effects of the Zeeman coupling on the zero mode are discussed.

Figure 1

A single two-dimensional Fermi surface associated with the Dirac Hamiltonian, $H_0\left(\vec{k}\right)=v_F\left(\widehat{z}\cdot \vec{k}\times \vec{\sigma }\right)-\mu$, on the metallic surface of a TI. $\widehat{z}$, the out-of-plane direction, is the spin quantization axis. The dark (dashes) arrows indicate the directions of the spin associated with the momentum state $\vec{k}$ on the Fermi surface of positive (negative) helicity when $\mu$ is positive (negative).

Figure 2

(Color online) The phases of zero mode for $h^2$ versus ${\mu }^2$ in the (a) $s$-wave pairing and (b) $p$-wave pairing cases. The left-hand sides with respect to the empty circles denote the existence of the zero mode while the right-hand sides, including the empty circle, indicate that the zero mode is not normalizable and hence does not exist. The blue regions stand for the fact that the wave function has, in addition to the decaying, an asymptotic oscillation part, which is gone when entering the green regions.

Figure 3

(Color online) Different phases for the zero mode in the mixed pairing symmetry, as a function of $\mu$ and the parameter ${\Delta }_{-}$, assuming ${\Delta }_{+}$ is positive. The dotted axis is to exclude the special cases with $\mu =0$. The gray and blue regions denote different eigenvalues of $A$ for the zero mode ${\Psi }_0$ as specified in text. At the dashed line $\mu <0$, ${\Delta }_{-}=0$, in between the two phases where ${\Delta }_{-}$ changes sign (as denoted by the green arrows) the localized zero mode does not exist.