Fermi Arc Criterion for Surface Majorana Modes in Superconducting Time-Reversal Symmetric Weyl Semimetals

Many clever routes to Majorana fermions have been discovered by exploiting the interplay between superconductivity and band topology in metals and insulators. However, realizations in semimetals remain less explored. We ask, “Under what conditions do superconductor vortices in time-reversal symmetric Weyl semimetals—three-dimensional semimetals with only time-reversal symmetry—trap Majorana fermions on the surface?” If each constant-$k_z$ plane, where $z$ is the vortex axis, contains equal numbers of Weyl nodes of each chirality, we predict a generically gapped vortex and derive a topological invariant $\nu =\pm 1$ in terms of the Fermi arc structure that signals the presence or absence of surface Majorana fermions. In contrast, if certain constant-$k_z$ planes contain a net chirality of Weyl nodes, the vortex is gapless. We analytically calculate $\nu$ within a perturbative scheme and provide numerical support with a lattice model. The criteria survive the presence of other bulk and surface bands and yield phase transitions between trivial, gapless, and topological vortices upon tilting the vortex. We propose $\mathrm{Li}({\mathrm{Fe}}_{0.91}{\mathrm{Co}}_{0.09})\mathrm{As}$ and ${\mathrm{Fe}}_{1+y}{\mathrm{Se}}_{0.45}{\mathrm{Te}}_{0.55}$ with broken inversion symmetry as candidates for realizing our proposals.

Figure 1

Schematic of the main result. Orange (blue) dots denote right (left)-handed WNs which produce right (left)-moving chiral MFs, colored red (green), inside the vortex. To determine the vortex phase, identify pairs of antichiral WNs at the same $k_z$ and project the line joining them onto the surface. If these lines (solid black), along with the FAs (red curves), form $M$ closed loops, the vortex is gapped and has a topological invariant $\nu =(-1{)}^M$ (a), whereas open arcs produce a gapless vortex (b).

Figure 2

Vortex topological phases predicted by Eq. (1) and computed numerically. The yellow mask (black dots) denotes a vortex predicted (computed) to be topological. Black lines separate the normal states: TWSMs with $N=1$, 2 quadruplets of WNs at $k_z=0$ and trivial with $N=0^{+}$ and topological with $N=0^{-}$ insulators. We fix $v_{x,y}=1.18$,.856, ${\beta }_{x,y,z}=.856$, 1.178, 3.0, $\mathrm{\Delta }(r)=0.42\tanh (0.3r)$, and $L_{x,y}=31$. Points $\mathbf{t}$, $\mathbf{m}$, and $\mathbf{b}$ are further studied in Fig. 3.

Figure 3

Left column: color plots of the lowest band for a 45-layer slab in the normal state for the points $\mathbf{t}$, $\mathbf{m}$, and $\mathbf{b}$ in Fig. 2 defined by $l=0.942$, $m_0=6.28$ (top), $l=0.972$, $m_0=5.48$ (middle) and $l=0.552$, $m_0=6.18$ (bottom). Red filled (empty) circles denote surface projections of right- (left-)handed WNs. Surface projections of geodesics (black lines) connecting antichiral WNs at $k_z=0$ form $M$ closed loops along with the FAs (red curves). Right column: probability densities of six lowest energy states along a $z$-oriented vortex in a $31\times 31\times 45$-site system. Bold and dotted lines mark states with energies $E<5.0\times {10}^{-3}$, considered “zero energy,” and $E>1.0\times {10}^{-2}$.

Figure 4

Topological phase transitions upon tilting the $M=2$ trivial vortex in Fig. 3 (bottom) obtained by diagonalization in real space (a),(c) and $k$ space (b),(d). (a) Tilting about the $x$ axis produces $M=3$ (inset) and gaps out one of the two MFs at ${\theta }_c\approx 0.06\pi$. (b) At ${\theta }_p=0.1\pi$, the bulk vortex is gapped, so the surviving MF in (a) at $\theta >{\theta }_c$ is protected and the vortex is topological. (c),(d) Analogous figures for tilting about $x+y=z=0$. In (c), a small gap opens for one MF at ${\theta }_c\approx 0.1\pi$ while the surface has open Fermi-geodesic arcs (inset). (d) The bulk vortex is gapless, indicating that the gap in (c) is a finite size gap.