Quasi-1D Topological Nodal Vortex Line Phase in Doped Superconducting 3D Dirac Semimetals

We study vortex bound states in three-dimensional (3D) superconducting Dirac semimetals with time reversal symmetry. We find that there exist robust gapless vortex bound states propagating along the vortex line in the $s$-wave superconducting state. We refer to this newly found phase as the quasi-1D nodal vortex line phase. According to the Altland-Zirnbauer classification, the phase is characterized by a topological index $(\nu ;N)$ at $k_z=0$ and $k_z=\pi$, where $\nu$ is the ${\mathcal{Z}}_2$ topological invariant for a 0D class-$D$ system and $N$ is the $\mathcal{Z}$ topological invariant for a 0D class-$A$ system. Furthermore, we show that the vortex end Majorana zero mode can coexist with the quasi-1D nodal phase in certain types of Dirac semimetals. The possible experimental observations and material realization of such nodal vortex line states are discussed.

Figure 1

(a) the Fermi surfaces in doped Dirac semimetals; (b)–(d) the sketch of different topological phases of the VBSs. (b) The VBSs in a trivial state are completely full gap. (c) In the topological nodal vortex line phase, there exist robust gapless VBSs which propagate along the whole vortex line. (d) In the topologically nontrivial full-gap state, the bulk VBSs are full gap, but there is a robust Majorana zero mode localized at the end of the vortex line.

Figure 2

(a) and (c) show the numerical results for the topological phase diagrams of the VBSs in the $(\mu ,k_z)$ space for the nontrivial and trivial Dirac semimetals, respectively. The blue points indicate the location of the Dirac points. In the green region, ${\mathcal{Z}}^{0D}$ topological index $Q=1$ and the edge represents the trajectory of the nodal points in the vortex line. On the red line at $k_z=0$, ${\mathcal{Z}}_2^{0D}$ topological index $\nu =-1$. (b) and (d) show the low-energy spectrum in different rotational subspaces (represented by different colors) as a function of $k_z$ at $\mu =0.44$ for the nontrivial Dirac semimetal and $\mu =0.20$ for the trivial Dirac semimetal, respectively. The superconducting order parameter is set to be ${\mathrm{\Delta }}_0=0.2$.

Figure 3

(a) shows the low-energy spectrum of the VBSs for the nontrivial Dirac semimetal in Eq. (2) with a $C_{4z}$-rotational symmetry breaking perturbation $t_{sb}\sin k_z{\mathrm{\Sigma }}_{11}$. (b) shows the density of states of the bulk VBSs for the $C_{4z}$-rotational symmetry broken phase in (a) (purple) and the nodal phase in Fig. 2 (green). The two peaks in (b) appear as a result of the gap opening in (a), as indicated by the black arrows. In the calculations, $t_{sb}$ is set to be 0.05 and all the other parameters are the same as that for Fig. 2.