Topologically Protected Landau Level in the Vortex Lattice of a Weyl Superconductor

The question whether the mixed phase of a gapless superconductor can support a Landau level is a celebrated problem in the context of $d$-wave superconductivity, with a negative answer: the scattering of the subgap excitations (massless Dirac fermions) by the vortex lattice obscures the Landau level quantization. Here we show that the same question has a positive answer for a Weyl superconductor: the chirality of the Weyl fermions protects the zeroth Landau level by means of a topological index theorem. As a result, the heat conductance parallel to the magnetic field has the universal value $G=\frac{1}{2}{g}_0\mathrm{\Phi }/{\mathrm{\Phi }}_0$, with $\mathrm{\Phi }$ as the magnetic flux through the system, ${\mathrm{\Phi }}_0$ as the superconducting flux quantum, and $g_0$ as the thermal conductance quantum.

Figure 1

Excitation spectrum of a nodal superconductor in zero magnetic field (black dashed curves) and in the mixed phase with a square lattice of Abrikosov vortices (red solid curves) [33]. (a) 2D $d$-wave superconductor. (b) 3D Weyl superconductor (with $k_z=\pi /3$ at the Weyl point). The momentum follows a path through the magnetic Brillouin zone of Fig. 2. The location of the zero-field Dirac and Weyl points is indicated by green arrows. The $n$th Landau level is expected at $E_n=\sqrt{n}{E}_1$, with $E_1=2\sqrt{\pi }{v}_F/d_0$. In the $d$-wave superconductor, the Landau levels are destroyed by the vortex lattice [6], while in the Weyl superconductor, they are protected by chiral symmetry.

Figure 2

Weyl superconductor in the mixed phase. (a) A Weyl semimetal-superconductor heterostructure (layers of a topological insulator, with perpendicular magnetization $\beta$, separated by $s$-wave superconducting spacer layers [20]). A magnetic field $B_0$ is applied perpendicular to the layers. The heterostructure has lattice constant $a_0$, while the square vortex array has lattice constant $d_0$ (with two $h/2e$ vortices per unit cell). (b),(c) Two different paths through the magnetic Brillouin zone of the vortex array.

Figure 3

Same as Fig. 1, but now as a function of $k_z$ for $k_x=0=k_y$ at the center of the Brillouin zone [47]. The color scale indicates the charge expectation value. The dashed curve is the dispersion (15) of the zeroth Landau level, calculated analytically for $K\ll 1$ (which explains the deviation from the numerics). The effective charge at $E=0$ is $\pm 0.73$, close to the analytical prediction of $\pm \kappa =\pm 1/\sqrt{2}$.

Figure 4

Color scale plot of $|\psi (x,y){|}^2$ in the zeroth Landau level of the Weyl superconductor [48]. The white dashed lines indicate the vortex array, with a pair of $h/2e$ vortices in each unit cell. On approaching a vortex core, when the separation $\delta r\to 0$, the density diverges as a power law $|\psi {|}^2\propto \delta r^{1/\sqrt{2}-1}$, in accord with Eq. (13).