Topological superconductivity in quantum Hall–superconductor hybrid systems

We develop a scenario to engineer a topological phase with Majorana edge states based on an integer quantum Hall (QH) system proximity coupled to a superconductor (SC). Due to the vortices in the SC order parameter, the SC–QH hybrid system is described by a Bloch problem with ten unpaired momenta, corresponding to the maxima and saddle points of the SC order parameter. For external potentials respecting the symmetry of the vortex lattice, the states with unpaired momenta have degeneracies such that the system always is in a trivial phase. However, an incommensurate potential can lift the degeneracies and drive the system into a topologically nontrivial phase.

Figure 1

(a) Real space structure of the $s$-wave pairing potential $\mathrm{\Delta }\left(\mathbf{r}\right)$ with triangular vortex lattice in the LLL approximation. The arrows denote the lattice vectors ${\mathbf{d}}_1=a_y{\mathbf{e}}_y$ and ${\mathbf{d}}_2=(a_x{\mathbf{e}}_x+a_y{\mathbf{e}}_y)/2$. (b) Brillouin zone of the rectangular unit cell comprising two vortices, and induced $p$-wave pairing potential ${\mathrm{\Delta }}_{\mathbf{k}}\sim {\partial }_x\mathrm{\Delta }(\mathbf{r}=l_B^2\mathbf{k}\times \widehat{z})$. Markers denote the unpaired momenta, which correspond to the $M$ points (dots) and the $K,K^{\prime }$ points (crosses) of the hexagonal Brillouin zone of the vortex lattice. By pairwise coupling the $M$ points, the system can by driven into a topologically nontrivial phase.

Figure 2

(a) Excitation spectrum $E_{\mathbf{k}}=\left|{\mathrm{\Delta }}_{\mathbf{k}}\right|$. All unpaired momenta have zero energy. (b) Coupling of unpaired momenta $(0,0)$ and $(0,\pi )$ by the external potential $V\left(\mathbf{r}\right)=Wcos(y{Q}_y/2)$ lifts their degeneracy, and (c) excitation spectrum as function of chemical potential for $W={\mathrm{\Delta }}_0/2$. Shaded: Bulk spectrum for periodic BC in $x$ direction. Lines: Majorana edge mode for open BC in $x$ direction.

Figure 3

Excitation spectrum as function of chemical potential for a potential $V\left(\mathbf{r}\right)$ with dominating Fourier coefficient for $\mathbf{q}=\pm {\mathbf{e}}_y{Q}_y/2$, and Gaussian disorder with $\overline{V^2}={\mathrm{\Delta }}_0^2/6,L_x=48a_x$, and $L_y=48a_y$. Shaded: Bulk spectrum for periodic BC in $x$ direction. Lines: Majorana edge mode for open BC in $x$ direction. Inset: Fourier coefficients of the potential $V\left(\mathbf{r}\right)$ as function of momentum $\mathbf{q}$.