Majorana lattices from the quantized Hall limit of a proximitized spin-orbit coupled electron gas

Motivated by recent experiments demonstrating intricate quantum Hall physics on the surface of elemental bismuth, we consider proximity coupling an $s$-wave superconductor to a two-dimensional electron gas with strong Rashba spin-orbit interactions in the presence of a strong perpendicular magnetic field. We focus on the high-field limit so that the superconductivity can be treated as a perturbation to the low-lying Landau levels. In the clean case, wherein the superconducting order parameter takes the form of an Abrikosov vortex lattice, we show that a lattice of hybridized Majorana modes emerges near the plateau transition of the lowest Landau level. However, unless magnetic-symmetry-violating perturbations are present, the system always has an even number of chiral Majorana edge modes and thus is strictly speaking Abelian in nature, in agreement with previous work on related setups. Interestingly, however, a weak topological superconducting phase can very naturally be stabilized near the plateau transition for the square vortex lattice. The relevance of our findings to potential near-term experiments on proximitized materials such as bismuth will be discussed.

Figure 1

Spatial dependence of the pairing potential $\left|\mathrm{\Delta }\right(\mathbf{r}\left)\right|$ for the square (left) and triangular (right) Abrikosov vortex lattice solutions [see Eqs. (4) and (6)]. Also shown are the respective primitive translation vectors ${\mathbf{a}}_{1,2}$ (separated by the angle $\theta$) and magnetic unit cells (dashed rectangles).

Figure 2

Landau-level energies ${\epsilon }_{N\pm }$ [Eq. (10)] as a function of reduced Rashba coupling $g_R$ [Eq. (8)] evaluated at zero Zeeman coupling, $g_Z=0$. Levels labeled by $N\ge 1$ are obtained by diagonalizing ${\mathcal{H}}_N$ [Eq. (9)] in the subspace $\{|N,\uparrow \rangle ,|N-1,\downarrow \rangle \}$; the $N=0$ level corresponds to the unpaired state $|0,\uparrow \rangle$ with energy ${\epsilon }_0=\hslash {\omega }_c(\frac{1}{2}+g_Z)$.

Figure 3

Pairing gap function ${\mathrm{\Delta }}_{00}\left(\mathbf{k}\right)$ for the square (left column) and triangular (right column) vortex lattices. (Top) The amplitude $|{\mathrm{\Delta }}_{00}\left(\mathbf{k}\right)|$ is plotted on the $z$ axis, while the phase $arg{\mathrm{\Delta }}_{00}\left(\mathbf{k}\right)$ is represented by color. (Bottom) We show the locations of the Dirac nodes of ${\mathrm{\Delta }}_{00}\left(\mathbf{k}\right)$ in the magnetic Brillouin zone (mBZ). For each lattice, the mBZ is rectangular with an $M$ point located at $\mathbf{M}\left(\theta \right)=\left(\frac{Q}{4sin\theta },\frac{Q}{2}\right)$.

Figure 4

Same as in Fig. 3, but now plotting the pairing gap function ${\mathrm{\Delta }}_{10}\left(\mathbf{k}\right)$ which enters Eq. (27) [after being renormalized by $\beta (g_R,g_Z)$, cf. Fig. 5]. Dirac nodes with positive (negative) phase winding are indicated in the bottom panels by blue circles (orange crosses). As explained in Sec. 4, the gap function for the square lattice case (left) gives rise to weak TSC phases.

Figure 5

Reduced pairing strength ${\tilde{\mathrm{\Delta }}}_0$ in units of ${\mathrm{\Delta }}_0\equiv C_0/\sqrt{2}$ [cf. Eq. (6)] as a function of $g_R$ and $g_Z$. $\beta (g_R,g_Z)$ is merely the product of wave function amplitudes for the eigenstate of ${\mathcal{H}}_1$ with energy ${\epsilon }_{1-}$. Without spin-orbit coupling, the effective pairing strength vanishes, i.e., $\beta (g_R=0,g_Z)=0$; while for modest $g_R$, ${\tilde{\mathrm{\Delta }}}_0\sim {\mathrm{\Delta }}_0$, e.g., $\left|\beta \right(g_R=1,g_Z=0\left)\right|\approx 0.35$.

Figure 6

Phase diagram in the vicinity of the plateau transition for the square vortex lattice encountered upon sweeping the chemical potential $\mu$ through the lowest Rashba-coupled Landau level of energy ${\epsilon }_{1-}$ ($\mu$ is measured relative to ${\epsilon }_{1-}$). Here, $t_b$ is the bandwidth of the Landau level [see Eq. (30)], and the magnetic-symmetry-violating term $V_{\mathrm{CDW}}=0$. For a completely flat Landau level ($t_b=0$), there is a direct Chern number $C=0\to 1$ transition at $\mu =0$, while at finite $t_b$ the transition splits into intermediate $C=-1$ or 2 states. All phases consist of Majorana lattices; in particular, the $C=-1,2$ states are weak TSCs (see Sec. 4).

Figure 7

Total density of states, $\mathrm{DOS}\left(E\right)$, of the Landau-level projected BdG Hamiltonian, Eq. (27), versus chemical potential $\mu$ at fixed Landau-level bandwidth $t_b=0.15{\tilde{\mathrm{\Delta }}}_0>0$ with $V_{\mathrm{CDW}}=0$. The black arrows indicate the critical points ${\mu }_{c1}=-2t_b$ and ${\mu }_{c2}=0$.

Figure 8

Phase diagram near the plateau transition for the square lattice, now including finite $V_{\mathrm{CDW}}$. Since this term breaks the magnetic translation symmetry [see Eq. (31)], the transitions in Fig. 6 get split into multiple transitions involving intermediate non-Abelian states with half-integer $C$, as labeled in the diagram. The left and right panels correspond to different signs of $t_b$, and the phase diagram is symmetric under $V_{\mathrm{CDW}}\leftrightarrow -V_{\mathrm{CDW}}$.

Figure 9

Unpaired Majorana zero mode ${\chi }_{\mathrm{disloc}}$ occurring at the core of a lattice dislocation in the intermediate $C=-1,2$ phases of the square vortex lattice (see Figs. 6 and 8). The model pairing potential $\left|\mathrm{\Delta }\right(\mathbf{r}\left)\right|$ shown here is merely meant to be illustrative and, in contrast to Fig. 1, is not an accurate rendition of the actual form, Eq. (4), used throughout.