Topological superconductivity in Landau levels

The intense search for topological superconductivity is inspired by the prospect that it hosts Majorana quasiparticles. We explore in this work the optimal design for producing topological superconductivity by combining a quantum Hall state with an ordinary superconductor. To this end, we consider a microscopic model for a topologically trivial two-dimensional $p$-wave superconductor exposed to a magnetic field and find that the interplay of superconductivity and Landau level physics yields a rich phase diagram of states as a function of $\mu /t$ and $\mathrm{\Delta }/t$, where $\mu ,t$, and $\mathrm{\Delta }$ are the chemical potential, hopping strength, and the amplitude of the superconducting gap. In addition to quantum Hall states and topologically trivial $p$-wave superconductor, the phase diagram also accommodates regions of topological superconductivity. Most importantly, we find that application of a nonuniform, periodic magnetic field produced by a square or a hexagonal lattice of $h/e$ fluxoids greatly facilitates regions of topological superconductivity in the limit of $\mathrm{\Delta }/t\to 0$. In contrast, a uniform magnetic field, a hexagonal Abrikosov lattice of $h/2e$ fluxoids, or a one-dimensional lattice of stripes produces topological superconductivity only for sufficiently large $\mathrm{\Delta }/t$.

Figure 1

(a) Experimental setup for a hybrid system with two-dimensional electron gas with $p$-wave superconducting grid on top under an external magnetic field. (b), (c) Schematic of a unit cell of $N\times N$ lattice points containing one flux quantum (b) of a square shape, (c) in stripe geometry. In each unit cell a flux quantum ${\varphi }_0$ penetrates through the yellow shaded zone of (b) $M\times M$ (c) $N\times M$ lattice points. The figures represent $N=4$ and $M=2$. The size of unit cell is $a$ and spacing between lattice points is $a_1=a/N$.

Figure 2

(a) Energy dispersion in the presence of uniform magnetic field $\left(M=8\right)$ and nonuniform magnetic fields $\left(M=4,6\right)$ for $N=8$ and $\mathrm{\Delta }=0$. (b) Energy dispersion near the Fermi energy for various values of chemical potential $\mu$ in the presence of uniform magnetic field $\left(M=8\right)$ with $N=8$ and $\mathrm{\Delta }=0.1$.

Figure 3

Phase diagram of states for various geometries. Each phase is characterized by its Chern number $\mathcal{N}$, with different Chern numbers shown in different colors. The gray hashed region is an insulator, the colored hashed regions represent even integer $\mathcal{N}$, which correspond to QH effect, and the solid colors depict the topological superconductor phase with odd Chern numbers. The different panels show: (a) a uniform magnetic field $\left(N=8,M=8\right)$, (b) a nonuniform periodic magnetic field of a square shape $\left(N=8,M=6\right)$, and (c) a nonuniform magnetic field in stripe geometry $\left(N=8,M=6\right)$.

Figure 4

(a) When a gap closes at a momentum ${\mathit{k}}_o$ that does not lie at a symmetry point, the energy dispersion at $-{\mathit{k}}_o$ also undergoes simultaneous gap closing. Consequently, Chern number changes by two on the phase boundary. The two red solid circles indicate the two momenta ${\mathit{k}}_o$ and $-{\mathit{k}}_o$ where the electron and the hole bands touch simultaneously for $\mu =0.38$ and $\mathrm{\Delta }=0.243$ with $N=M=8$. (b) The energy dispersions along the blue dashed line in (a) for $\mathrm{\Delta }=0.243$ and several values of $\mu$ with $N=M=8$. The semimetal phase with electron and hole pockets occurs at $\mu =0.38$ (which is the phase boundary where $\mathcal{N}$ changes by 2) and $\mu =0.36$.

Figure 5

Phase diagram under (a) uniform magnetic field $\left(N=8,M=8\right)$; (b) nonuniform magnetic field $\left(N=8,M=6\right)$. On the boundary when Chern number changes by two, we observe a semimetallic phase on the finite region, which is displayed by black thick solid lines.

Figure 6

Scaled energy dispersion $N^{2}E$ for $N=8,16,32$ under (a) uniform magnetic field $\left(N=M\right)$; (b) nonuniform magnetic field $\left(M/N=3/4\right)$.

Figure 7

Phase diagram in the plane of scaled chemical potential $N^2\mu$ and scaled SC pairing parameter $N\mathrm{\Delta }$ for $N=8,16,32$ under nonuniform magnetic field $\left(M/N=3/4\right)$.

Figure 8

(a) Schematic of a unit cell and (b) phase diagram for a triangular lattice of $h/2e$ fluxoids of hexagonal shape $\left(N=8,M=6\right)$. One unit cell is composed of $N\times 2N$ lattice points and two $h/2e$ fluxoids in hexagonally shaped yellow-shaded regions. The figure in (a) is an example of $N=4$ and $M=2$.

Figure 9

(a) Schematic of a unit cell and (b) phase diagram under periodic $h/e$ fluxoids of hexagonal shape $\left(N=8,M=6\right)$ in triangular lattice. One unit cell is composed of $N\times N$ lattice points and a yellow-shaded zone of hexagon shape with $3M^2/2$ triangles contain $h/e$ fluxoids. The figure in (a) is an example of $N=4$ and $M=2$.

Figure 10

The top panel (a) shows the phase diagram for a superconductor with $p_x$-wave pairing and the middle panel (b) with $p_y$-wave pairing, with the magnetic field configuration having stripes along the $x$ direction $\left(N=8,M=6\right)$. The bottom panel (c) shows the phase diagram for a square lattice of ${\varphi }_0$ flux $\left(N=8,M=6\right)$ with $p_x$-wave pairing.