Topological phases of noncentrosymmetric superconductors: Edge states, Majorana fermions, and non-Abelian statistics

The existence of edge states and zero-energy modes in vortex cores is a hallmark of topologically nontrivial phases realized in various condensed-matter systems such as the fractional quantum Hall states, $p+ip$ superconductors, and ${\mathit{Z}}_2$ insulators (quantum spin Hall state). We examine this scenario for two-dimensional noncentrosymmetric superconductors which allow the parity mixing of Cooper pairs. It is found that even when the $s$-wave pairing gap is nonzero, provided that the superconducting gap of spin-triplet pairs is larger than that of spin-singlet pairs, gapless edge states and zero-energy Majorana modes in vortex cores emerge, characterizing topological order. Furthermore, it is shown that for Rashba superconductors, the quantum spin Hall effect produced by gapless edge states exists even under an applied magnetic field which breaks time-reversal symmetry provided that the field direction is perpendicular to the propagating direction of the edge modes. This result making a sharp contrast to the ${\mathit{Z}}_2$ insulator is due to an accidental symmetry inherent in the Rashba model. It is also demonstrated that in the case with magnetic fields, the non-Abelian statistics of vortices is possible under a particular but realistic condition.

Figure 1

Schematic picture of phase diagram of 2D NCS with Rashba coupling.

Figure 2

The energy spectra of the 2D NCS with edges at $i_x=0$ and $i_x=50$ in the absence of magnetic field. Here $k_y$ denotes the momentum in the $y$ direction. We take $t=1$, $\mu =-3$, and $\alpha =0.6$. (a) NCS with dominating $p$-wave paring. ${\Delta }_t=0.6$ and ${\Delta }_s=0.1$. (b) NCS with purely $s$-wave paring. ${\Delta }_t=0$ and ${\Delta }_s=0.6$.

Figure 3

The energy spectra of 2D NCS with edges at $i_x=0$ and $i_x=50$ in phases I–IV in Fig. 1a. Here $k_y$ denotes the momentum in the $y$ direction. We take $t=1$, $\mu =-3$, $\alpha =0.6$, ${\Delta }_t=0.6$, and ${\Delta }_s=0.1$. $H_z$ is (I) ${\mu }_B{H}_z=0$, (II) ${\mu }_B{H}_z=1.5$, (III) ${\mu }_B{H}_z=5$, and (IV) ${\mu }_B{H}_z=8$.

Figure 4

The energy spectra of 2D NCS with edges at $i_x=0$ and $i_x=50$ in phases ${\text{I}}^{\prime }$, ${\text{II}}^{\prime }$, ${\text{III}}^{\prime }$, and ${\text{IV}}^{\prime }$ in Fig. 1b. Here $k_y$ denotes the momentum in the $y$ direction. We take $t=1$, $\mu =-1$, $\alpha =0.6$, ${\Delta }_t=0.6$, and ${\Delta }_s=0.1$. $H_z$ is $\left({\text{I}}^{\prime }\right)$ ${\mu }_B{H}_z=0$, $\left({\text{II}}^{\prime }\right)$ ${\mu }_B{H}_z=1$, $\left({\text{III}}^{\prime }\right)$ ${\mu }_B{H}_z=3$, and $\left({\text{IV}}^{\prime }\right)$ ${\mu }_B{H}_z=5$.

Figure 5

The dependence of gapless edge states in phase I on the direction of the magnetic field. We take $t=1$, $\mu =-3$, $\alpha =0.6$, ${\Delta }_t=0.6$, and ${\Delta }_s=0,1$. (a) ${\mu }_B{H}_x=0.15$, ${\mu }_B{H}_y=0$, and ${\mu }_B{H}_z=0$. (b) ${\mu }_B{H}_x=0$, ${\mu }_B{H}_y=0.15$, and ${\mu }_B{H}_z=0$. A similar dependence is obtained for gapless edge states in phase ${\text{I}}^{\prime }$.

Figure 6

Hall bar with six terminals used for the detection of edge states. Solid and dotted gray lines with arrows represent propagating edge modes with different spin chiralities corresponding to two SO split bands. These modes propagate in opposite directions.

Figure 7

(Color online) The density of quasiparticles for the purely $p$-wave state plotted on the $81\times 81$ $xy$ plane for some low energies $E$. The top and bottom panels are, respectively, the plots of ${\left|u_{\uparrow }\left(\mathit{r}\right)\right|}^2+{\left|u_{\downarrow }\left(\mathit{r}\right)\right|}^2$ and ${\left|v_{\uparrow }\left(\mathit{r}\right)\right|}^2+{\left|v_{\downarrow }\left(\mathit{r}\right)\right|}^2$ for $\alpha =t$, ${\Delta }_t=0.05t$, ${\mu }_B{H}_z=0$, and $\mu =-3.75t$. $E=4.87\times {10}^{-4}t$ (left), $E=7.67\times {10}^{-4}t$ (middle), and $E=1.023\times {10}^{-3}t$ (right). For $E=4.87\times {10}^{-4}t$ and $E=1.023\times {10}^{-3}t$, edge states appear.

Figure 8

(Color online) The density of quasiparticles for the purely $p$-wave state plotted on the $81\times 81$ $xy$ plane for low energies. The top and bottom panels are, respectively, the plots of ${\left|u_{\uparrow }\left(\mathit{r}\right)\right|}^2+{\left|u_{\downarrow }\left(\mathit{r}\right)\right|}^2$ and ${\left|v_{\uparrow }\left(\mathit{r}\right)\right|}^2+{\left|v_{\downarrow }\left(\mathit{r}\right)\right|}^2$. $\alpha =t$, ${\Delta }_t=0.05t$, ${\mu }_B{H}_z=0.04t$, and $\mu =-4.0t$. $E=0.00123t$ (left), $E=0.0182t$ (middle), and $E=0.0315t$ (right). Edge states appear for $E=0.0182t$ and $E=0.0315t$.

Figure 9

(Color online) The density of quasiparticles for the $s+p$-wave state plotted on the $81\times 81$ $xy$ plane for low energies. The top and bottom panels are, respectively, the plots of ${\left|u_{\uparrow }\left(\mathit{r}\right)\right|}^2+{\left|u_{\downarrow }\left(\mathit{r}\right)\right|}^2$ and ${\left|v_{\uparrow }\left(\mathit{r}\right)\right|}^2+{\left|v_{\downarrow }\left(\mathit{r}\right)\right|}^2$. $\alpha =t$, ${\Delta }_t=0.05t$, ${\Delta }_s=-0.02t$, ${\mu }_B{H}_z=0$, and $\mu =-4.0t$. $E=0.0023t$ (left), $E=0.039t$ (middle), and $E=0.0434t$ (right). Edge states appear for $E=0.0434t$.