Non-Abelian topological orders and Majorana fermions in spin-singlet superconductors

The non-Abelian topological order for superconductors is characterized by the existence of zero-energy Majorana fermions in edges of systems and in a vortex of a macroscopic condensate, which obey the non-Abelian statistics. This paper is devoted to an extensive study on the non-Abelian topological phase of spin-singlet superconductors with the Rashba spin-orbit interaction proposed in our previous paper [M. Sato, Y. Takahashi, and S. Fujimoto, Phys. Rev. Lett. 103, 020401 (2009)]. We mainly consider the $s$-wave pairing state and the $d+id$ pairing state. In the case of $d+id$-wave pairing, Majorana fermions appear in almost all parameter regions of the mixed state under an applied magnetic field, provided that the Fermi level crosses $k$ points in the vicinity of the $\Gamma$ point or the M point in the Brillouin zone while in the case of $s$-wave pairing, a strong magnetic field, the Zeeman energy of which is larger than the superconducting gap is required to realize the topological phase. We clarify that Majorana fermions in Rashba spin-singlet superconductors are much more stable than those realized in spin-triplet $p+ip$ superconductors in certain parameter regions. We also investigate the topological number which ensures the topological stability of the phase in detail. Furthermore, as a by-product, we found that topological order is also realized in conventional spin (or charge)-density wave states with the Rashba spin-orbit interaction, for which massless Dirac fermions appear in the edge of the systems and charge fractionalization occurs.

Figure 1

(Color online) The diagrams of the non-Abelian topological phase for spin-singlet NCS superconductors. (a) $s$-wave case. (b) $d+id$-wave case with $\Delta \left(\mathbf{k}\right)={\Delta }_d^{\left(1\right)}\left(\text{cos}{k}_y-\text{cos}{k}_x\right)+i{\Delta }_d^{\left(2\right)}\text{sin}{k}_x\text{sin}{k}_y$. (c) $d+id$-wave case with $\Delta \left(\mathbf{k}\right)={\Delta }_d^{\left(1\right)}\left({\text{sin}}^2{k}_x-{\text{sin}}^2{k}_y\right)+i{\Delta }_d^{\left(2\right)}\text{sin}{k}_x\text{sin}{k}_y$. The topological numbers $\mathbf{\left(}{\left(-1\right)}^{{\nu }_{\text{Ch}}};I\left(0\right),I\left(\pi \right)\mathbf{\right)}$ are given only in the phases supporting a non-Abelian topological order. In each case, there are four different non-Abelian topological phases. In the $s$-wave case, the non-Abelian topological phase is realized only when the Zeeman magnetic field satisfies ${\left({\mu }_{\text{B}}{H}_z\right)}^2>{\left({\Delta }_s\right)}^2$ but in the $d+id$-wave cases, the non-Abelian topological phase can be realized even for a small but nonzero $H_z$.

Figure 2

(Color online) Topological phase transition at ${\mu }_{\text{B}}{H}_z={\Delta }_s$. BZ(I) and BZ(II) indicate the Brillouin zones after and before the transition, respectively. The gap closes at $\left(k_x,k_y\right)=\left(0,0\right)$ when ${\mu }_{\text{B}}{H}_z={\Delta }_s$. ${\text{S}}_i$ and ${\text{S}}_i^{\prime }$ $\left(i=1,2\right)$ are the side faces of the rectangle in which the top and bottom faces are BZ(I) and BZ(II). For simplicity, only ${\text{S}}_1$ and ${\text{S}}_1^{\prime }$ are explicitly indicated.

Figure 3

(Color online) The energy spectra of the 2D $s$-wave NCS superconductor with open edges at $i_x=0$ and $i_x=30$ for $\mu <-2t$. Here $k_y$ denotes the momentum in the $y$ direction, and $k_y∊\left[-\pi ,\pi \right]$. We take $t=1$, $\mu =-2.5$, $\lambda =0.5$, and ${\Delta }_s=1$. The Zeeman magnetic field $H_z$ is (I) ${\mu }_{\text{B}}{H}_z=0$, (II) ${\mu }_{\text{B}}{H}_z=2$, (III) ${\mu }_{\text{B}}{H}_z=3$, and (IV) ${\mu }_{\text{B}}{H}_z=7$. The cases of (I)–(IV) correspond to, respectively, the four regions in Table , (a). The non-Abelian phases are (II) and (III). The two gapless modes found in (II) correspond to edge modes for two open edges, respectively. Thus, they are chiral. The same is also true for the gapless modes in (III).

Figure 4

(Color online) The energy spectra of the 2D $s$-wave NCS superconductor with open edges at $i_x=0$ and $i_x=30$ for $-2t<\mu <0$. Here $k_y$ denotes the momentum in the $y$ direction, and $k_y∊\left[-\pi ,\pi \right]$. We take $t=1$, $\mu =-1$, $\lambda =0.5$, and ${\Delta }_s=1$. The Zeeman magnetic field $H_z$ is (I) ${\mu }_{\text{B}}{H}_z=0$, (II) ${\mu }_{\text{B}}{H}_z=2$, (III) ${\mu }_{\text{B}}{H}_z=4$, and (IV) ${\mu }_{\text{B}}{H}_z=6$. The cases of (I)–(IV) correspond to, respectively, the four regions in Table , (b). The non-Abelian phase is (III).

Figure 5

(Color online) The energy spectra of the 2D $d+id$-wave NCS superconductor [$\Delta \left(\mathbf{k}\right)={\Delta }_d^{\left(1\right)}\left(\text{cos}{k}_y-\text{cos}{k}_x\right)+i{\Delta }_d^{\left(2\right)}\text{sin}{k}_x\text{sin}{k}_y$] with open edges at $i_x=0$ and $i_x=30$ for $\mu <-2t+{\left({\Delta }_d^{\left(1\right)}\right)}^2/2t$. Here $k_y$ denotes the momentum in the $y$ direction and $k_y∊\left[-\pi ,\pi \right]$. We take $t=1$, $\mu =-2.5$, $\lambda =0.5$, ${\Delta }_d^{\left(1\right)}=0.5$, and ${\Delta }_d^{\left(2\right)}=0.8$. The Zeeman magnetic field $H_z$ is (I) ${\mu }_{\text{B}}{H}_z=0$, (II) ${\mu }_{\text{B}}{H}_z=2$, (III) ${\mu }_{\text{B}}{H}_z=3$, and (IV) ${\mu }_{\text{B}}{H}_z=7$. The cases of (I)–(IV) correspond to, respectively, the four regions in Table , (a). The non-Abelian phases are (II) and (III).

Figure 6

(Color online) The energy spectra of the 2D $d+id$-wave NCS superconductor [$\Delta \left(\mathbf{k}\right)={\Delta }_d^{\left(1\right)}\left(\text{cos}{k}_y-\text{cos}{k}_x\right)+i{\Delta }_d^{\left(2\right)}\text{sin}{k}_x\text{sin}{k}_y$] with open edges at $i_x=0$ and $i_x=30$ for $-2t+{\left({\Delta }_d^{\left(1\right)}\right)}^2/2t<\mu <0$. Here $k_y$ denotes the momentum in the $y$ direction and $k_y∊\left[-\pi ,\pi \right]$. We take $t=1$, $\mu =-1$, $\lambda =0.5$, ${\Delta }_d^{\left(1\right)}=0.5$, and ${\Delta }_d^{\left(2\right)}=0.8$. The Zeeman magnetic field $H_z$ is (I) ${\mu }_{\text{B}}{H}_z=0$, (II) ${\mu }_{\text{B}}{H}_z=1.6$, (III) ${\mu }_{\text{B}}{H}_z=3.1$, and (IV) ${\mu }_{\text{B}}{H}_z=6$. The cases of (I)–(IV) correspond to, respectively, the four regions in Table , (b). The non-Abelian phase is (III).

Figure 7

(Color online) The energy spectra of the 2D $d+id$-wave NCS superconductor [$\Delta \left(\mathbf{k}\right)={\Delta }_d^{\left(1\right)}\left({\text{sin}}^2{k}_x-{\text{sin}}^2{k}_y\right)+i{\Delta }_d^{\left(2\right)}\text{sin}{k}_x\text{sin}{k}_y$] with open edges at $i_x=0$ and $i_x=30$ for $\mu <-2t$. Here $k_y$ denotes the momentum in the $y$ direction and $k_y∊\left[-\pi ,\pi \right]$. We take $t=1$, $\mu =-2.5$, $\lambda =0.5$, ${\Delta }_d^{\left(1\right)}=1$, and ${\Delta }_d^{\left(2\right)}=1$. The Zeeman magnetic field $H_z$ is (I) ${\mu }_{\text{B}}{H}_z=0$, (II) ${\mu }_{\text{B}}{H}_z=2$, (III) ${\mu }_{\text{B}}{H}_z=3.5$, and (IV) ${\mu }_{\text{B}}{H}_z=7$. The cases of (I)–(IV) correspond to, respectively, the four regions in Table , (a). The non-Abelian phases are (II) and (III).

Figure 8

(Color online) The energy spectra of the 2D $d+id$-wave NCS superconductor [$\Delta \left(\mathbf{k}\right)={\Delta }_d^{\left(1\right)}\left({\text{sin}}^2{k}_x-{\text{sin}}^2{k}_y\right)+i{\Delta }_d^{\left(2\right)}\text{sin}{k}_x\text{sin}{k}_y$] with open edges at $i_x=0$ and $i_x=30$ for $-2t<\mu <0$. Here $k_y$ denotes the momentum in the $y$ direction and $k_y∊\left[-\pi ,\pi \right]$. We take $t=1$, $\mu =-1$, $\lambda =0.5$, ${\Delta }_d^{\left(1\right)}=1$, and ${\Delta }_d^{\left(2\right)}=1$. The Zeeman magnetic field $H_z$ is (I) ${\mu }_{\text{B}}{H}_z=0$, (II) ${\mu }_{\text{B}}{H}_z=2$, (III) ${\mu }_{\text{B}}{H}_z=3.5$, and (IV) ${\mu }_{\text{B}}{H}_z=6$. The cases of (I)–(IV) correspond to, respectively, the four regions in Table , (b). The non-Abelian phase is (III).

Figure 9

(Color online) The energy spectra of the NCS SDW state with edges at $i_x=0$ and $i_x=30$. Here $k_y$ denotes the momentum in the $y$ direction and $k_y∊\left[-\pi ,\pi \right]$. We take $t=1=\alpha =1={\Delta }_{\text{S}}=1$. The Zeeman magnetic field $H_z$ is ${\mu }_{\text{B}}{H}_z=1.5$. At $k_y=0,\pm \pi /2,\pi$, we have gapless states on each edge. These gapless edge modes are chiral.

Figure 10

(Color online) The energy spectra of the NCS CDW state with edges at $i_x=0$ and $i_x=30$. Here $k_y$ denotes the momentum in the $y$ direction and $k_y∊\left[-\pi ,\pi \right]$. We take $t=1$, $\mu =0$, $\lambda =0.5$, and ${\Delta }_{\text{C}}=1$. The Zeeman magnetic field $H_z$ is ${\mu }_{\text{B}}{H}_z=1.5$.