Non-Abelian topological order in noncentrosymmetric superconductors with broken time-reversal symmetry

We consider two-dimensional noncentrosymmetric superconductors, in which the order parameter is a mixture of $s$-wave and $p$-wave parts, in the presence of an externally induced Zeeman splitting. We derive the conditions under which the system is in a non-Abelian phase. By considering the nondegenerate zero-energy Majorana solutions of the Bogoliubov-de Gennes (BdG) equations for a vortex and by constructing a topological invariant, we show that the condition for the non-Abelian phase to exist is completely independent of the triplet pairing amplitude. The existence condition for the non-Abelian phase derived from the real-space solutions of the BdG equations involves the Pfaffian of the BdG Hamiltonian at $k=0$, which is completely insensitive to the magnitude of the $p$-wave component of the order parameter. We arrive at the same conclusion by using the appropriate topological invariant for this case. This is in striking contrast to the analogous condition for the time-reversal invariant topological phases, in which the amplitude of the $p$-wave component must be larger than the amplitude of the $s$-wave piece of the order parameter. As a by-product, we establish the intrinsic connection between the Pfaffian of the BdG Hamiltonian at $k=0$ (which arises at the BdG approach) and the relevant $\mathbb{Z}$ topological invariant.

Figure 1

(Color online) Plot shows how the energy-gap scales with the ratio ${\Delta }_t/{\Delta }_s$ for $V_{\mathcal{Z}}=6.0$, $\mu =4.0$, $\alpha =1.0$, and different values of ${\Delta }_s$ (all energy scales are in units of $\eta ={\hslash }^2/2m$). The discontinuity in the gap for ${\Delta }_s=3.0$, ${\Delta }_s=2.0$, and ${\Delta }_s=1.5$ are due to the global minimum of the gap function switching from $k\ne 0$ to $k=0$ (see inset of Fig. 2). Eventually, for very large values of ${\Delta }_s$, the global minimum always sits at $k=0$ and there is no gap closing at any value of ${\Delta }_t/{\Delta }_s$.

Figure 2

Plot shows how the ratio ${\Delta }_t/{\Delta }_s$ at gap closing scales with the Zeeman splitting $V_{\mathcal{Z}}$ for $\alpha =1$, $\mu =1$, and ${\Delta }_s=0.5$ (in units of $\eta ={\hslash }^2/2m$). Note that, since $V_{\mathcal{Z}}^2>{\mu }^2+{\Delta }_s^2$ for the range of parameters in the plot, the topological invariant $C_1=1$ in spite of the gap closing. Inset shows the dependence on ${\Delta }_s$ of the ratio ${\left({\Delta }_t/{\Delta }_s\right)}_0$ at which global minimum of the gap shifts from $k_x,k_y\ne 0$ to the origin. For large ${\Delta }_s$, the global minimum is always at $k_x=k_y=0$. Following values have been used: $V_{\mathcal{Z}}=6.0$, $\mu =4.0$, and $\alpha =1.0$ (in units of $\eta ={\hslash }^2/2m$).

Figure 3

(Color online) Nature of roots for Eq. (8) for the following parameters: $\alpha =1$, ${\Delta }_s=1$, ${\Delta }_t=1$, and (a) $\mu =4$, $V_{\mathcal{Z}}=2$, and $\lambda =+1$, (b) $\mu =4$, $V_{\mathcal{Z}}=2$, and $\lambda =-1$, (c) $\mu =2$, $V_{\mathcal{Z}}=4$, and $\lambda =+1$, and (d) $\mu =2$, $V_{\mathcal{Z}}=4$, and $\lambda =-1$ (in units of $\eta ={\hslash }^2/2m$).