Type-I and type-II topological nodal superconductors with $s$-wave interaction

Topological nodal superconductors with protected gapless points in momentum space are generally realized based on unconventional pairings. In this work we propose a minimal model to realize these topological nodal phases with only $s$-wave interaction. In our model the linear and quadratic spin-orbit couplings along the two orthogonal directions introduce anisotropic effective unconventional pairings in momentum space. This model may support different nodal superconducting phases characterized by either an integer winding number in BDI class or a ${\mathbb{Z}}_2$ index in D class at the particle-hole invariant axes. In the vicinity of the nodal points the effective Hamiltonian can be described by either type-I or type-II Dirac equations, and the Lifshitz transition from type-I nodal phases to type-II nodal phases can be driven by external in-plane magnetic fields. We show that these nodal phases are robust against weak impurities, which only slightly renormalizes the momentum-independent parameters in the impurity-averaged Hamiltonian, thus these phases are possible to be realized in experiments with real semi-Dirac materials. The smoking-gun evidences to verify these phases based on scanning tunneling spectroscopy method are also briefly discussed.

Figure 1

A typical single gapless point in the whole Brillouin zone in semi-Dirac metals.

Figure 2

Phase diagram for the physical model in Eq. (3). The line $X=4$ represents the phase transition at $k_x=\pi$, and the lines $Y=X-4$ and $Y=X$ represent the transition at $k_x=0$. The shadowed regimes mark the topological gapped trivial phase ${\mathrm{TG}}^{\text{I}}(0,0)$. In the nodal TSCs, the symbols $\pm$ in the black circles denote the winding number of the nodal points.

Figure 3

Edge states for a strip with width $L=100$ along $k_y$ direction. (a)–(c) The Majorana flat bands in ${\mathrm{TG}}^{\text{I}}(2,2)$, ${\mathrm{TG}}^{\text{I}}(2,0)$, and ${\mathrm{TG}}^{\text{I}}(0,2)$ phases, respectively. The corresponding spectra are shown in (d)–(f).

Figure 4

Phase diagrams in the presence of magnetic field $h_z$ and $h_y$. (a) and (b) The results with $\left|Z\right|<1$ and $\left|Z\right|>1$, respectively. All topological TSCs are denoted by ${\mathrm{TG}}^{\text{I}}(n,m)$ and the shadowed regimes mark the trivial phase ${\mathrm{TG}}^{\text{I}}(0,0)$.

Figure 5

(a) Band structure for a typical type-II Dirac point. (b) Phase diagram for the two nodal points in the model with $h_x$ and $h_z$. Parameters are $\beta =1.2\alpha$, $\gamma =\alpha$, $\mathrm{\Delta }=0.2\alpha$, $\mu =0.6\alpha$, $h_y=0.3\alpha$. In (a) $h_x=0.5\alpha$, $h_z=-0.1\alpha$, $k_y=0.353$.

Figure 6

(a) The DOS in the bulk and (b) the LDOS at the surface for the fully gapped trivial phase (red solid line), type-I nodal TSCs (blue dashed line) and type-II nodal TSCs (purple dotted points). For type-I TSCs, $\mathrm{\Delta }=1.2\alpha$, $\mu =1.8\alpha$, $h_x=0$, and for the type-II TSCs, $\mathrm{\Delta }=1.2\alpha$, $\mu =1.8\alpha$, $h_x=0.3\alpha$, and for the fully gapped phase, $\mathrm{\Delta }=3.5\alpha$, $\mu =1.0\alpha$. Other parameters are $\beta =1.2\alpha$, $\gamma =\alpha$. The DOS in the bulk is determined by $\rho \left(\omega \right)=-\frac{1}{\pi }{\sum }_{\mathbf{k}}\text{Tr}\left[\text{Im}{\mathcal{G}}_{\text{r}}(\omega ,\mathbf{k})\right]$, where the retarded Green's function is defined as ${\mathcal{G}}_{\text{r}}(\omega ,\mathbf{k})=\mathcal{G}(i\omega \to \omega +i\delta ,\mathbf{k})$. The LDOS at the surface is computed via the Green function iteration method in Ref. [66].

Figure 7

Self-energy $\mathrm{\Sigma }\left(i\omega \right)$ from random impurity scattering, in which only the lowest diagram is considered. ${\mathcal{G}}_0$ and $\mathcal{G}$ represent the free Green's function and impurity-averaged Green's function, respectively.