Stability of flat-band edge states in topological superconductors without inversion center

Nodal superconductors without inversion symmetry exhibit nontrivial topological properties, manifested by topologically protected flat-band edge states. Here we study the effects of edge roughness and strong edge disorder on the flat-band states using large-scale numerical simulations. We show that the bulk-edge correspondence remains valid for rough edges and demonstrate that midgap states generically appear at the boundary of nodal noncentrosymmetric superconductors, for almost all edge orientations. Moderately strong nonmagnetic disorder shifts some of the edge states away from zero energy, but does not change their total number. Strong spin-independent edge disorder, on the other hand, leads to the appearance of new weakly disordered midgap states in the layers adjacent to the disordered edge, i.e., at the interface between the bulk topological superconductor and the one-dimensional Anderson insulator formed by the strongly disordered edge layers. Furthermore, we show that magnetic impurities, which lift the time-reversal symmetry protection of the flat-band states, lead to a rapid decrease of the number of edge states with increasing disorder strength.

Figure 1

(a) Spin-orbit split Fermi surfaces and nodal points of the superconducting gap ${\Delta }_{\mathbf{k}}$, Eq. (1). Solid black dots indicate the location of the eight nodal points, where the gap of the quasiparticle spectrum closes. (b) Energy- and layer-resolved density of states ${\rho }_y\left(\omega \right)$ of the ($d_{xy}+p$)-wave superconductor in a ribbon geometry with (01) edge and width ${\mathcal{N}}_y=70$ lattice sites. (c) and (d) Edge band structure of the ($d_{xy}+p$)-wave superconductor in a ribbon geometry with a (01) and a (11) edge, respectively, as a function of edge momentum. Zero-energy flat bands appear at the (01) edge connecting the projected nodal points of the two helical Fermi surfaces. These flat bands give rise to a divergent density of states at $\omega =0$ [see panel (b)]. Note that there are also dispersing edge states, which lead to a feature in the edge density of states at $\omega \simeq \pm 0.3$.

Figure 2

Density plot of the edge-state wave function amplitudes $|{\Psi }_L{|}^2$ in a ($d_{xy}+p$)-wave superconducting dot with (a) smooth and (b) rough boundaries. Edge states are present both at an irregular but smooth boundary (a) and at a boundary with short-range disorder (b). Panel (c) shows the average number of edge states for an ensemble of randomly shaped superconducting dots with smooth (blue circles) and rough (red squares) boundaries as a function of circumference $L$ of the dot [45]. Here, the edge states are separated from the bulk states according to criterion (15) and by additionally requiring that the energy of the states is smaller than $0.1|{\Delta }_{\pm }|$ in absolute value. The solid black line represents the analytical approximation given by Eq. (17).

Figure 3

Layer-resolved local density of states ${\rho }_y\left(\omega \right)$ plotted for the first four outermost layers of a ($d_{xy}+p$)-wave superconducting ribbon with (01) edges in the presence of (a)–(e) “Gaussian” edge disorder and (f)–(j) “unitary” edge disorder (for details see text). The insets show the width $\Gamma$ and the area $A$ of the Lorentzian peaks at $\omega =0$ as a function of layer index $y$. The number in the lower left corner of the insets indicates the total area of the zero-bias peak as obtained by summing $A$ over the first four layers. In the clean case, $v_{\mathrm{imp}}=0$, the edge states penetrate only about two layers into the bulk [panel (a)]. For strong disorder the outermost layer shows signatures of localization, while new weakly disordered states appear in the second and third inward layers [panels (e) and (j)].

Figure 4

Local density of states summed over the four outermost layers, ${\rho }_{\mathrm{edge}}\left(\omega \right)=\frac{1}{4}{\sum }_{y=1}^4{\rho }_y\left(\omega \right)$, of a ($d_{xy}+p$)-wave superconducting ribbon with (01) edges in the presence of nonmagnetic [panels (a) and (f)] and magnetic impurities [panels (b)–(e) and (g)–(j)] in the two outermost layers. Two different disorder distributions are considered: (a)–(e) “Gaussian” disorder and (f)–(j) “unitary” disorder with ${\rho }_{\mathrm{imp}}=0.2$ (for details see text). Individual traces are vertically offset by $0.02$ from one another for clarity. The insets show the width $\Gamma$ and the area $A$ of the Lorentzian peaks at $\omega =0$ as a function of disorder strengths ${\gamma }_{\mathrm{imp}}$ and $v_{\mathrm{imp}}$, respectively.

Figure 5

Scheme of the recursive Green's function method. To accommodate next-nearest-neighbor hopping an effective building block of two columns is considered, as well as an effective coupling $V$. The local Green's function $G(\omega ;i)$ is calculated by attaching to the block $i$ the left ($G^L$) and right ($G^R$) ribbons according to Eq. (B7). The blue sites represent random on-site disorder potentials, defined by $H_{\mathrm{imp}}^{\beta }\left(x\right)$.