Complete zero-energy flat bands of surface states in fully gapped chiral noncentrosymmetric superconductors

Noncentrosymmetric superconductors can support flat bands of zero-energy surface states in part of their surface Brillouin zone. This requires that they obey time-reversal symmetry and have a sufficiently strong triplet-to-singlet-pairing ratio to exhibit nodal lines in the bulk. These bands are protected by a winding number that relies on chiral symmetry, which is realized as the product of time-reversal and particle-hole symmetry. We reveal a way to stabilize a flat band in the entire surface Brillouin zone, while the bulk dispersion is fully gapped. This idea could lead to a robust platform for quantum computation and represents an alternative route to strongly correlated flat bands in two dimensions, besides twisted bilayer graphene. The necessary ingredient is an additional spin-rotation symmetry that forces the direction of the spin-orbit-coupling vector not to depend on the momentum component normal to the surface. We define a winding number that leads to flat zero-energy surface bands due to bulk-boundary correspondence. We discuss under which conditions this winding number is nonzero in the entire surface Brillouin zone and verify the occurrence of zero-energy surface states by exact numerical diagonalization of the Bogoliubov–de Gennes Hamiltonian for a slab. In addition, we consider how a weak breaking of the additional symmetry affects the surface band, employing first-order perturbation theory and a quasiclassical approximation. We find that the surface states and the bulk gap persist for weak breaking of the additional symmetry but that the band does not remain perfectly flat. The broadening of the band strongly depends on the deviation of the spin-orbit-coupling vector from its unperturbed direction as well as on the spin-orbit-coupling strength and the triplet-pairing amplitude.

Figure 1

Parameter regime which ensures a nonzero winding number in the entire sBZ for the model of Sec. 3b. The boundary of the region in the three-dimensional space $(\tilde{\mu },\tilde{\lambda },{\mathrm{\Delta }}^s/\left(c_z{\mathrm{\Delta }}^t\right))$ where this is possible is shown in semi-transparent gray. The colored planes show the maximal possible value of ${\tilde{t}}_{xy}$ such that the winding number is still nonzero everywhere in the sBZ.

Figure 2

Energy and IPR of the surface state and the bulk state with the lowest energy at given ${\mathbf{k}}_{\parallel }=(k_x,k_y)$ in an ideal system with nonzero winding number $W_{\perp ,\sigma }\left({\mathbf{k}}_{\parallel }\right)$ in the full sBZ. (a) Surface-state energy and lowest bulk energy, illustrating the size of the gap. (b) Lowest bulk energy as a density plot. (c) IPR of the corresponding state, where low IPR corresponds to a delocalized state and high IPR corresponds to a strongly localized state. (d) Energy and (e) IPR of the surface state. Note that in panels (b) to (e) the same color scheme is used for different orders of magnitude.

Figure 3

Lowest eigenvalue of the Hamiltonian as a function of the thickness $Z$ for various values of the triplet pairing amplitude ${\mathrm{\Delta }}^t$. For sufficiently large values of ${\mathrm{\Delta }}^t$, the winding number ensures the existence of a zero-energy surface state in the limit of an infinitely thick slab, while for a slab of finite thickness, the energy of the surface state decreases exponentially. If ${\mathrm{\Delta }}^t$ is too small, there are no zero-energy surface states and the energy approaches a finite value. The inset shows the same data for small ${\mathrm{\Delta }}^t$ on a double logarithmic scale. The red line indicates the transition between zero-energy surface states and states at finite energy.

Figure 4

Energy and IPR of the surface state and the bulk state with the lowest energy at given ${\mathbf{k}}_{\parallel }=(k_x,k_y)$ for the $C_4$ point group and a SOC vector with nonzero components ${\mathbf{l}}_{\mathbf{k}}^{\parallel }$ with $c_{1,0,0}^x=c_{0,1,0}^x=0.025,c_{0,0,1}^z=1$. (a) Surface-state energy from first-order perturbation theory (blue) and from exact diagonalization (orange, obscured by and indistinguishable from the blue surface) and lowest bulk energy from exact diagonalization (green) and from the minimal gap (red, nearly obscured by the green surface), illustrating the size of the gap. (b) Lowest bulk energy as a density plot. (c) IPR of the corresponding state, where low IPR corresponds to a delocalized state and high IPR corresponds to a strongly localized state. (d) Energy and (e) IPR of the surface state.

Figure 5

Surface-state energy $E_0^e$ calculated by exact diagonalization of the slab BdG Hamiltonian for the point groups (a) $C_2$, (b) $C_4$, (c) $D_2$, (d) $C_s$, and (e) $C_{2v}$.

Figure 6

Relative difference between the surface-state energy $E_0^e$ calculated by exact diagonalization of the slab BdG Hamiltonian and the surface-state energy $E_0^p$ calculated within perturbation theory with Eq. (71) for the point groups (a) $C_2$, (b) $C_4$, (c) $D_2$, (d) $C_s$, and (e) $C_{2v}$.