Majorana flat bands in $s$-wave gapless topological superconductors

We demonstrate how the nontrivial interplay between spin-orbit coupling and nodeless $s$-wave superconductivity can drive a fully gapped two-band topological insulator into a time-reversal invariant gapless topological superconductor supporting symmetry-protected Majorana flat bands. We characterize topological phase diagrams by a ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ partial Berry-phase invariant, and show that, despite the trivial crystal geometry, no unique bulk-boundary correspondence exists. We trace this behavior to the anisotropic quasiparticle bulk gap closing, linear vs quadratic, and argue that this provides a unifying principle for gapless topological superconductivity. Experimental implications for tunneling conductance measurements are addressed, relevant for lead chalcogenide materials.

Figure 1

(a) Phase diagram of $H$ [Eq. (1)] for $\mu =0$, $t=\lambda =1$. Each phase is labeled by the partial Berry-phase parities $(P_{B,0},P_{B,\pi })$. The topological numbers do not change under $\Delta \mapsto -\Delta$. (b) and (c) Sketch of the spectrum of $H$ with $\Delta =0$ in a TI phase, and with $\Delta \ne 0$ in a TS flat-band phase, respectively.

Figure 2

Excitation spectrum of $H$ [Eq. (1)] for $\mu =0,t=\lambda =u_{cd}=1$. Top (bottom) panels correspond to BC1 (BC2), whereas right vs left columns correspond to $(P_{B,0},P_{B,\pi })=(1,1)$ vs $(1,-1)$. System size: $N_x=N_z=40$.

Figure 3

(a) and (b) LDOS and DOS for $H$ [Eq. (1)] for $\mu =0,t=\lambda =u_{cd}=1$, $\Delta =4$. Insets: The jagged lines signify cuts of the system, implying that in BC1 (BC2) the OBC is along the $\widehat{x}$ ($\widehat{z}$). In (a), the brown (orange) arrows indicate a continuum of TR pairs of Majoranas on each boundary, propagating along opposite directions. (c) and (d) LDOS and DOS for a gapped TS in 2D and 1D. System size: $(N_x,N_z)=(80,400)$ (a), $(N_x,N_z)=(400,80)$ (b), $(N_x,N_z)=(80,400)$ (c), and $N_x=80$ (d).

Figure 4

Excitation spectrum of $H+H_{\nu }$ for $\mu =0,t=\lambda =1,u_{cd}=4,\Delta =2$. (a) and (b) In-plane vs out-of-plane field. (c) and (d) LDOS for increasing field strength along $\widehat{z}$ vs $\widehat{y}$. System size: $(N_x,N_z)=(40,40)$ [(a),(b)], $(N_x,N_z)=(40,400)$ [(c),(d)].