Bulk boundary correspondence and the existence of Majorana bound states on the edges of 2D topological superconductors

The bulk-boundary correspondence establishes a connection between the bulk topological index of an insulator or superconductor, and the number of topologically protected edge bands or states. For topological superconductors in two dimensions, the first Chern number is related to the number of protected bands within the bulk energy gap, and is therefore assumed to give the number of Majorana band states in the system. Here we show that this is not necessarily the case. As an example, we consider a hexagonal-lattice topological superconductor based on a model of graphene with Rashba spin-orbit coupling, proximity-induced $s$-wave superconductivity, and a Zeeman magnetic field. We explore the full Chern number phase diagram of this model, extending what is already known about its parity. We then demonstrate that, despite the high Chern numbers that can be seen in some phases, these do not strictly always contain Majorana bound states.

Figure 1

Numerically determined topological phase diagrams for Eq. (1) using Eq. (2), where $\nu$ is the Chern number. The parameters are: (a) $\alpha =\mathrm{\Delta }=0.5t$, (b) $3\alpha =\mathrm{\Delta }=1.5t$, (c) $\alpha =0.5t$ and $\mathrm{\Delta }=0.4t$, and (d) $\alpha =\mathrm{\Delta }=0.1t$. Solid black lines show the phase boundaries caused by the gap closings at the TRI momenta and the blue dashed line corresponds to the gap closes at the Dirac point. These are not, however, the only phase boundaries.

Figure 2

Numerically determined topological phase diagrams for Eq. (1) using Eq. (2), where $\nu$ is the Chern number (red squares), compared to the gap closing (blue circles) where ${\epsilon }_G$ is the gap. (a) shows a transition from $\nu =-2\to 4$ as a function of $B$ (accompanied by a gap closing at the Dirac point) with the gap calculated for a system of length and width $L=L_w=L_{\parallel }=500$. The parameters are $\mathrm{\Delta }=\alpha =0.5t$ and $\mu =0$. This is a cut through the phase diagram in Fig. 1. (b) shows transitions through $\nu =1\to -5\to -2$ as a function of $\alpha$ (accompanied by gap closings at points in the BZ) with the gap calculated for $L=500$ as for (a). The parameters are $\mathrm{\Delta }=0.6tB=1.5t$ and $\mu =1.3t$.

Figure 3

Topological phase diagrams for Eq. (1). The parameters are (a) $3\alpha =\mathrm{\Delta }=1.5t$ and (b) $\alpha =\mathrm{\Delta }=0.1t$. Compare with Figs. 1 and 1, respectively. The red regions satisfy ${(-1)}^{\nu }=-1$ and the white regions satisfy ${(-1)}^{\nu }=1$. The solid black lines show the phase boundaries and $\nu$ is the Chern number.

Figure 4

(a), (b) The band structures of zigzag (a) and armchair (b) nanoribbons in a regime with $\nu =4$. The parameters are $\alpha =\mathrm{\Delta }=0.5t$, $\mu =0.1t$, and $B=1.4t$. (c), (d) The band structures corresponding to $\nu =-5$ for zigzag (c) and armchair (d) nanoribbons. The parameters are $\alpha =\mathrm{\Delta }=0.5t$, $\mu =1.5t$, and $B=1.3t$. In (d), three pairs of edge bands are crossing at $k_{\parallel }=0$ and there are three MBS per edge in this case. The topologically protected bands localized on one edge are depicted by a dashed red line, while those localized on the other edge are represented by a dotted green line. The K and K' points are marked in the figures for reference.

Figure 5

(a), (b) The bandstructure of zigzag (a) and armchair (b) nanoribbons in a regime with $\nu =-2$. The parameters are $\alpha =\mathrm{\Delta }=\mu =0.5t$, and $B=1.5t$. In (b), extra unprotected crossings can be seen at $k_{\parallel }\ne 0$, which can be removed by continuously deforming the bands, see Fig. 7.

Figure 6

The energy scaling of the lowest energy state in each band normalized by the mean level spacing, $\epsilon /\lambda$. Here we consider a zigzag nanoribbon (left-hand panel) and an armchair nanoribbon (right-hand panel) in a $\nu =-5$ phase, same as in Figs. 4 and 4. We take $L=L_w=L_{\parallel }$ to be the length and width of system. The low energy states at $k_{\parallel }=0$ (denoted in blue) show an exponential scaling to zero as $\epsilon /\lambda \sim L^2{e}^{-L/\tilde{L}}$ (dashed blue line); the other low energy states at $k_{\parallel }\ne 0,\pi$, (denoted in red and black) do not scale to zero but show a weaker positive divergence, $ln\epsilon /\lambda \sim L$ (dashed red and black lines).

Figure 7

(a), (b) The band structures of two armchair nanoribbons: (a) in a regime with $\nu =-2$ and (b) in a regime with $\nu =4$. The parameters are the same as for Figs. 5 and 4, respectively, with a disorder strength of $s=0.3\mathrm{t}$ (a) and $s=0.1\mathrm{t}$ (b). The bulk bands are those for the clean systems, provided here for reference.