Edge Majoranas on locally flat surfaces: The cone and the Möbius band

In this paper, we investigate the edge Majorana modes in the simplest possible $p{}_x+i{p}_y$ superconductor defined on surfaces with different geometries, the annulus, the cylinder, the Möbius band, and a cone (by cone we mean a cone with the tip cut away so it is topologically equivalent to the annulus and cylinder), and with different configurations of magnetic fluxes threading holes in these surfaces. In particular, we shall address two questions: Given that, in the absence of any flux, the ground state on the annulus does not support Majorana modes while the one on the cylinder does, how is it possible that the conical geometry can interpolate smoothly between the two? Given that in finite geometries edge Majorana modes have to come in pairs, how can a $p{}_x+i{p}_y$ state be defined on a Möbius band, which has only one edge? We show that the key to answering these questions is that the ground state depends on the geometry, even though all the surfaces are locally flat. In the case of the truncated cone, there is a nontrivial holonomy, while the nonorientable Möbius band must necessarily support a domain wall.

Figure 1

The embedding given by Eq. (7).

Figure 2

A Möbius band is formed by gluing together the two edges of a rectangle with a twist. Because one twists the strip before gluing, the Möbius band is nonorientable and has only a single edge.

Figure 3

Illustration of the two different types of line defects on a Möbius band supporting a chiral superconductor. The edge of the strip is shown in blue, and the line defect is shown in red. The circular arrows indicate the chirality of a $p$-wave superconducting order parameter that is constant along the strip, except at the defect.

Figure 4

(a) The Möbius band with a domain wall configuration (in red), as in Fig. 3. The arrow depicts a flux quantum threaded through the hole. (b) The Möbius band from (a), but cut along the domain wall. After this cut, the band turns into a cylinder with a $4\pi$ twist, and the flux quantum winds around the band twice.

Figure 5

(a) One of two inequivalent ways to thread a flux through the surface of the Möbius band. The flux line only encircles the edge and not the line defect. (b) The situation obtained by cutting the Möbius band from (a) along the line defect. It is explicitly visible that the flux line winds once around the edge and not around the line defect. (c) One of two inequivalent ways to thread a flux through the surface of the Möbius band. The flux line encircles both the edge and the line defect. (d) The situation obtained by cutting the Möbius band from (c) along the line defect. It is now clear that the flux line winds once around the edge and twice around the domain wall.