Edge Majoranas on locally flat surfaces - the cone and the M\"obius band

TL;DR

This paper explores how edge Majorana modes in a $p_x+ip_y$ superconductor are affected by different geometries, including cones and M"obius bands, revealing the role of topology and holonomy in their existence.

Contribution

It demonstrates that geometry and topology, such as non-trivial holonomy and non-orientability, influence the presence and nature of Majorana modes in locally flat superconducting surfaces.

Findings

Conical geometry supports non-trivial holonomy affecting Majorana modes.

M"obius band's non-orientability requires a domain wall for Majorana modes.

Ground state properties depend on surface topology despite local flatness.

Abstract

In this paper, we investigate the edge Majorana modes in the simplest possible $p{}_{x}+ip_{y}$ superconductor defined on surfaces with different geometry - the annulus, the cylinder, the M\"obius 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 configuration 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 $p{}_{x}+ip_{y}$ state be defined on a M\"obius 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 non-trivial holonomy, while the non-orientable M\"obius band must necessarily support a domain wall.