Zero energy and chiral edge modes in a p-wave magnetic spin model

TL;DR

This paper explores zero energy vortex and chiral edge modes in a fermionic representation of the Kitaev honeycomb model, revealing how branch cuts influence Majorana modes and edge states, with exact solutions on a cylinder.

Contribution

It introduces a Jordan-Wigner based representation and branch cut framework to analyze Majorana zero modes and chiral edge modes in the Kitaev honeycomb model, providing exact solutions and penetration depth calculations.

Findings

Exact edge energy eigensolutions derived for a cylinder geometry.

Penetration depth depends on edge mode momentum.

Branch cuts determine the character of chiral edge modes.

Abstract

In this work we discuss the formation of zero energy vortex and chiral edge modes in a fermionic representation of the Kitaev honeycomb model. We introduce the representation and show how the associated Jordan-Wigner procedure naturally defines the so called branch cuts that connect the topological vortex excitations. Using this notion of the branch cuts we show how to, in the non-Abelian phase of the model, describe the Majorana zero mode structure associated with vortex excitations. Furthermore we show how, by intersecting the edges between Abelian and non-Abelian domains, the branch cuts dictate the character of the chiral edge modes. In particular we will see in what situations the exact zero energy Majorana edge modes exist. On a cylinder, and for the particular instances where the Abelian phase of the model is the full vacuum, we have been able to exactly solve for the systems edge energy eigensolutions and derive a recursive formula that exactly describes the edge mode structure. Penetration depth is also calculated and shown to be dependent on the momentum of the edge mode. These solutions also describe the overall character of the fully open non-Abelian domain and are excellent approximations at moderate distances from the corners.