Coupled Majorana modes in a dual vortex of the Kitaev honeycomb model

TL;DR

This paper investigates coupled Majorana modes in a Kitaev honeycomb model with vortices, revealing how dual vortices bind finite-energy fermionic modes and analyzing their wavefunctions and coupling analytically and numerically.

Contribution

It provides an analytical and numerical study of Majorana modes bound to dual vortices in the Kitaev model, highlighting their coupling and energy spectrum in different parameter regimes.

Findings

Dual vortices bind finite-energy fermionic modes.

Majorana wavefunctions are analytically computed in different limits.

Numerical results confirm analytical predictions with high accuracy.

Abstract

The Kitaev model is exactly solvable in terms of Majorana fermions hopping on a honeycomb lattice and coupled to a static $\mathbb{Z}_2$ gauge field, giving the possibility of $\pi$-vortices in hexagonal plaquettes. In the vortex-full sector and in the presence of a time-reversal-breaking three-spin term of strength $\kappa$, the energy spectrum is gapped and the ground state possesses an even Chern number. An isolated vortex-free plaquette acts as a ``dual vortex'' and binds a fermionic mode at finite energy $\epsilon$ in the bulk gap. This mode is equivalent to two coupled Majorana zero modes located on the same dual vortex. In a continuum approximation, we analytically compute the Majorana wavefunctions and their coupling $\epsilon$ in the two limits of small or large $\kappa$. The analytical approach is confirmed by numerical perturbation theory directly on the lattice. The latter is in excellent agreement with the full numerics on a finite-size system. We contrast our results with states bound to an isolated vortex in a topological superconductor with even Chern number.