Near zero modes in condensate phases of the Dirac theory on the honeycomb lattice

TL;DR

This paper studies zero-energy bound states in fermionic condensate phases on the honeycomb lattice, revealing their non-uniqueness, explicit symmetry breaking effects, and their implications for topological protection and non-Abelian statistics.

Contribution

It provides exact wavefunctions for vortex zero modes and analyzes their degeneracy, showing these modes are not topologically protected nor exhibit non-Abelian statistics.

Findings

Zero modes are not unique to specific phases or half filling.

Vortex zero modes have even degeneracy, pairing into ordinary fermions.

Zero modes are split on the lattice due to explicit symmetry breaking.

Abstract

We investigate a number of fermionic condensate phases on the honeycomb lattice, to determine whether topological defects (vortices and edges) in these phases can support bound states with zero energy. We argue that topological zero modes bound to vortices and at edges are not only connected, but should in fact be \emph{identified}. Recently, it has been shown that the simplest s-wave superconducting state for the Dirac fermion approximation of the honeycomb lattice at precisely half filling, supports zero modes inside the cores of vortices (P. Ghaemi and F. Wilczek, 2007). We find that within the continuum Dirac theory the zero modes are not unique neither to this phase, nor to half filling. In addition, we find the \emph{exact} wavefunctions for vortex bound zero modes, as well as the complete edge state spectrum of the phases we discuss. The zero modes in all the phases we examine have even-numbered degeneracy, and as such pairs of any Majorana modes are simply equivalent to one ordinary fermion. As a result, contrary to bound state zero modes in $p_x+i p_y$ superconductors, vortices here do \emph{not} exhibit non-Abelian exchange statistics. The zero modes in the pure Dirac theory are seemingly topologically protected by the effective low energy symmetry of the theory, yet on the original honeycomb lattice model these zero modes are split, by explicit breaking of the effective low energy symmetry.