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

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.

Final version including numerics, accepted for publication in PRB

References in corpus (8)

• The electronic properties of graphene
• A topological Dirac insulator in a quantum spin Hall phase : Experimental observation of first strong topological insulator
• Bipolar supercurrent in graphene
• Electron fractionalization in two-dimensional graphenelike structures
• Zero modes of two-dimensional chiral p-wave superconductors
• An Index Theorem for the Majorana Zero Modes in Chiral P-Wave Superconductors
• Vortices, zero modes and fractionalization in bilayer-graphene exciton condensate
• Zero-energy states and fragmentation of spin in the easy-plane antiferromagnet on a honeycomb lattice

Cited by in corpus (1)

• Supersolidity of cold-atom Bose-Fermi mixtures in optical lattices