Bound-State energies and phase transitions in graphene-like structures

Graphene has been actively researched because its low energy electronic Hamiltonian is the relativistic Dirac equation with vanishing rest mass. Graphene was first fabricated in 2004 by Geim and Novoselov. Although graphene is a semi-metal, electronic applications require knowledge of how to change its phase from a semi-metal to an insulator. For spinless fermions on graphene, fermion density imbalance, coupling between its Dirac points, and directed next nearest neighbor hopping can lead to charge density wave, Kekule bond density wave, and quantum Hall insulating phases. Furthermore, topological defects such as line defects and vortices allow bound state solutions within the gap giving rise to fractional charge. The results are not only applicable to graphene, but can also be applied in general to fermions on a hexagonal lattice. Another example where a Dirac linear dispersion is found is for spinless fermions at one-third filling on the Lieb lattice. Fermion density imbalance, staggered nearest neighbour hopping, and directed next nearest neighbor hopping can change this lattice from a semi-metal to an insulating phase characterized by a charge density wave, staggered hopping, broken pi/2 rotation symmetry, or broken time reversal symmetry. By adding and adjusting the strengths of nearest and next nearest neighbour interactions, many of these interesting phases can be energetically favourable in mean-field theory.