In semiconducting armchair graphene ribbons a chiral lattice deformation can induce pairs of topological gap states with opposite energies. Near the critical value of the deformation potential these kink and antikink states become almost degenerate with zero energy and have a fractional charge one-half. Such a semiconducting armchair ribbon represents a one-dimensional topological insulator with nearly zero energy end states. Using data collapse of numerical results we find that the shape of the kink displays an anomalous power-law dependence on the width of the local lattice deformation. We suggest that these gap states may be probed in optical measurements. However, "metallic" armchair graphene ribbons with a gap induced by many-electron interactions have no gap states and are not topological insulators.Comment: 4 pages, 3 figure

In neutral graphene dots, the Fermi level coincides with the Dirac points. We have investigated in the presence of a magnetic field several unusual properties of single electron states near the Fermi level of such a rectangular-shaped graphene dot with two zigzag and two armchair edges. We find that a quasidegenerate level forms near zero energy and the number of states in this level can be tuned by the magnetic field. The wave functions of states in this level are all peaked on the zigzag edges with or without some weight inside the dot. Some of these states are magnetic field-independent surface states while the others are field-dependent. We have found a scaling result from which the number of magnetic field-dependent states of large dots can be inferred from those of smaller dots.

Analytical solutions of the Coulomb impurity problem of graphene in the absence of a magnetic field show that when the dimensionless strength of the Coulomb potential g reaches a critical value the solutions become supercritical with imaginary eigenenergies. Application of a magnetic field is a singular perturbation, and no analytical solutions are known except at a denumerably infinite set of magnetic fields. We find solutions of this problem by numerical diagonalization of large Hamiltonian matrices. Solutions are qualitatively different from those of zero magnetic field. All energies are discrete and no complex energies allowed. We have computed the finite-size scaling function of the probability density containing s-wave component of Dirac wavefunctions. This function depends on the coupling constant, regularization parameter, and the gap. In the limit of vanishing regularization parameter our findings are consistent with the expected values exponent ν which determines of the asymptotic behavior of the wavefunction near r = 0.

We find that a repulsive potential of graphene in the presence of a magnetic field has bound states that are peaked inside the barrier with tails extending over ℓ(N + 1), where ℓ and N are the magnetic length and Landau level(LL) index. We have investigated how these bound states affect scaling properties of the induced density of filled Landau levels of massless Dirac fermions. For chiral fermions we find, in strong coupling regime, that the density inside the repulsive potential can be greater than the value in the absence of the potential while in the weak coupling regime we find negative induced density. Similar results hold also for non-chiral fermions. As one moves from weak to strong coupling regimes the effective coupling constant between the potential and electrons becomes more repulsive, and then it changes sign and becomes attractive. Different power-laws of induced density are found for chiral and non-chiral fermions.PACS numbers:

We have investigated a new feature of impurity cyclotron resonances common to various localized potentials of graphene. A localized potential can interact with a magnetic field in an unexpected way in graphene. It can lead to formation of anomalous boundstates that have a sharp peak with a width R in the probability density inside the potential and a broad peak of size magnetic length ℓ outside the potential. We investigate optical matrix elements of anomalous states and find that they are unusually small and depend sensitively on the magnetic field. The effect of many-body interactions on their optical conductivity is investigated using a self-consistent time-dependent Hartree-Fock approach. For a completely filled Landau level we find that an excited electron-hole pair, originating from the optical transition between two anomalous impurity states, is nearly uncorrelated with other electron-hole pairs, although it displays substantial exchange self-energy effects. This absence of correlation is a consequence of a small vertex correction in comparison to the difference between renormalized transition energies computed within the one electron-hole pair approximation. However, an excited electron-hole pair originating from the optical transition between a normal and an anomalous impurity state can be substantially correlated with other electron-hole states with a significant optical strength.