Phase diagram and crossover phases of topologically ordered graphene zigzag nanoribbons: role of localization effects.

We computed the phase diagram of zigzag graphene nanoribbons as a function of on-site repulsion, doping, and disorder strength. The topologically ordered phase undergoes topological phase transitions into crossover phases, which are new disordered phases with non-universal topological entanglement entropy that exhibits significant variance. We explored the nature of non-local correlations in both the topologically ordered and crossover phases. In the presence of localization effects, strong on-site repulsion and/or doping weaken non-local correlations between the opposite zigzag edges of the topologically ordered phase. In one of the crossover phases, bothe-/2solitonic fractional charges and spin-charge separation were absent; however, charge-transfer correlations between the zigzag edges were possible. Another crossover phase contains solitonice-/2fractional charges but lacks charge transfer correlations. We also observed properties of non-topological, strongly disordered, and strongly repulsive phases. Each phase on the phase diagram exhibits a different zigzag-edge structure. Additionally, we investigated the tunneling of solitonic fractional charges under an applied voltage between the zigzag edges of undoped topologically ordered zigzag ribbons, and found that it may lead to a zero-bias tunneling anomaly.

Soliton fractional charge of disordered graphene nanoribbon.

We investigate the properties of the gap-edge states of half-filled interacting disordered zigzag graphene nanoribbons, and find that the midgap states can display a quantized fractional charge of 1/2. These gap-edge states can be represented by topological kinks with their site probability distribution divided between the left and right zigzag edges with different chiralities. In addition, there are numerous spin-split gap-edge states, similar to those in a Mott-Anderson insulator.

Resonant, non-resonant and anomalous states of Dirac electrons in a parabolic well in the presence of magnetic fields.

We report on several new basic properties of a parabolic dot in the presence of a magnetic field. The ratio between the potential strength and the Landau level (LL) energy spacing serves as the coupling constant of this problem. In the weak coupling limit the energy spectrum in each Hilbert subspace of an angular momentum consists of discrete LLs of graphene. In the intermediate coupling regime non-resonant states form a closely spaced energy spectrum. We find, counter-intuitively, that resonant quasi-bound states of both positive and negative energies exist in the spectrum. The presence of resonant quasi-bound states of negative energies is a unique property of massless Dirac fermions. As the strong coupling limit is approached resonant and non-resonant states transform into anomalous states, whose probability densities develop a narrow peak inside the well and another broad peak under the potential barrier. These properties may investigated experimentally by measuring optical transition energies that can be described by a scaling function of the coupling constant.