Abstract
Midgap states induced by edges or by vacancies in low dimensions are examined. For semi-infinite nanowires in tight-binding limit, edge states are found within the gaps of corresponding bulk spectrum. The presence of these midgap states reflects an underlying generalized supersymmetry. This supersymmetric structure is a generalized rotational symmetry among sublattices and results in a universal tendency: all midgap states tend to vanish with periods commensurate with the underlying lattice. A few implications are also discussed. At the same time, the fractional charges could be found from edges. This anomalous accumulated charges could be understood as the remnant charge from an unit cell masked by the edge effect. By the generalized supersymmetric method and the combination rules used in the content, some relations between the accumulated charges for different edges have been found. For a modern research about the defect, we come to study the magnetism in graphene. Through Lippmann-Schwinger equation of scattering theory, we could write down the localized wavefunctions for the defect states on graphene. By calculating the Coulomb and exchange energies of two defects, it shows that the spin orientations is depending on the separating distance. We also use the random matrix theory to estimate the magnetization of the graphene with impurities. The density of state is characterized by Wigner semi-circle law. The formed impurity band supports ferromagnetism with induced magnetic moment depending on quasi-particle lifetime and defect density.