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Structure factors of s=1/2 spin chains and magnetism at the edges of graphene ribbons

In this thesis we study two different one dimensional systems. The first project is on the transverse dynamical structure factors of the XXZ spin chain and the second project is on magnetism of zigzag edges of graphene nano-ribbons. In chapter 2, we apply field theory methods, first developed to study x-ray edge singularities, to interacting one-dimensional systems in order to include band curvature effects and study edge singularities at arbitrary momentum. We point out that spin chains with uniform Dzyaloshinskii-Moriya interactions provide an opportunity to test these theories since these interactions may be exactly eliminated by a gauge transformation that shifts the momentum. However, this requires an extension of these x-ray edge methods to the transverse spectral function of the XXZ spin chain in a magnetic field. In chapter 3, by considering the Hubbard model in the weak coupling limit, U t, for bearded as well as zigzag edges, we argue for existence of magnetic edges. We first present an argument based on Lieb’s theorem. Then, projecting the Hubbard interactions onto the flat edge band, we prove that the resulting onedimensional model has a fully polarized ferromagnetic ground state. We also study excitons and the effects of second neighbor hopping as well as a potential energy term acting on the edge only, proposing a simple and possibly exact phase diagram with the magnetic moment varying smoothly to zero. Finally, we consider corrections of second order in U, arising from integrating out the gapless bulk Dirac excitations.