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We present a weak-relativistic limit comparison between the Kohn-Sham-Dirac equation and its approximate form containing the exchange coupling, which is used in almost all relativistic codes of density-functional theory. For these two descriptions, an exact expression of the Dirac Green's function in terms of the non-relativistic Green's function is first derived and then used to calculate the effective Hamiltonian, i.e., Pauli Hamiltonian, and effective velocity operator in the weak-relativistic limit. We point out that, besides neglecting orbital magnetism effects, the approximate Kohn-Sham-Dirac equation also gives relativistic corrections which differ from those of the exact Kohn-Sham-Dirac equation. These differences have quite serious consequences: in particular, the magnetocrystalline anisotropy of an uniaxial ferromagnet and the anisotropic magnetoresistance of a cubic ferromagnet are found from the approximate Kohn-Sham-Dirac equation to be of order $1/c^2$, whereas the correct results obtained from the exact Kohn-Sham-Dirac equation are of order $1/c^4$ . We give a qualitative estimate of the order of magnitude of these spurious terms

A model to treat the anomalous Hall effect is developed. Based on the Kubo formalism and on the Dirac equation, this model allows the simultaneous calculation of the skew-scattering and sidejump contributions to the anomalous Hall conductivity. The continuity and the consistency with the weak-relativistic limit described by the Pauli Hamiltonian is shown. For both approaches, Dirac and Pauli, the Feynman diagrams, which lead to the skew-scattering and the side-jump contributions, are underlined. In order to illustrate this method, we apply it to a particular case: a ferromagnetic bulk compound in the limit of weak-scattering and free-electrons approximation. Explicit expressions for the anomalous Hall conductivity for both skew-scattering and side-jump mechanisms are obtained. Within this model, the recently predicted "spin Hall effect" appears naturally.

Transport through a metallic carbon nanotube is considered, where electrons are injected in the bulk by a scanning tunneling microscope tip. The charge current and noise are computed both in the absence and in the presence of one dimensional Fermi liquid leads. For an infinite homogeneous nanotube, the shot noise exhibits effective charges different from the electron charge. Noise correlations between both ends of the nanotube are positive, and occur to second order only in the tunneling amplitude. The positive correlations are symptomatic of an entanglement phenomenon between quasiparticles moving right and left from the tip. This entanglement involves many body states of the boson operators which describe the collective excitations of the Luttinger liquid. I. INTRODUCTIONOver the years, the study of current noise and noise correlations has become a respected and useful diagnosis for transport measurements on mesoscopic conductors. Theoretically, noise was first computed mostly for non-interacting systems 1 . However, it soon became clear that low frequency noise could be used to isolate the quasiparticle charge 2,3 and to study the statistical correlations 4,5 in specific quasi one-dimensional correlated electron systems, such as the edge waves in the quantum Hall effect. In these chiral Luttinger liquids, the charge of the collective excitations along the edges corresponds to the electron charge multiplied by the filling factor.Attention is now turning towards conductors -individual nano-objects -which occur naturally, and which can be connected to current/voltage probes in order to perform a transport experiment. The crucial advantage of such nanoobjects is that they are essentially free of defects and in some circumstances they have an inherent one dimensional character. Carbon nanotubes constitute the archetype of such 1D nano-objects: single wall armchair nanotubes have metallic behavior, with two propagating modes at the Fermi level. Incidentally, electronic correlations are known to play an important role in such systems. Carbon nanotubes seem to constitute good candidates to study Luttinger liquid behavior. In particular, their tunneling density of states -and thus the tunneling I(V ) characteristics is known to have a power law behavior 6,7,8 in accordance with Luttinger liquid theory.Luttinger models for nanotubes differ significantly from their quantum Hall effect counterpart, because of their nonchiral character. Forward and backward fields describing collective excitations effectively mix, because the interactions between electrons are spread along the whole length of the nanotube. For this reason, a straightforward transposition of the results obtained for chiral edge system proves difficult. Nevertheless, non-chiral Luttinger liquids can be described with chiral fields 9,10 . Such chiral fields correspond to excitations with anomalous (non-integer) charge, which has eluded detection so far.In the present work, we propose an experimental geometry which allows to probe directly the underlying charg...