Abstract:
The lowest eigenenergies of few, strongly interacting electrons in a one--dimensional ring are studied in the presence of an impurity barrier. The persistent current $\:I\:$, periodic in an Aharonov--Bohm flux penetrating the ring, is strongly influenced by the electron spin. The impurity does not remove discontinuities in $\:I\:$ at zero temperature. The total electron spin of the ground state oscillates with the flux. Strong electron--electron interaction enhances $\:I\:$, albeit not up to the value of the clean ring which itself is smaller than $\:I\:$ for free electrons. $\:I\:$ disappears on a temperature scale that depends exponentially on the electron density. In the limit of very strong interaction the response to small fluxes is diamagnetic.

Abstract:
Coherent Rashba spin precession along interacting multi-mode quantum channels is investigated, revisiting the theory of coupled Tomonaga-Luttinger liquids. We identify susceptibilities as the key-parameters to govern exponents and Rashba precession lengths. In semiconducting quantum wires spins of different transport channels are found to {\em dephase} in their respective precession angles with respect to one another, as a result of the interaction. This could explain the experimental difficulty to realize the Datta Das transistor. In single walled carbon nanotubes, on the other hand, interactions are predicted to suppress dephasing between the two flavor modes at small doping.

Abstract:
Dispersionless (flat) electronic bands are investigated regarding their conductance properties. Due to "caging" of carriers these bands are usually insulating at partial filling, at least on the non-interacting level. Considering the specific example of a $\mathcal{T}_3$--lattice we study long-range Coulomb interactions. A non-trivial dependence of the conductivity on flat band filling is obtained, exhibiting an infinite number of zeros. Near these zeros, the conductivity rises linearly with carrier density. At densities half way in between adjacent conductivity-zeros, strongly enhanced conductivity is predicted, accompanying a solid-solid phase transition.

Abstract:
Rashba precession of spins moving along a one-dimensional quantum channel is calculated, accounting for Coulomb interactions. The Tomonaga--Luttinger model is formulated in the presence of spin-orbit scattering and solved by Bosonization. Increasing interaction strength at decreasing carrier density is found to {\sl enhance} spin precession and the nominal Rashba parameter due to the decreasing spin velocity compared with the Fermi velocity. This result can elucidate the observed pronounced changes of the spin splitting on applied gate voltages which are estimated to influence the interface electric field in heterostructures only little.

Abstract:
Low energy spectra of isotropic quantum dots are calculated in the regime of low electron densities where Coulomb interaction causes strong correlations. The earlier developed pocket state method is generalized to allow for continuous rotations. Detailed predictions are made for dots of shallow confinements and small particle numbers, including the occurance of spin blockades in transport.

Abstract:
Rashba precession of spins moving along a one-subband quantum channel is calculated. The quantitative influence of unoccupied higher subbands depends on the shape of the transversal confinement and can be accounted for perturbatively. Coulomb interactions are included within the Tomonaga--Luttinger model with spin-orbit coupling incorporated. Increasing interaction strength at decreasing carrier density is found to {\sl enhance} spin precession.

Abstract:
The length over which electron spins reverse direction due to the Rashba effect when injected with an initial polarization along the axes of a quantum wire is investigated theoretically. A soft wall confinement of the wire renormalizes the spin-orbit parameter (and the effective mass) stronger than hard walls. Electron-electron interactions enhance the Rashba precession while evidence is found that the coupling between transport channels may suppress it.

Abstract:
We describe the theory of few Coulomb-correlated electrons in a magnetic quantum dot formed in graphene. While the corresponding nonrelativistic (Schr\"odinger) problem is well understood, a naive generalization to graphene's "relativistic" (Dirac-Weyl) spectrum encounters divergencies and is ill-defined. We employ Sucher's projection formalism to overcome these problems. Exact diagonalization results for the two-electron quantum dot, i.e., the artificial helium atom in graphene, are presented.

Abstract:
The spin sector of charge-spin separated single mode quantum wires is studied, accounting for realistic microscopic electron-electron interactions. We utilize the ladder approximation (LA) to the interaction vertex and exploit thermodynamic relations to obtain spin velocities. Down to not too small carrier densities our results compare well with existing quantum Monte-Carlo (QMC) data. Analyzing second order diagrams we identify logarithmically divergent contributions as crucial which the LA includes but which are missed, for example, by the self-consistent Hartree-Fock approximation. Contrary to other approximations the LA yields a non-trivial spin conductance. Its considerably smaller computational effort compared to numerically exact methods, such as the QMC method, enables us to study overall dependences on interaction parameters. We identify the short distance part of the interaction to govern spin sector properties.