We study the nonrelativistic limit of the N = 2 supersymmetric Chern-Simons matter system. We show that in addition to Galilean invariance the model admits a set of symmetries generated by fermionic charges, which can be interpreted as an extended Galilean supersymmetry . The system also possesses a hidden conformal invariance and then the full group of symmetries is the extended superconformal Galilean group. We also show that imposing extended superconformal Galilean symmetry determines the values of the coupling constants in such a way that their values in the bosonic sector agree with the values of Jackiw and Pi for which self-dual equation exist. We finally analyze the second quantized version of the model and the two-particle sector. * 1 (2.7b) * where for any vector V = (V 1 , V 2 ), V ± = V 1 ± iV 2 .

We perform a perturbative analysis of the Aharonov-Bohm problem to one loop in a field-theoretic formulation, and show that contact interactions are necessary for renormalizability. In general, the classical scale invariance of this problem is broken quantum mechanically. There exists however a critical point for which this anomaly disappears.

We present an analytic solution to the problem of the Rashba spin-orbit coupling in semiconductor quantum dots. We calculate the exact energy spectrum, wave-functions, and spin-flip relaxation times. We discuss various effects inaccessible via perturbation theory. In particular, we find that the effective gyromagnetic ratio is strongly suppressed by the spin-orbit coupling. The spin-flip relaxation rate has a maximum as a function of the spin-orbit coupling and is therefore suppressed in both the weak-and strong coupling limits.PACS numbers: 71.70. Ej, 73.21.La, 72.25.Rb In recent years, spin-orbit (SO) effects in quantum dots attracted much attention, as it has become clear that these effects play a crucial role in the novel field of spintronics 1 . The combination of a confined geometry and SO coupling has interesting consequences for the electron spectrum 2 . Since future quantum computation devices would have to control coherent spin states over sufficiently long timescales 3 , it is important to understand spin relaxation mechanisms, most of which are rooted in the SO coupling. SO interactions can arise in quantum dots by various mechanisms related to electron confinement and symmetry breaking and are generally known as the Rashba term 4 and the Dresselhaus term 5 . For most experimental realisations, quantum dots can be described as effectively two-dimensional systems in a confining potential which is usually modelled as hard-wall or harmonic confinement.To our knowledge all existing theoretical studies of SO effects in such systems rely on perturbative schemes or numerical simulations. The purpose of this paper is to provide an exact solution of the quantum mechanical problem of the combined effects of the SO coupling, confinement, and magnetic field.In the bulk (i.e. without confinement), the problem was solved in the original paper 4 , see also 6 , while the effects of the confining potential (but without SO coupling) were studied in Ref.7 . We shall show here how these exact solutions can be combined and generalised.The one-particle Hamiltonian describing an electron in a two-dimensional quantum dot is of the form:where m is the effective electron mass, g is the effective gyromagnetic ratio, and µ B is the Bohr magneton. A constant magnetic field B is introduced via the Zeeman term above and the Peierls substitution, p = −i∇ − e c A (we use the axial gauge, A ρ = 0 and A ϕ = Bρ/2); α R is the strength of the SO coupling, the Pauli matrices are defined as standard, the electron charge is e = −|e|, and we seth = 1. In (1), V (ρ) is the (symmetric) confining potential. In this paper we will consider a hard-wall confining potential, i.e. V (ρ) = 0 for ρ < R and V (ρ) = ∞ for ρ > R, R being the radius of the dot. We have included the Rashba term rather than the Dresselhaus term, which would be of the form α D (p x σ x − p y σ y ). (The Rashba term maybe dominant, since the coupling strength α R can be varied by system design, e.g., values of α R ≃ 6 × 10 −9 eV cm were reported for InAlAs/InGaAs structure...

We study the real-time dynamics of quantum models with long-range interactions coupled to a heat-bath within the closed-time path-integral formalism. We show that quantum fluctuations depress the transition temperature. In the subcritical region there are two asymptotic time-regimes with (i) stationary, and (ii) slow aging dynamics. We extend the quantum fluctuation-dissipation theorem to the nonequilibrium case in a consistent way with the notion of an effective temperature that drives the system in the aging regime. The classical results are recovered forh → 0.