This article surveys a number of theoretical problems and open questions in the field of twodimensional dilute Bose gases with weak repulsive interactions. In contrast to three dimensions, in two dimensions the formation of long-range order is prohibited by the Bogoliubov-Hohenberg theorem, and Bose-Einstein condensation is not expected to be realized. Nevertheless, the first experimental indications supporting the formation of a condensate in low dimensional systems have been recently obtained. This unexpected behaviour appears to be due to the non-uniformity introduced into a system by the external trapping potential. Theoretical predictions, made for homogeneous systems, require therefore careful reexamination. We survey a number of popular theoretical treatments of the dilute weakly interacting Bose gas and discuss their regions of applicability. The possibility of Bose-Einstein condensation in a two-dimensional gas, the validity of the perturbative t-matrix approximation and the diluteness condition are issues that we discuss in detail.

We perform a detailed quantum dynamical study of nonequilibrium Josephson oscillations between interacting Bose-Einstein condensates confined in a finite-size double-well trap. We find that the Josephson junction can sustain multiple undamped Josephson oscillations up to a characteristic time scale tau(c) without quasipartcles being excited in the system. This may explain recent related experiments. Beyond a characteristic time scale tau(c) the dynamics of the junction is governed by fast, quasiparticle-assisted Josephson tunneling as well as Rabi oscillations between the discrete quasiparticle levels. We predict that an initially self-trapped state of the Bose-Einstein condensates will be destroyed by these fast dynamics.

We interpret the recent observation of a zero-bias anomaly in spin-1 quantum dots in terms of an underscreened Kondo effect. Although a spin-1 quantum dots are expected to undergo a two-stage quenching effect, in practice the log normal distribution of Kondo temperatures leads to a broad temperature region dominated by underscreened Kondo physics. General arguments, based on the asymptotic decoupling between the partially screened moment and the leads, predict a singular temperature and voltage dependence of the conductance G and differential conductance g, resulting in dg/dT ∼ 1/T and dG/dV ∼ 1/V . Using a Schwinger boson approach, we show how these qualitative expectations are borne out in a detailed many body calculation.PACS numbers: 72.15. Qm, 73.63.Kv, 75.20.Hr Single-electron transistors (SETs) offer the intriguing opportunity to probe and explore classes of strongly correlated electron behavior associated with the Kondo effect that are difficult to access in bulk materials [1,2,3,4,5]. The possibility of observing a break-down in Landau Fermi liquid behavior that accompanies the overscreened two-channel Kondo effect in quantum dots has been a subject of particular recent interest [6,7]. In this paper, we propose that singular deviations from Landau Fermi liquid behavior associated with the underscreened Kondo effect, hitherto unobserved in bulk materials, will develop in conventional quantum dots with even numbers of electrons and a triplet ground-state [8,9,10]. These deviations from conventional Fermi liquid behavior are predicted to lead to singular voltage, field and temperature dependences in the conductance.The Kondo effect in quantum dots with odd numbers of electrons, predicted more than fifteen years ago, [2,3] is now well-established by experiment [4,5]. Subsequent observations have shown that zero-bias anomalies associated with a Kondo effect can also occur in quantum dots with even occupancies, where Hund's coupling between the electrons can lead to novel degeneracies, through the formation of higher spin states, or the accidental degeneracy of singlet and triplet states. Zero-bias anomalies in integer spin quantum dots were first reported by Schmid et al. [8]. Sasaki et al [9] later discovered a zerobias anomaly in even electron quantum dots, associated with the degeneracy point between singlet and triplet states, tuned by a small magnetic field. Most recently, Kogan et al [10] have shown that the singlet-triplet excitation energy in lateral quantum dots can be tuned by the gate voltage, explicitly demonstrating that the zero bias anomaly develops once the triplet state drops below the singlet configuration. Pustilnik and Glazman [11] have analyzed the lowtemperature Fermi liquid physics of higher spin quantum dots. Their analysis shows that lateral quantum dot in a triplet configuration develops two screening channels which fully screen the local moment at the lowest temperatures. Using the Landauer formula, they deduce the conductance G of the Fermi liquid which develops to bewhere δ 1 ...

We discuss the physics of a of a spin-1 quantum dot, coupled to two metallic leads and develop a simple model for the temperature dependence of its conductance. Such quantum dots are described by a two-channel Kondo model with asymmetric coupling constants and the spin screening of the dot by the leads is expected to proceed via a two-stage process. When the Kondo temperatures of each channel are widely separated, on cooling, the dot passes through a broad cross-over regime dominated by underscreened Kondo physics. A singular, or non-fermi liquid correction to the conductance develops in this regime. At the lowest temperatures, destructive interference between resonant scattering in both channels leads to the eventual suppression of the conductance of the dot. We develop a model to describe the growth, and ultimate suppression of the conductance in the two channel Kondo model as it is screened successively by its two channels. Our model is based upon large-N approximation in which the localized spin degrees of freedom are described using the Schwinger boson formalism.

We study the effect of a strong electric field on the fluctuation conductivity within the timedependent Ginzburg-Landau theory for the case of arbitrary dimension. Our results are based on the analytical derivation of the velocity distribution law for the fluctuation Cooper pairs, from the Boltzmann equation. Special attention is drawn to the case of small nonlinearity of conductivity, which can be investigated experimentally. We obtain a general relation between the nonlinear conductivity and the temperature derivative of the linear Aslamazov-Larkin conductivity, applicable to any superconductor. For the important case of layered superconductors we derive an analogous relation between the small nonlinear correction for the conductivity and the fluctuational magnetoconductivity. On the basis of these relations we provide new experimental methods for determining both the lifetime constant of metastable Cooper pairs above Tc and the coherence length. A systematic investigation of the 3rd harmonic of the electric field generated by a harmonic current can serve as an alternative method for the examination of the metastable Cooper-pair relaxation time.