We study the equilibrium properties of a quantum dot connected to a bulk lead by a single-mode quantum point contact. The ground state energy and other thermodynamic characteristics of the grain show periodic dependence on the gate voltage (Coulomb blockade). We consider the case of almost perfect transmission, and show that the oscillations exist as long as the transmission coefficient of the contact is less than unity. Near the points where the dot charge is half-integer the thermodynamic characteristics show a non-analytic behavior identical to that of the two-channel spin-1 2 Kondo model. In particular, at any transmission coefficient the capacitance measured between the gate and the lead shows periodic logarithmic singularities as a function of the gate voltage.

The conductance of a weakly interacting electron gas in the presense of a single scatterer is found at arbitrary strength of the scattering potential. At weak interaction, a simple renormalization group approach can be used instead of the standard bosonization procedure. Our technique allows to take into account the backscattering of electrons that leads to a non-Luttinger-liquid behavior of the low-temperature conductance. In the presence of magnetic field, the backscattering may give rise to a peak in differential conductance at bias equal to the Zeeman energy.

The superconductivity in very thin rings is suppressed by quantum phase slips. As a result, the amplitude of the persistent current oscillations with flux becomes exponentially small, and their shape changes from sawtooth to a sinusoidal one. We reduce the problem of low-energy properties of a superconducting nanoring to that of a quantum particle in a sinusoidal potential and show that the dependence of the current on the flux belongs to a one-parameter family of functions obtained by solving the respective Schrödinger equation with twisted boundary conditions.

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We study the superconductivity in small grains in the regime when the quantum level spacing δε is comparable to the gap ∆. As δε is increased, the system crosses over from superconducting to normal state. This crossover is studied by calculating the dependence of the ground state energy of a grain on the parity of the number of electrons. The states with odd numbers of particles carry an additional energy ∆P , which shows non-monotonic dependence on δε. Our predictions can be tested experimentally by studying the parity-induced alternation of Coulomb blockade peak spacings in grains of different sizes.The standard BCS theory [1] gives a good description of the phenomenon of superconductivity in large samples. However, it was noticed by Anderson [2] in 1959 that as the size of a superconductor becomes smaller, and the quantum level spacing in the sample δε approaches the superconducting gap ∆, the BCS theory fails. The interest to the superconductivity in such ultrasmall grains was renewed by recent experiments by Ralph, Black and Tinkham [3,4], who fabricated and studied nanometerscale aluminum grains. In qualitative agreement with the prediction [2], they demonstrated [4] the existence of superconducting gap in relatively large grains, with estimated level spacings δε ≈ 0.02 and 0.08 meV smaller than the superconducting gap ∆ ≈ 0.31 meV, whereas no signs of superconductivity were observed [3] in smaller grains, δε ≈ 0.7 meV. These experiments raise a theoretical question about the nature of the crossover from superconducting to normal state in ultrasmall particles with level spacings δε ∼ ∆.This problem was addressed in two recent theoretical papers. J. von Delft et al.[5] explored the BCS gap equation in a finite-size system with equidistant discrete energy levels, and found that as the level spacing is increased, the superconducting gap of the grain vanishes at a certain critical value of δε, which is of order ∆ and depends on the parity of the total number of electrons in the grain. Smith and Ambegaokar [6] extended the treatment of Ref.[5] to take into account Wigner-Dyson fluctuations of the energy levels in the grain.It is worth noting that the theories [5,6] treat the superconductivity in small grains within the selfconsistent mean field approximation for the superconducting order parameter. Although this approximation works well for large systems, one should expect the quantum fluctuations of the order parameter to grow when the level spacing δε reaches ∆. In this paper we present a theory of superconductivity in ultrasmall grains which includes the effects of quantum fluctuations of the order parameter.We show that the corrections to the mean field results which are small in large grains, δε ≪ ∆, become important in the opposite limit, δε ≫ ∆.The superconducting gap ∆ studied in Refs. [5,6] is not well defined in the presence of quantum fluctuations. Therefore, we must first identify an observable physical quantity which characterizes the superconducting properties of small grains. The most convenient s...