Josephson effect between superconducting nanograins with discrete energy levels

We investigate the Josephson effect between two coupled superconductors, coupled by the tunneling of pairs of electrons, in the regime that their energy level spacing is comparable to the bulk superconducting gap, but neglecting any charging effects. In this regime, BCS theory is not valid, and the notion of a superconducting order parameter with a well-defined phase is inapplicable. Using the density matrix renormalization group, we calculate the ground state of the two coupled superconductors and extract the Josephson energy. The Josephson energy is found to display a reentrant behavior (decrease followed by increase) as a function of increasing level spacing. For weak Josephson coupling, a tight-binding approximation is introduced, which illustrates the physical mechanism underlying this reentrance in a transparent way. The DMRG method is also applied to two strongly coupled superconductors and allows a detailed examination of the limits of validity of the tight-binding model

The Density Matrix Renormalization Group for finite Fermi systems

The Density Matrix Renormalization Group (DMRG) was introduced by Steven White in 1992 as a method for accurately describing the properties of one-dimensional quantum lattices. The method, as originally introduced, was based on the iterative inclusion of sites on a real-space lattice. Based on its enormous success in that domain, it was subsequently proposed that the DMRG could be modified for use on finite Fermi systems, through the replacement of real-space lattice sites by an appropriately ordered set of single-particle levels. Since then, there has been an enormous amount of work on the subject, ranging from efforts to clarify the optimal means of implementing the algorithm to extensive applications in a variety of fields. In this article, we review these recent developments. Following a description of the real-space DMRG method, we discuss the key steps that were undertaken to modify it for use on finite Fermi systems and then describe its applications to Quantum Chemistry, ultrasmall superconducting grains, finite nuclei and two-dimensional electron systems. We also describe a recent development which permits symmetries to be taken into account consistently throughout the DMRG algorithm. We close with an outlook for future applications of the method.Comment: 48 pages, 17 figures Corrections made to equation 19 and table