Scale-dependent correction to the dynamical conductivity of a disordered system at unitary symmetry

Anderson localization has been studied extensively for more than half a century. However, while our understanding has been greatly enhanced by calculations based on a small epsilon expansion in d = 2 + epsilon dimensions in the framework of non-linear sigma models, those results can not be safely extrapolated to d = 3. Here we calculate the leading scale-dependent correction to the frequency-dependent conductivity sigma(omega) in dimensions d <= 3. At d = 3 we find a leading correction Re{sigma(omega)} ~ |omega|, which at low frequency is much larger than the omega^2 correction deriving from the Drude law. We also determine the leading correction to the renormalization group beta-function in the metallic phase at d = 3.

Superconducting proximity effect in quantum wires without time-reversal symmetry

We study the superconducting proximity effect in a quantum wire with broken time-reversal (TR) symmetry connected to a conventional superconductor. We consider the situation of a strong TR-symmetry breaking, so that Cooper pairs entering the wire from the superconductor are immediately destroyed. Nevertheless, some traces of the proximity effect survive: for example, the local electronic density of states (LDOS) is influenced by the proximity to the superconductor, provided that localization effects are taken into account. With the help of the supersymmetric sigma model, we calculate the average LDOS in such a system. The LDOS in the wire is strongly modified close to the interface with the superconductor at energies near the Fermi level. The relevant distances from the interface are of the order of the localization length, and the size of the energy window around the Fermi level is of the order of the mean level spacing at the localization length. Remarkably, the sign of the effect is sensitive to the way the TR symmetry is broken: In the spin-symmetric case (orbital magnetic field), the LDOS is depleted near the Fermi energy, whereas for the broken spin symmetry (magnetic impurities), the LDOS at the Fermi energy is enhanced.

Hybridization of wave functions in one-dimensional localization

A quantum particle can be localized in a disordered potential, the effect known as Anderson localization. In such a system, correlations of wave functions at very close energies may be described, due to Mott, in terms of a hybridization of localized states. We revisit this hybridization description and show that it may be used to obtain quantitatively exact expressions for some asymptotic features of correlation functions, if the tails of the wave functions and the hybridization matrix elements are assumed to have log-normal distributions typical for localization effects. Specifically, we consider three types of one-dimensional systems: a strictly one-dimensional wire and two quasi-one-dimensional wires with unitary and orthogonal symmetries. In each of these models, we consider two types of correlation functions: the correlations of the density of states at close energies and the dynamic response function at low frequencies. For each of those correlation functions, within our method, we calculate three asymptotic features: the behavior at the logarithmically large Mott length scale, the low-frequency limit at length scale between the localization length and the Mott length scale, and the leading correction in frequency to this limit. In the several cases, where exact results are available, our method reproduces them within the precision of the orders in frequency considered.