Abstract:
We investigate the quasiparticle density of states in disordered d-wave superconductors. By constructing a quantum map describing the quasiparticle dynamics in such a medium, we explore deviations of the density of states from its universal form ($\propto E$), and show that additional low-energy quasiparticle states exist provided (i) the range of the impurity potential is much larger than the Fermi wavelength [allowing to use recently developed semiclassical methods]; (ii) classical trajectories exist along which the pair-potential changes sign; and (iii) the diffractive scattering length is longer than the superconducting coherence length. In the classically chaotic regime, universal random matrix theory behavior is restored by quantum dynamical diffraction which shifts the low energy states away from zero energy, and the quasiparticle density of states exhibits a linear pseudogap below an energy threshold $E^* \ll \Delta_0$.

Abstract:
Andreev billiards are finite, arbitrarily-shaped, normal-state regions, surrounded by superconductor. At energies below the superconducting gap, single-quasiparticle excitations are confined to the normal region and its vicinity, the essential mechanism for this confinement being Andreev reflection. This Paper develops and implements a theoretical framework for the investigation of the short-wave quantal properties of these single-quasiparticle excitations. The focus is primarily on the relationship between the quasiparticle energy eigenvalue spectrum and the geometrical shape of the normal-state region, i.e., the question of spectral geometry in the novel setting of excitations confined by a superconducting pair-potential. Among the central results of this investigation are two semiclassical trace formulas for the density of states. The first, a lower-resolution formula, corresponds to the well-known quasiclassical approximation, conventionally invoked in settings involving superconductivity. The second, a higher-resolution formula, allows the density of states to be expressed in terms of: (i) An explicit formula for the level density, valid in the short-wave limit, for billiards of arbitrary shape and dimensionality. This level density depends on the billiard shape only through the set of stationary-length chords of the billiard and the curvature of the boundary at the endpoints of these chords; and (ii) Higher-resolution corrections to the level density, expressed as a sum over periodic orbits that creep around the billiard boundary. Owing to the fact that these creeping orbits are much longer than the stationary chords, one can, inter alia, hear the stationary chords of Andreev billiards.

Abstract:
Andreev billiards are finite, arbitrarily-shaped, normal-state regions, surrounded by superconductor. At energies below the superconducting energy gap, single-quasiparticle excitations are confined to the normal region and its vicinity, the mechanism for confinement being Andreev reflection. Short-wave quantal properties of these excitations, such as the connection between the density of states and the geometrical shape of the billiard, are addressed via a multiple scattering approach. It is shown that one can, inter alia, hear the stationary chords of Andreev billiards.

Abstract:
The prediction of intrinsic spin Hall currents by Murakami \textit{et al.} and Sinova \textit{et al.} raised many questions about methods of detection and the effect of disorder. We focus on a contact between a Rashba type spin orbit coupled region with a normal two-dimensional electron gas and show that the spin Hall currents, though vanishing in the bulk of the sample, can be recovered from the edges. We also show that the current induced spin accumulation in the spin orbit coupled system diffuses into the normal region and contributes to the spin current in the leads.

Abstract:
We address frequency-dependent quantum transport through mesoscopic conductors in the semiclassical limit. By generalizing the trajectory-based semiclassical theory of dc quantum transport to the ac case, we derive the average screened conductance as well as ac weak-localization corrections for chaotic conductors. Thereby we confirm respective random matrix results and generalize them by accounting for Ehrenfest time effects. We consider the case of a cavity connected through many leads to a macroscopic circuit which contains ac-sources. In addition to the reservoir the cavity itself is capacitively coupled to a gate. By incorporating tunnel barriers between cavity and leads we obtain results for arbitrary tunnel rates. Finally, based on our findings we investigate the effect of dephasing on the charge relaxation resistance of a mesoscopic capacitor in the linear low-frequency regime.

Abstract:
Detection of current-induced spin accumulation via ferromagnetic contacts is discussed. Onsager's relations forbid that in a two-probe configuration spins excited by currents in time-reversal symmetric systems can be detected by switching the magnetization of a ferromangetic detector contact. Nevertheless, current-induced spins can be transferred as a torque to a contact magnetization and affect the charge currents in many-terminal configurations. We demonstrate the general concepts by solving the microscopic transport equations for the diffuse Rashba system with magnetic contacts.

Abstract:
The low-energy quasiparticle states of a disordered d-wave superconductor are investigated theoretically. A class of such states, formed via tunneling between the Andreev bound states that are localized around extended impurities (and result from scattering between pair-potential lobes that differ in sign) is identified. Its (divergent) contribution to the total density of states is determined by taking advantage of connections with certain one-dimensional random tight-binding models. The states under discussion should be distinguished from those associated with nodes in the pair potential.

Abstract:
We investigate spin transport in diffusive graphene nanoribbons with both clean and rough zigzag edges, and long-range potential fluctuations. The long-range fields along the ribbon edges cause the local doping to come close to the charge neutrality point forming $p$-$n$ junctions with localized magnetic moments, similar to the predicted magnetic edge of clean zigzag graphene nanoribbons. The resulting random edge magnetization polarizes charge currents and causes sample-to-sample fluctuations of the spin currents obeying universal predictions. We show furthermore that, although the average spin conductance vanishes, an applied transverse in-plane electric field can generate a finite spin conductance. A similar effect can also be achieved by aligning the edge magnetic moments through an external magnetic field.

Abstract:
The issue of probing the pseudogap regime of the cuprate superconductors, specifically with regard to the existence and nature of superconducting pairing correlations of d-wave symmetry, is explored theoretically. It is shown that if the d-wave correlations believed to describe the superconducting state persist into the pseudogap regime, but with pair-potential phase-fluctuations that destroy their long-range nature, then the low-energy quasiparticle states observed near extended impurities in the truly superconducting state should also persist as resonances in the pseudogap regime. The scattering of quasiparticles by these phase-fluctuations broadens what was (in the superconducting state) a sharp peak in the single-particle spectral function at low energy, as we demonstrate within the context of a simple model. This peak and its broadening is, in principle, accessible via scanning tunneling spectroscopy near extended impurities in the pseudogap regime. If so, such experiments would provide a probe of the extent to which d-wave superconducting correlations persist upon entering the pseudogap regime, thus providing a stringent diagnostic of the phase-fluctuation scenario.

Abstract:
Over the years, supersymmetric quantum mechanics has evolved from a toy model of high energy physics to a field of its own. Although various examples of supersymmetric quantum mechanics have been found, systems that have a natural realization are scarce. Here, we show that the extension of the conventional Cooper-pair box by a 4pi-periodic Majorana-Josephson coupling realizes supersymmetry for certain values of the ratio between the conventional Josephson and the Majorana- Josephson coupling strength. The supersymmetry we find is a "hidden" minimally bosonized supersymmetry that provides a non-trivial generalization of the supersymmetry of the free particle and relies crucially on the presence of an anomalous Josephson junction in the system. We show that the resulting degeneracy of the energy levels can be probed directly in a tunneling experiment and discuss the various transport signatures. An observation of the predicted level degeneracy would provide clear evidence for the presence of the anomalous Josephson coupling.