The interplay of superconductivity and disorder has intrigued scientists for several decades. Disorder is expected to enhance the electrical resistance of a system, whereas superconductivity is associated with a zero-resistance state. Although, superconductivity Superconductivity-the occurrence of the zero-resistance state-has been a central issue in solid-state physics for nearly a hundred years. About half a century after its discovery Bardeen, Cooper and Schreiffer 13 (BCS) explained its microscopic foundation. BCS theory

The localization problem of electronic states in a two-dimensional quantum spin Hall system (that is, a symplectic ensemble with topological term) is studied by the transfer matrix method. The phase diagram in the plane of energy and disorder strength is exposed, and demonstrates "levitation" and "pair annihilation" of the domains of extended states analogous to that of the integer quantum Hall system. The critical exponent nu for the divergence of the localization length is estimated as nu congruent with 1.6, which is distinct from both exponents pertaining to the conventional symplectic and the unitary quantum Hall systems. Our analysis strongly suggests a different universality class related to the topology of the pertinent system.

We study a delocalization transition for non-interacting quasiparticles moving in two dimensions, which belongs to a new symmetry class. This symmetry class can be realised in a dirty, gapless superconductor in which time reversal symmetry for orbital motion is broken, but spin rotation symmetry is intact. We find a direct transition between two insulating phases with quantized Hall conductances of zero and two for the conserved quasiparticles. The energy of quasiparticles acts as a relevant, symmetry-breaking field at the critical point, which splits the direct transition into two conventional plateau transitions.The variety of universality classes possible in singleparticle models of disordered conductors is now appreciated to be quite rich. Three of these classes were identified early in the development of weak localization theory [1]: they are distinguished by the behavior of the system under time-reversal, and by its spin properties, and are termed orthogonal, unitary and symplectic, in analogy with Dyson's classification of random matrice ensembles. Further alternatives can arise by two different mechanisms. First, in certain contexts, most notably the integer quantum Hall effect, the non-linear σ model describing a two-dimensional system may admit a topological term [2], which results in the existence of extended states at isolated energies in an otherwise localized spectrum. Physically, such systems have more than one distinct insulating phase, each characterised by its number of edge states, and separated from other phases by delocalization transitions. Second, it may happen that the Hamiltonian has an additional, discrete symmetry, absent from Dyson's classification. This is the case in two-sublattice models for localization, if the Hamiltonian has no matrix elements connecting states that belong to the same sublattice [3,4]. It is also true of the Bogoliubov-de Gennes formalism for quasiparticles in a superconductor with disorder [5,6]. One consequence of this extra symmetry is that, at a delocalization transition, critical behavior can appear not only in two-particle properties such as the conductivity, but also in single-particle quantities, such as the density of states.Universality classes in systems with extra discrete symmetries of this kind have attracted considerable attention from various directions. A general classification, systematizing earlier discussions [3,5], has been set out by Altland and Zirnbauer [6], who examined mesoscopic normal-superconducting systems as zero-dimensional realisations of some examples. Very recently, quasiparticle transport and weak localization has been studied in disordered, gapless superconductors in higher dimensions, with applications to normal-metal/superconductor junctions [7], and to thermal and spin conductivity in high temperature superconductors [8][9][10]. Separately, the behaviour of massless Dirac fermions in two space dimensions, scattered by particle-hole symmetric disorder in the form of a random vector potential, has been investigated intensive...

We study the universality class for localization which arises from models of non-interacting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation symmetries. Two-dimensional systems in this category, which is known as class D, can display phases with three different types of quasiparticle dynamics: metallic, localized, or with a quantized (thermal) Hall conductance. Correspondingly, they can show a variety of delocalization transitions. We illustrate this behavior by investigating numerically the phase diagrams of network models with the appropriate symmetry, and for the first time show the appearance of the metallic phase.PACS numbers: 73.20.Fz, 72.15.Rn The properties of quasiparticles in disordered superconductors have been a subject of much recent interest. Within a mean field treatment of pairing, the quasiparticles are noninteracting fermions, governed by a quadratic Hamiltonian which may contain effects of disorder in both the normal part and the superconducting gap function. Such Hamiltonians are representatives of a set of universality classes different from the three classes which are familiar both in normal disordered conductors and in the Wigner-Dyson random matrix ensembles. A list of additional random matrix ensembles, determined by these new symmetry classes, has been established relatively recently [1]. These additional random matrix ensembles describe zero-dimensional problems, and are appropriate to model a small grain of a superconductor in the ergodic limit. In the corresponding higherdimensional systems from the same symmetry classes, there can be transitions between metallic, localized, or quantized Hall phases for the quasiparticles [2][3][4]. The associated changes in quasiparticle dynamics must be probed by energy transport or (in singlet superconductors) spin transport, rather than charge transport, since quasiparticle charge density is not conserved [2]. There are various possibilities for behavior, depending on the particular symmetry class considered. These have been studied theoretically using nonlinear sigma model methods [2], numerically [3], and in quasi-one dimensional models [5]. An important question not addressed in such work so far, and which will not be considered here, is whether the self-consistent solution to the gap equation in the presence of disorder affects the universal statistical properties of the ensembles.In this paper we present extensive numerical results on a symmetry class with particularly rich phase diagram in two dimensions, class D. The symmetry may be realized in superconductors with broken time-reversal invariance, and either broken spin-rotation invariance (as in d-wave superconductors with spin-orbit scattering) or spinless or spin-polarized fermions (as in certain p-wave states). The nonlinear sigma model for class D [1] has been shown, in the two-dimensional case, to flow under the renormalization group to weaker values of the coupling constant [6][7][8][9]. The coupling constant is proportional to the i...

When an asymmetric double dot is hybridized with itinerant electrons, its singlet ground state and lowly excited triplet state cross, leading to a competition between the Zhang-Rice mechanism of singlet-triplet splitting in a confined cluster and the Kondo effect (which accompanies the tunneling through quantum dot under a Coulomb blockade restriction). The rich physics of an underscreened S = 1 Kondo impurity in the presence of low-lying triplet-singlet excitations is exposed and estimates of the magnetic susceptibility and the electric conductance are presented, together with applications for molecule chemisorption on metallic substrates.

An effective theory is constructed for analyzing a generic phase transition between the quantum spin Hall and the insulator phases. Occurrence of degeneracies due to closing of the gap at the transition are carefully elucidated. For systems without inversion symmetry the gap-closing occurs at ± k0( = G/2) while for systems with inversion symmetry, the gap can close only at wave-numbers k = G/2, where G is a reciprocal lattice vector. In both cases, following a unitary transformation which mixes spins, the system is represented by two decoupled effective theories of massive twocomponent fermions having masses of opposite signs. Existence of gapless helical modes at a domain wall between the two phases directly follows from this formalism. This theory provides an elementary and comprehensive phenomenology of the quantum spin Hall system.

We show that the Kondo effect can be induced by an external magnetic field in quantum dots with an even number of electrons. If the Zeeman energy B is close to the single-particle level spacing Delta in the dot, the scattering of the conduction electrons from the dot is dominated by an anisotropic exchange interaction. A Kondo resonance then occurs despite the fact that B exceeds by far the Kondo temperature T(K). As a result, at low temperatures T<

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