Localisation 2011

The following sections are included:

• PREFACE

• CONTENTS

Field transmission coefficients for microwave radiation between arrays of points on the incident and output surfaces of random samples are analyzed to yield the underlying quasi-normal modes and transmission eigenchannels of each realization of the sample. The linewidths, central frequencies, and transmitted speckle patterns associated with each of the modes of the medium are found. Modal speckle patterns are found to be strongly correlated leading to destructive interference between modes. This explains distinctive features of transmission spectra and pulsed transmission. An alternate description of wave transport is obtained from the eigenchannels and eigenvalues of the transmission matrix. The maximum transmission eigenvalue, τ₁ is near unity for diffusive waves even in turbid samples. For localized waves, τ₁ is nearly equal to the dimensionless conductance, which is the sum of all transmission eigenvalues, g = Στ_n. The spacings between the ensemble averages of successive values of lnτ_n are constant and equal to the inverse of the bare conductance in accord with predictions by Dorokhov. The effective number of transmission eigenvalues N_eff determines the contrast between the peak and background of radiation focused for maximum peak intensity. The connection between the mode and channel approaches is discussed.

We have calculated the lowest energy quantized spectra of Intrinsically Localized Modes (ILMs) for the Fermi-Pasta-Ulam lattices. The quantized ILM spectra are composed of resonances in the two-phonon continuum and branches of infinitely long-lived excitations that are bound states formed from even numbers of phonons. For quartic anharmonicity and one atom per unit cell, the calculated ILMs are consistent with the results of previous calculations using the number conserving approximation. However, by contrast the ILM spectrum of the lattice with cubic interactions couples resonantly with the singlephonon spectrum and cannot be calculated within a number conserving approximation. Furthermore we argue that, by introducing a sufficiently strong cubic non-linearity, the quantized ILMs can be observed directly through the single-phonon inelastic neutron scattering spectrum. We compare our theoretical predictions with the recent experimental observation of breathers in NaI by Manley et al.

It is well known that the metal-insulator transition in two dimensions for non-interacting fermions takes place at infinitesimal disorder. In contrast, the superconductor-to-insulator transition takes place at a finite critical disorder (on the order of V_c ∼ 2t), where V is the typical width of the distribution of random site energies and t is the hopping scale. In this article we compare the localization/delocalization properties of one and two particles. Whereas the metal-insulator transition is a consequence of singleparticle Anderson localization, the superconductor-insulator transition (SIT) is due to pair localization – or, alternatively, fluctuations of the phase conjugate to pair density. The central question we address is how superconductivity emerges from localized single-particle states. We address this question using inhomogeneous mean field theory and quantum Monte Carlo techniques and make several testable predictions for local spectroscopic probes across the SIT. We show that with increasing disorder, the system forms superconducting blobs on the scale of the coherence length embedded in an insulating matrix. In the superconducting state, the phases on the different blobs are coherent across the system whereas in the insulator long-range phase coherence is disrupted by quantum fluctuations. As a consequence of this emergent granularity, we show that the single-particle energy gap in the density of states survives across the transition, but coherence peaks exist only in the superconductor. A characteristic pseudogap persists above the critical disorder and critical temperature, in contrast to conventional theories. Surprisingly, the insulator has a two-particle gap scale that vanishes at the SIT despite a robust single-particle gap.

We discuss the competition between the Kondo effect, the spin glass state and a magnetic order observed in disordered Cerium systems. We present firstly the experimental situation of disordered alloys such as CeNi_{1 − x}Cu_x and then the different theoretical approaches based on the Kondo lattice model, with different descriptions of the intersite exchange interaction for the spin glass. After the gaussian approach of the Sherrington-Kirkpatrick model, we discuss the Mattis and the van Hemmen models. Then, we present simple cluster calculations in order to describe the percolative evolution of the clusters from the cluster spin glass to the inhomogeneous ferromagnetic order recently observed in CeNi_{1 − x}Cu_x disordered alloys and finally we discuss the effect of random and transverse magnetic field.

We consider transport properties of disordered two-dimensional electron gases under high perpendicular magnetic field, focusing in particular on the peak longitudinal conductivity at the quantum Hall plateau transition. We use a local conductivity model, valid at temperatures high enough such that quantum tunneling is suppressed, taking into account the random drift motion of the electrons in the disordered potential landscape and inelastic processes provided by electron-phonon scattering. A diagrammatic solution of this problem is proposed, which leads to a rich interplay of conduction mechanisms, where classical percolation effects play a prominent role. The scaling function for is derived in the high temperature limit, which can be used to extract universal critical exponents of classical percolation from experimental data.

In Ref. 1, we reported an estimate of the critical exponent for the divergence of the localization length at the quantum Hall transition that is significantly larger than those reported in the previous published work of other authors. In this paper, we update our finite size scaling analysis of the Chalker-Coddington model and suggest the origin of the previous underestimate by other authors. We also compare our results with the predictions of Lötken and Ross.

The tunneling between the Laughlin state and its quasihole excitations are studied by using the Jack polynomial. We find a universal analytical formula for the tunneling amplitude, which can describe both bulk and edge quasihole excitations. The asymptotic behavior of the tunneling amplitude reveals the difference and the crossover between bulk and edge states. The effects of the realistic coulomb interaction with a background-charge confinement potential and disorder are also discussed. The stability of the tunneling amplitude manifests the topological nature of fractional quantum Hall liquids.

We discuss the role of rare fluctuation effects in quantum condensed matter systems. In particular, we present recent numerical results of the effect of resonant states in Anderson's original model of electron localization. We find that such resonances give rise to anomalous behavior of eigenstates not just far in the Lifshitz tail, but rather for a substantial fraction of eigenstates, especially for intermediate disorder. The anomalous behavior includes non-analyticity in various properties as a characteristic. The effect of dimensionality on the singularity, which is present in all dimensions, is described, and the behavior for bounded and unbounded disorder is contrasted.

We report a numerical analysis of Anderson localization in a model of a doped semiconductor. The model incorporates the disorder arising from the random spatial distribution of the donor impurities and takes account of the electron-electron interactions between the carriers using density functional theory in the local density approximation. Preliminary results suggest that the model exhibits a metal-insulator transition.

We employed a first-principles theory – the supersymmetric field theory – formulated for wave transport in very general open media to study static transport of waves in quasione- dimensional localized samples. We predicted analytically and confirmed numerically that in these systems, localized waves display an unconventional diffusive phenomenon. Different from the prevailing self-consistent local diffusion model, our theory is capable of capturing all disorder-induced resonant transmissions, which give rise to significant enhancement of local diffusion inside a localized sample. Our theory should be able to be generalized to two- and three-dimensional open media, and open a new direction in the study of Anderson localization in open media.

At the Anderson metal-insulator transition the eigenstates develop multifractal fluctuations. Therefore their properties are intermediate between being extended and localized. As a result these wave functions are power-law correlated, which causes a substantial suppression of the local density of states at some random positions, resembling random local pseudogaps at the Fermi energy. Consequently the Kondo screening of magnetic moments is suppressed when a magnetic impurity happens to be at such a position. Due to these unscreened magnetic moments the critical exponents and multifractal dimensions at the metal-insulator transition take their smaller, unitary ensemble values for exchange couplings not exceeding a certain critical value J* ≈ .3D, where D is the band width. Here we present numerical calculations of the distribution of Kondo temperatures for the critical Power-law Band Random Matrix (PBRM) ensemble, whose properties are similar to that of the Anderson transition with the advantage of using a continuous parameter for tuning the generalized multifractal dimensions of the eigenstates.

We study the (Anderson) metal-insulator transition (MIT) in tight binding models (TBM) of disordered systems using the scaling behavior of the typical density of states (GDOS) as localization criterion. The GDOS is obtained as the geometrical mean value of the local density of states (LDOS) averaged over many different lattice sites and disorder realizations. The LDOS can efficiently be obtained within the kernel polynomial method (KPM). To check the validity and accuracy of the method, we apply it here to the standard Anderson model of disordered systems, for which the results (for instance for the critical disorder strength of the Anderson transition) are well known from other methods.

We have estimated the critical exponent describing the divergence of the localization length at the metal-quantum spin Hall insulator transition. The critical exponent for the metal-ordinary insulator transition in quantum spin Hall systems is known to be consistent with that of topologically trivial symplectic systems. However, the precise estimation of the critical exponent for the metal-quantum spin Hall insulator transition proved to be problematic because of the existence, in this case, of edge states in the localized phase. We have overcome this difficulty by analyzing the second smallest positive Lyapunov exponent instead of the smallest positive Lyapunov exponent. We find a value for the critical exponent ν = 2.73 ± 0.02 that is consistent with that for topologically trivial symplectic systems.

Of late, the study of BCS-BEC crossover in the presence of weak random impurity is an interesting issue. In this proceedings we study the effect of this disorder which is included through the Nozières and Smith-Rink theory of superconducting fluctuations. In the weak regime, the random potential leaves an effect on the superconducting order parameter but it spares the chemical potential. Here we present the exact behavior of the mean field quantities as a function of the disorder by self-consistently solving the coupled equations.

We present a comparative study of the localization effects induced by harmonic trapping and random onsite disorder in a 2D s-wave superconductor. Performing a numeric computation of the Bogoliubov-de Gennes equations in the presence of parabolic trap and random disorder, we obtain the eigensolutions as a function of trap depth and disorder strength. While the wavefunctions demonstrate localization tendencies for moderate disorder strengths, with finally yielding an insulating behavior at large disorder, the harmonic confinement effects are seen to be vastly more drastic in inducing localization effects.

The aim here is to show an example for localization and relevant low-dimensional superfluids in nuclear system. Due to a particular property of tensor force originating from the One-Pion-Exchange (OPE) between two nucleons, dense nuclear medium undergoes a layer confinement of the nucleons on one hand and also pion condensation (PC) for pion field mediating two-nucleon interaction on the other hand. The localization is characterized by a layered structure with a specific spin-isospin ordering. In that situation, the pairing problem has a two-dimensional (2D) character, i.e., low-dimensional superfluid realized in the hadronic matter with strong interactions. The pairing description suitable to the 2D nature is presented and possible realization of superfluidity in neutron stars is discussed, togeter with its effect on the cooling scenarios.

We theoretically investigate the effects of intercalation of a thin Ti layer between graphene (Gr) and metal surface (Au, Ag, Al, Pt, and Pd) on structural and electronic properties by using the first-principles total energy calculations. We find that the strongest binding energy is realized when 1 monolayer (ML) of Ti is intercalated between Gr and a metal surface independent of the metal atoms, which is 0.08-0.15 eV larger than that for Gr absorbed on a Ti(0001) surface. As the number of Ti layers increases, the binding energy monotonically decreases and converges to that for Gr adsorbed onto the Ti surface. We show that the origin of the enhancement of binding energy can be classified into two classes by considering the affinity of Ti for the metal surfaces.

The notion of chiral symmetry for the conventional Dirac cone is generalized to include the tilted Dirac cones, where the generalized chiral operator turns out to be nonhermitian. It is shown that the generalized chiral symmetry generically protects the zero modes (n = 0 Landau level) of the Dirac cone even when tilted. The present generalized symmetry is equivalent to the condition that the Dirac Hamiltonian is elliptic as a differential operator, which provides an explicit relevance to the index theorem.

We study that stability of edge localized states in semi-infinite graphene with a corner edge of the angles 60°, 90°, 120° and 150°. We adopt a nearest-neighbor tight-binding model to calculate the local density of states (LDOS) near each corner edge using Haydock's recursion method. The results of the LDOS indicate that the edge localized states stably exist near the 60°, 90°, and 150° corner, but locally disappear near the 120° corner. By constructing wave functions for a graphene ribbon with three 120° corners, we show that the local disappearance of the LDOS is caused by destructive interference of edge states and evanescent waves.

We report our recent numerical study on the effects of dephasing on a perfectly conducting channel (PCC), its presence believed to be dominant in the transport characteristics of a zigzag graphene nanoribbons (GNR) and of a metallic carbon nanotubes (CNT). Our data confirms an earlier prediction that a PCC in GNR exhibits a peculiar robustness against dephasing, in contrast to that of the CNT. By studying the behavior of the conductance as a function of the system's length we show that dephasing destroys the PCC in CNT, whereas it stabilizes the PCC in GNR. Such opposing responses of the PCC against dephasing stem from a different nature of the PCC in these systems.

The dependence of spin relaxation on the direction of the quantum wire under Rashba and Dresselhaus (linear and cubic) spin-orbit coupling is studied. Comprising the dimensional reduction of the wire in the diffusive regime, the lowest spin relaxation and dephasing rates for (001) and (110) systems are found.

We present numerical and analytical studies of the crossover between weak antilocalization and weak localization in monolayer graphene and their influence on thermopower. By the use of the recursive Green's function method, we find that these quantum corrections result in an enhancement of thermopower, which can be observed in the resulting magnetic field dependence. This magneto thermopower strongly depends on the size and strength of the impurities as well as on the back gate voltage of the system and the impurity concentration. We show in detail the crossover of these localization effects with these parameters. Using the disorder parameters of the numerical calculation, we find quantitative agreement with the analytical calculations.

We study the Ruderman-Kittel-Katsuya-Yoshida (RKKY) interactions in graphene controlling the gate voltage and applying nonmagnetic disorder. It is found that oscillations of the RKKY interactions in undoped graphene are characterized by the interference of two neighbor Dirac nodes K and K′ in the first Brillouin zone and decays with R⁻³ distance dependence. In the slightly doped graphene, a beating pattern, which consists of two characteristic wavevectors (K – K and k_F), starts to appear. The distance dependence in this regime shows a crossover from the R⁻³ to R⁻². We present the effect of weak disorder on the RKKY interactions in diffusive regime. The arithmetic averaged interaction over disorder configurations decreases exponentially at distances exceeding the elastic mean free path, while the geometrical average(typical) value has the same power-law as the clean limit.

The Heisenberg spin 1/2 chain is revisited in the perturbative RG approach with special focus on the transition of the critical exponents. We give a compact review that first order RG in the couplings is sufficient to derive the exact transition from ν = 1 to ν = 2/3, if the boson radius obtained in the bosonization procedure is replaced by the exact radius obtained in the Bethe approach. We explain the fact, that from the bosonization procedure alone, the critical exponent can not be derived correctly in the isotropic limit J_z → J. We further state that this fact is important if we consider to bosonize the antiferromagnetic super spin chain for the quantum Hall effect.