Abstract:
We investigate the static and dynamical behavior of 1D interacting fermions in disordered Hubbard chains, contacted to semi-infinite leads. The chains are described via the repulsive Anderson-Hubbard Hamiltonian, using static and time-dependent lattice density-functional theory. The dynamical behavior of our quantum transport system is performed via an integration scheme available in the literature, which we modify via the recursive Lanczos method, to increase its efficiency. To quantify the degree of localization due to disorder and interactions, we adapt the definition of the inverse participation ratio to obtain an indicator which is both suitable for quantum transport geometries and which can be obtained within density-functional theory. Lattice density functional theories are reviewed and, for contacted chains, we analyze the merits and limits of the coherent-potential approximation in describing the spectral properties, with interactions included via lattice density functional theory. Our approach appears to able to capture complex features due to the competition between disorder and interactions. Specifically, we find a dynamical enhancement of delocalization in presence of a finite bias, and an increase of the steady-state current induced by inter-particle interactions. This behavior is corroborated by results for the time-dependent densities and for the inverse participation ratio. Using short isolated chains with interaction and disorder, a brief comparative analysis between time-dependent density-functional theory and exact results is then given, followed by general conclusive remarks.

Abstract:
Localized states in one-dimensional open disordered systems and their connection to the internal structure of random samples have been studied. It is shown that the localization of energy and anomalously high transmission associated with these states are due to the existence inside the sample of a transparent (for a given resonant frequency) segment with the minimal size of order of the localization length. A mapping of the stochastic scattering problem in hand onto a deterministic quantum problem is developed. It is shown that there is no one-to-one correspondence between the localization and high transparency: only small part of localized modes provides the transmission coefficient close to one. The maximal transmission is provided by the modes that are localized in the center, while the highest energy concentration takes place in cavities shifted towards the input. An algorithm is proposed to estimate the position of an effective resonant cavity and its pumping rate by measuring the resonant transmission coefficient. The validity of the analytical results have been checked by extensive numerical simulations and wavelet analysis.

Abstract:
The perturbation theory is developed for joint statistics of the advanced and retarded Green's functions of the 1D Schrodinger equation with a piecewise-constant random potential. Using this method, analytical expressions are obtained for spectral dependence of the degree of localization and for the limiting (at $t\rightarrow\infty$) probability to find the particle at the point it was located at $t = 0$ (Andeson criterion). Definition of the localization length is introduced. The computer experiments confirming correctness of the calculations are described.

Abstract:
The role of a prominent photonic bandgap (PBG) on the phenomenon of transverse localization of light in a semi-infinite lossless waveguide lattice consisting of evanescently coupled disordered one-dimensional optical waveguides has been investigated numerically. The interplay between the underlying photonic bandgap due to inherent periodicity of the optical system and various levels of deliberately induced transverse disorder in its refractive index periodicity has been studied. We show that the PBG indeed plays an important role and its simultaneous presence could catalyze realization of localized light even when strength of disorder is not sufficiently strong to independently cause localization of light. An important outcome of this study revealed that PBG could be gainfully exploited to tailor the spectral window for localization of light in potential applications like lasing in a disordered optical lattice.

Abstract:
The explicit analytical expression for the distribution function of parametric derivatives of energy levels ("level velocities") with respect to a random change of scattering potential is derived for the chaotic quantum systems belonging to the quasi 1D universality class (quantum kicked rotator, "domino" billiard, disordered wire, etc.).

Abstract:
We report the enhancement of the effect of transverse localization of light (TL) in presence of a weak longitudinal modulation of refractive index in disordered waveguide lattices. In our chosen lattices, tunneling inhibition along length favors to achieve the diffraction-free propagation along with the simultaneous presence of transverse disorder. Results will be useful to tune the threshold value of disorder to achieve localized light.

Abstract:
This paper is part of a broader study whose main goal is the study of the finite-energy spectral properties of the non-perturbative one-dimensional (1D) Hubbard model and the evaluation of finite-energy correlation-function expressions. Here we study the deviations from the ground state values of double occupancy which result from creation or annihilation of holons, spinons, and pseudoparticles. The band-momentum dependence of the obtained double-occupancy spectra provides important information on the degree of localization/delocalization of the real-space lattice electron site distribution configurations associated with the pseudoparticles. We also study the band-momentum, on-site electronic repulsion, and electronic density dependence of the pseudoparticle energy bands. The shape of these bands plays an important role in the finite-energy spectral properties of the model. Such a shape defines the form of the lines in the momentum-energy/frequency plane where the peaks and edges of the one-electron and two-electron spectral weight of physical operators are located. Our findings are useful for the study of the one-electron and two-electron spectral weight distribution of physical operators.

Abstract:
The competition between the Mott transition and the Anderson localization in one dimensional electron systems is studied based upon the bosonization and the renormalization group method. The beta function is calculated up to the second order in the strength of diagonal disorder by using a replica trick. It is found that the sufficiently strong forward scattering by random impurities destroys the Mott-Hubbard gap, and the backward scattering gives rise to the Anderson localization for the resulting gapless state. On the other hand, if the Umklapp interaction is strong enough, the Mott insulating state still overwhelms the Anderson localization.

Abstract:
A method is proposed for manipulating with diagrammatic expansion of Green's function of Frenkel's exciton random walks on the perfect lattice. The method allows one to select diagrams, to supply diagrams with factors containing information about the number of sites the diagram has passed through, etc. Simple problems related to the defect lattices are considered using the proposed method. The new criterion of localization of Frenkel exciton - the number of sites covered by the wave function - is established. The number of sites covered by the zero state of 1D non-diagonally disordered chain is studied. It is shown that this problem can be solved by calculating the random walks Green's function with modified diagrammatic expansion. By means of the developed method an exact analitical expression for the number of sites covered by the zero-state is obtained and zero-state is shown to be localized.

Abstract:
We report on the experimental observation of reduced light energy transport and disorder-induced localization close to a boundary of a truncated one-dimensional (1D) disordered photonic lattice. Our observations uncover that near the boundary a higher level of disorder is required to obtain similar localization than in the bulk.