From closed to open one-dimensional Anderson model: Transport versus spectral statistics

Abstract

Using the phenomenological expression for the level spacing distribution with only one parameter, 0 ≤ β ≤ ∞, covering all regimes of chaos and complexity in a quantum system, we show that transport properties of the one-dimensional Anderson model of finite size can be expressed in terms of this parameter. Specifically, we demonstrate a strictly linear relation between β and the normalized localization length for the whole transition from strongly localized to extended states. This result allows one to describe all transport properties in the open system entirely in terms of the parameter β and strength of coupling to continuum. For non-perfect coupling, our data show a quite unusual interplay between the degree of internal chaos defined by β, and degree of openness of the model. The results can be experimentally tested in single-mode waveguides with either bulk or surface disorder.

Localization length versus level repulsion in one-dimensional driven disordered quantum wires

“…and the parameters are very close to the ones quoted by Sorathia et al 37 , which provides justification for the use of the ensemble unfolding for Anderson models.…”

“…However, in disordered lattice models, there is no simple classical analogue but the β parameter is still very useful to study localization and can even be used to define the metal-insulator transition 35,36 . In the 1-D Anderson model there is a simple linear relationship between l ∞ /L and β (L is the total length of the system), implying that both quantities are different measures of the same property 37 . The parameter β in finite Anderson chains separates from its value in the randommatrix ensembles and takes all possible values between 0 and ∞ depending on disorder and chain length.…”

Superradiance at the localization-delocalization crossover in tubular chlorosomes

“…The parameter β behaves as a function of the energy in a complementary way to the IPR. This complementarity is tightly related to the sensitivity of the β parameter to the ratio between the localization length and the total length of the system [13,14]. In the localized region, the IPR saturates for sufficiently large tubes, indicative of the finite localization length of these states.…”

“…One of the important results in the field is the statistical relationship between the spectral repulsion parameter β and the wave function localization. The β parameter can be used to locate the metal-insulator transtion in three dimensions [10] and it has been shown that there are specific scaling laws between localization and the repulsion parameter in finite, disordered onedimensional systems [11][12][13][14]. Localization in systems with dipolar interactions like the natural light-harvesting complexes have been shown to be much weaker than for the Anderson model with only nearest neighbor coupling [15].…”

Transport through quasi-one-dimensional wires with correlated disorder

“…[6,7], outside the resonances the analytical results for the transmission coefficient and its variance, obtained for 1D continuous random potentials, can be safely used for tight-binding and KronigPenny models as well, provided the energy is not too close to the resonances. Moreover, it was shown that with specific methods one can effectively modify the theory and describe the global properties of the transmission within the whole energy region, including the resonance regions 6 .…”