Localization properties of a one-dimensional tight-binding model with nonrandom long-range intersite interactions

Abstract

We perform both analytical and numerical studies of the one-dimensional tight-binding Hamiltonian with stochastic uncorrelated on-site energies and nonfluctuating long-range hopping integrals J mn = J / ͉m − n͉ . It was argued recently ͓A. Rodríguez et al., J. Phys. A 33, L161 ͑2000͔͒ that this model reveals a localizationdelocalization transition with respect to the disorder magnitude provided 1 Ͻ Ͻ 3 / 2. The transition occurs at one of the band edges ͑the upper one for J Ͼ 0 and the lower one for J Ͻ 0͒. The states at the other band edge are always localized, which hints at the existence of a single mobility edge. We analyze the mobility edge and show that, although the number of delocalized states tends to infinity, they form a set of null measure in the thermodynamic limit, i.e., the mobility edge tends to the band edge. The critical magnitude of disorder for the band edge states is computed versus the interaction exponent by making use of the conjecture on the universality of the normalized participation number distribution at the transition.

Non-Anderson critical scaling of the Thouless conductance in 1D

“…Furthermore, the scaling of the dependence of the critical disorder strength on |ε| is given by γ c ∝ |ε| 2 3 , in accordance with the earlier study in Ref. 26 and in disagreement with the expectation γ c ∝ |ε| from the perturbative RG analysis. At the same time, the numerical result for dynamical exponent, z = 1 2 + O(ε), agrees well with RG predictions.…”

“…5) which also contradicts the predictions of the analytical RG approach but is consistent with earlier results for a similar one-dimensional model. 26…”

“…The singular dispersion ∝ |k ± π/2| α sign(k ± π/2) near the nodes corresponds to the power-law hopping ∝ 1/|x| 1+α sign x in coordinate space at large distances x. Disorder-driven transitions in similar models with powerlaw hopping, corresponding to the dispersion ∝ k α which is even with respect to the momentum inversion k → −k), have been studied numerically in Refs. [23][24][25][26][27]. Because the dispersion considered here is odd with respect to inverting the momentum measured from the node and the momentum states with energies above and below the nodes have equal weights, the energies of the nodes are not renormalised if the system is exposed to quenched short-range-correlated disorder with zero average.…”

Quantum walk and Anderson localization of rotational excitations in disordered ensembles of polar molecules

“…The n n 1 3 | − ′| dependence of the tunnelling amplitudes is particularly interesting [20]. Beyond allowing for direct transitions between distant lattice sites, it leads to nonanalyticity of particle dispersion at low particle velocities [21]. It may also lead to interesting long-range correlations in the localization dynamics of multiple interacting particles [46,47].…”

“…For example, the scaling hypothesis providing the relation between quantum localization in systems of different dimensionality [16] still awaits experimental confirmation. Equally important and widely debated are the effects of inter-particle interactions [17] and dissipative forces [18] on localization in far-from-equilibrium systems or the role of long-range tunnelling amplitudes [19][20][21][22] allowing particles to transition between distant lattice sites.…”