Non-locally correlated disorder and delocalization in one dimension (I): Density of states

Abstract

We study delocalization transition in a one-dimensional electron system with quenched disorder by using supersymmetric (SUSY) methods. Especially we focus on effects of nonlocal correlation of disorder, for most of studies given so far considered δ-function type white noise disorder. We obtain wave function of the "lowest-energy" state which dominates partition function in the limit of large system size. Density of states is calculated in the scaling region. The result shows that delocalization transition is stable against nonlocal short-ranged correlation of disorder. Especially states near the band center are enhanced by the correlation of disorder which partially suppresses random fluctuation of disorder. Physical picture of the localization and the delocalization transition is discussed. Quenched disorder plays important roles in various physical phenomena. Anderson localization is one of these examples[1]. Especially in two and lower dimensions almost all states are localized, and extended delocalized states are isolated points in physical parameter region. The transition between quantum Hall plateaus is such an example. For the transition between integer quantum Hall plateaus, useful model, named network model, has been proposed[2], which incorporates effects of localization and quantum tunneling in a strong magnetic field. Numerical studies indicate that the network model belongs to the same universality class of the transition between integer quantum Hall plateaus. However it is rather difficult to solve the network model analytically because there exists no controllable parameter for perturbative expansion and also because of nature of quenched disorder itself. Compared with the twodimensional (2D) systems, one dimensional (1D) systems are more tractable. These 1D systems include Dyson's study on random strength harmonic springs[3], random Ising model by McCoy and Wu[4], and more recently random exchange spin chains[5]and random hopping tight-binding (RHTB) model [6,7,8].Recently supersymmetric (SUSY) methods appear useful for handling the quenched disorder. SUSY methods are applied to the network model [9], the 1D RHTB model [7] etc. In this paper we shall revisit the 1D RHTB model by applying the SUSY methods. Especially we shall consider nonlocally correlated quenched disorder and study stability of the delocalization transition which exists in the case of the δ-function-type white noise disorder. The model contains two parameters which control magnitude of fluctuation and correlation length of disorder. We expect that we can get detailed physical picture of (de)localization transition from calculations of density of states, Green's functions, etc. as a function of the above parameters. Another motivation of the present work is rather technical, i.e., we show how to use the SUSY methods for nonlocally correlated disorder systems.

Amplitude scaling behavior of band center states of Frenkel exciton chains with correlated off-diagonal disorder

“…We have used a negative eigenvalue counting technique [25] which provides a simple and physically transparent analysis of the IDOS and the ILL. We investigated the question of scaling when there was correlated off-diagonal disorder such as might occur, for example, when the offdiagonal interaction depended on the distance between two ions and thus would be affected by small displacements of the ions from their equilibrium positions [15]. Our numerical data indicate that in the correlated case the scaling behavior found by Ziman [13] is followed only for a limited range of c and very small E. We compared our results with the scaling predictions of Ziman [13] and with the similar tight binding correlated electronic case [7,8]. We computed the asymptotic amplitude factors as a function of concentration and found that they are equal to the uncorrelated variance at c ¼ 0:5 multiplied by the factor c=ð1 À cÞ (see Appendix A).…”

“…In particular, the asymptotic amplitude factor is not symmetric about c ¼ 0:5 while the ILL diverges for concentrations approaching 1 in contrast to what is observed in the uncorrelated case. This amplitude factor can explain only the first order expansion in a Dyson-type singularity as supported by the SUSY model [7,8]. As shown in Appendix A, if the random walk observation of Eggarter et al [14] is implemented, with correlation, the obtained amplitude factor can reproduce the data within a shrinking energy range as E !…”

“…Many efforts have also been spent on the same off-diagonal tight-binding system using supersymmetric methods (SUSY) [6] where the interactions are formally similar to our dipolar interactions, but the authors eventually worked with continuous variables, whereas ours are discrete. Later, correlated disorder, in particular the exponential type, was included for the same tight binding model using the SUSY [7,8]. Our results can thus be compared with what has been developed for the SUSY methods.…”

“…To see whether longer chains and/or lower energies improve the situation, we checked further and found that the longer chain did not improve the agreement appreciably; but as the energy is lowered, the results follow the scaling behavior more closely up to a certain limiting c. Similar disagreement between the data and the Dysonian (with a single term) singularity were encountered in the SUSY [7] off-diagonal tight binding case for continuous exponential correlations. The analytical investigations [8] revealed that the spectra for the exponential correlation can be expanded for the IDOS as…”

“…For the first order expanded coupling we have J n ¼ 1 þ Ax n with unit lattice constant where A is the strength of the disorder and x n is a random sign variable that is exponentially correlated hx n x m i ¼ e ÀjmÀnj=lðcÞ where lðcÞ is the correlation length as a function of ''wrong sign'' concentration c (see below). Thus the dipolar coupling J n has a non-fluctuating part and a fluctuating part as in Dirac fermions with random-varying mass [7,8].…”