Localization in a one‐dimensional random potential with spatial correlation

Abstract

The r61e of oscillating vertices usually neglected in the Berezinskii diagram technique is analyzed in connection with a random potential, whose spatial correlation is generated by a Markov chain. It appears that the consideration of some of the oscillating vertices is necessary so that the theory can remark the spatial correlation. Correlation mainly leads to an increase of the localization length in comparison with an uncorrelated potential. However, there is a region of the parameter, where the localization length decreases. The Berezinskii diagram technique is used in a modified form.Die Rolle der oszillierenden Vertizes, die ublicherweise in der Berezinskii-Diagrammtechnik vernachlassigt werden, wird im Zusammenhang mit einem zufiilligen Potential, dessen riinmliche Korrelation durch eine Markovkette erzeugt wird, analysiert. Es zeigt sich, daR die Beriicksichtigung einiger oszillierender Vertizes notwendig ist, damit die Theorie die raumliche Korrelation bemerken kann. Korrelation fiihrt meistens zu einer Vergrofierung der Lokalisierungslange gegeniiber dem unkorrelierten Fall. Es gibt jedoch ein Parametergebiet, in dem die Lokalisierungslange verkltinert wird. Die Berezinskii-Diagrammtechnik wird in einer modifizierten Form verwendet.

The generalized localization lengths in one-dimensional systems with correlated disorder

“…First of all in the most extended limit, q → 1, Sstr (q) ≈ (1 − q)/2 should hold [22]. In contrast, according to (10)…”

Exact Solutions for One-Dimensional Systems with Short-Range Order

“…In ref. [13,14] the influence of correlation on the localization length l/y(w2) was discussed for a tight-binding system with weak disorder. Here we consider the haxmonic system with p = q and M << 1, where light and heavier particles occur both with probability 1/2.…”

“…Of special interest is the binary chain with light (L) and heavy (H) atoms, which has grown from one side, and where the sticking probability of a new atom only depended on the last atom of the chain. This is described by a Markov property p(L IL) = 1 -p(H 1 L) = p , p(L 1 H) = 1p(H 1 H) = p. In this system the fractions of light and heavy atoms are where q = 1p , Q = 1p , and correlations decay exponentially Tight-binding models with weak disorder described by such transition probabilities were discussed by several authors [13,14]. It is the purpose of the present letter to show that exact solutions exist, if one goes away from binary to exponential distributions, just as it was done in the uncorrelated case.…”

Resistance statistics in one-dimensional systems with correlated disorder

“…The correlated random alloy has been investigated by several authors, 4,5 showing, among other results, that the localization length defined as the inverse of the Lyapunov exponent is not generally proportional to the spatial correlation length of the potential.…”

“…[1][2][3] However, different behaviors are often found when certain rules or correlations are imposed on the disorder. For example, memory effects may lead to unexpected behaviors of the eigenstates localization lengths; e.g., they can be inversely proportional to the correlation length of the disorder in some regions of the spectrum of electronic 4 and harmonic 5 chains. Moreover, there are disordered heterostructures exhibiting properties similar to those of aperiodic ordered structures with same composition 6 or short-range correlation.…”