[Paper Review] Single parameter scaling of one-dimensional systems with real-space long-range correlated disorder

Greg Petersen, Nancy Sandler|arXiv (Cornell University)|Jun 15, 2012

Theoretical and Computational Physics3 citations

TL;DR

This study investigates single parameter scaling (SPS) in one-dimensional disordered systems with real-space long-range correlated disorder, showing that SPS breaks down for energies between the band center and edge when the disorder strength-to-correlation exponent ratio $ W/(teta) < 1 $. The critical and fractal exponents $ u $ and $ D $ become dependent on the correlation exponent $ \alpha $, indicating that correlations fundamentally alter localization physics beyond simple rescaling of disorder strength.

ABSTRACT

Advances in material growth methods have renewed the interest in localization of one-dimensional systems in the presence of scale-free long-range correlated disorder potentials. We analyze the validity of single parameter scaling for the β-function away from the band center, in the presence of correlations. A renormalized disorder strength emerges reducing the regime of validity of the single parameter scaling hypothesis. Analysis of localization lengths and participation ratios leads to correlation dependent critical and fractal exponents, consistent with the extended Harris criterion.

Motivation & Objective

• To assess the validity of single parameter scaling (SPS) for the $ \beta $-function in one-dimensional systems with real-space long-range correlated disorder.
• To investigate the crossover between two scaling regimes—those influenced by effective disorder strength $ W_{\text{eff}} $ and those by effective white-noise disorder—via localization length analysis.
• To characterize the nature of localized eigenstates using the participation ratio and fractal dimension $ D $, and to determine its dependence on the correlation exponent $ \alpha $.
• To test whether the extended Harris criterion holds for critical and fractal exponents in the presence of power-law correlated disorder.
• To resolve inconsistencies in prior scaling theories by linking SPS violation to non-analytic behavior in the localization length $ \xi $ near band edges and center.

Proposed method

• Numerical diagonalization of the one-dimensional Anderson Hamiltonian with power-law correlated on-site energies $ \epsilon_n $, where $ \langle \epsilon_n \epsilon_0 \rangle \propto (1+n)^{-\alpha} $.
• Use of transfer matrix methods to compute localization lengths $ \xi $, enabling analysis of the $ \beta $-function and identification of scaling regimes.
• Calculation of the inverse participation ratio (IPR) via $ A_m = \frac{1}{N} \frac{\left( \sum |\psi_m(n)|^2 \right)^2}{\sum |\psi_m(n)|^4} $, with fitting to $ A_m \sim (L/a)^{-D} $ to extract fractal dimension $ D $.
• Systematic variation of $ \alpha $, $ W/t $, and energy $ E/t $ to probe the dependence of $ \nu $ and $ D $ on correlation strength and energy position.
• Application of the extended Harris criterion to interpret the $ \alpha $-dependence of $ \nu $ and $ D $, comparing results to uncorrelated disorder limits.
• Analysis of the energy threshold $ E_{\text{crossover}} $ where scaling behavior changes, linked to non-analyticities in $ \xi(W/t) $.

Experimental results

Research questions

• RQ1 Does single parameter scaling (SPS) hold for the $ \beta $-function in one-dimensional systems with real-space long-range correlated disorder, particularly away from the band center?
• RQ2 How does the effective disorder strength $ W_{\text{eff}} $ modify the scaling behavior, and what is its role in defining the crossover between scaling regimes?
• RQ3 What is the dependence of the localization length critical exponent $ \nu $ on the correlation exponent $ \alpha $, and does it follow predictions from the extended Harris criterion?
• RQ4 How does the fractal dimension $ D $ of localized eigenstates vary with $ \alpha $, and does it differ between band-center and band-edge states?
• RQ5 Why does the SPS hypothesis break down in the presence of long-range correlations, and how is this related to non-analytic behavior in the localization length?

Key findings

• Single parameter scaling (SPS) is violated for energies between the band center and edge when $ W/(t\alpha) < 1 $, with the violation threshold $ E_{\text{SPS}} $ coinciding with the onset of band-edge anomalies in the localization length expansion.
• The localization length critical exponent $ \nu $ exhibits a non-monotonic dependence on $ \alpha $, with distinct functional forms for $ \alpha < 1 $ and $ \alpha > 1 $, indicating a crossover at $ \alpha = 1 $.
• The fractal dimension $ D $ of localized eigenstates becomes $ \alpha $-dependent for $ \alpha < 1 $, especially at the band center, where $ D $ decreases with increasing $ \alpha $, implying more extended wave functions.
• For $ \alpha < 1 $, the number of extended-like states increases significantly, particularly near the band center, suggesting a continuous interpolation toward perfectly extended states in the $ \alpha \to 0 $ limit.
• The anomalous enhancement of localization length near band edges cannot be explained by a simple rescaling of disorder strength $ W \to W_{\text{eff}} $, indicating that correlations affect the nature of localized states in a non-trivial, energy-dependent way.
• The results are consistent with the extended Harris criterion, confirming that $ \nu $ and $ D $ acquire $ \alpha $-dependence in correlated systems, challenging the universality of SPS in realistic disordered materials with long-range correlations.