[Paper Review] A numerical study of wave-function and matrix-element statistics in the Anderson model of localization

Ville Uski, B. Mehlig+1 more|arXiv (Cornell University)|Nov 9, 1998

Quantum chaos and dynamical systems3 references4 citations

TL;DR

This paper presents a numerical investigation of wave-function and matrix-element statistics in the 2D and 3D Anderson model under weak disorder, focusing on the transition from GOE to GUE symmetry via an Aharonov-Bohm flux and deviations from random matrix theory (RMT) at increasing disorder. Using exact diagonalization and statistical analysis of eigenstates, the study confirms semiclassical and non-linear sigma model predictions for wave-function amplitude distributions, showing good agreement with analytical results and revealing g⁻¹ corrections to Porter–Thomas statistics that deviate from naive scaling expectations.

ABSTRACT

We have calculated wave functions and matrix elements of the dipole operator in the two- and three-dimensional Anderson model of localization and have studied their statistical properties in the limit of weak disorder. In particular, we have considered two cases. First, we have studied the fluctuations as an external Aharonov-Bohm flux is varied. Second, we have considered the influence of incipient localization. In both cases, the statistical properties of the eigenfunctions are non-trivial, in that the joint probability distribution function of eigenvalues and eigenvectors does no longer factorize. We report on detailed comparisons with analytical results, obtained within the non-linear sigma model and/or the semiclassical approach.

Motivation & Objective

• To investigate the statistical properties of wave functions and matrix elements in the Anderson model under weak disorder.
• To examine the transition from GOE to GUE statistics induced by an Aharonov-Bohm flux, particularly the non-trivial joint statistics of eigenvalues and eigenvectors.
• To analyze deviations from Porter–Thomas statistics in the metallic regime as disorder increases, focusing on g⁻¹ corrections to RMT predictions.
• To compare numerical results with analytical predictions from the non-linear sigma model and semiclassical approaches.

Proposed method

• Numerical diagonalization of the 2D and 3D Anderson Hamiltonian using a modified Lanczos algorithm for systems up to 27×27 and 13×13×13.
• Calculation of eigenstates and matrix elements of the dipole operator, with statistical analysis of wave-function amplitudes and level velocities.
• Use of smoothed variances Cv(ε,φ) and Cm(ε,φ) to study fluctuations in level velocities and diagonal matrix elements as functions of energy and flux.
• Application of the non-linear sigma model to derive g⁻¹ corrections to RMT, with comparison to numerical data via fitting Eq. (8).
• Averaging over 400 disorder realizations in 3D to compute the distribution f(t) of wave-function amplitudes and its deviation ∆f(t) from Porter–Thomas.
• Use of δ-function energy smearing with width ǫ to define the distribution f(t), and comparison with analytical expressions for the GOE/GUE transition and g⁻¹ corrections.

Experimental results

Research questions

• RQ1 How do wave-function amplitude statistics evolve during the GOE to GUE transition driven by an Aharonov-Bohm flux?
• RQ2 To what extent do numerical results for matrix-element and level-velocity fluctuations agree with semiclassical and RMT predictions in the transition regime?
• RQ3 How do deviations from Porter–Thomas statistics in wave-function amplitudes scale with disorder strength W in the 3D Anderson model?
• RQ4 Do the g⁻¹ corrections to RMT predicted by the non-linear sigma model match numerical observations, and how does the fit parameter a₃ scale with W?

Key findings

• Numerical results for wave-function amplitude distributions in the GOE to GUE transition show excellent agreement with analytical predictions from Ref. [10], including the characteristic suppression of the peak and enhancement of tails.
• The distribution of level velocities ceases to be Gaussian in the transition regime, though numerical precision limits definitive confirmation of the predicted deviations.
• Deviations from Porter–Thomas statistics in the 3D Anderson model are well described by the g⁻¹ correction formula in Eq. (8), with enhanced probability at small and large amplitudes and reduced probability near the peak.
• The fit parameter a₃/g in Eq. (8) scales as W² rather than the expected W⁴, indicating a discrepancy with the naive l ∼ W⁻² scaling of the mean free path.
• The zeroes of the first-order correction term in Eq. (8) are well reproduced by the numerical data across all disorder strengths W = 1 to 5.
• The study confirms the validity of non-linear sigma model predictions for g⁻¹ corrections, while highlighting the need for higher-precision data at weak disorder to resolve scaling anomalies.