[Paper Review] Dynamical Scaling Properties of Electrons in Quantum Systems with Multifractal Eigenstates

Jianxin Zhong, Zhenyu Zhang+3 more|arXiv (Cornell University)|Nov 7, 2000

Theoretical and Computational Physics3 citations

TL;DR

This paper establishes a direct link between the multifractal nature of electron eigenstates and their dynamical spreading in quantum systems. It proposes that the scaling exponent β of the root mean square displacement r(t) ∼ t^β is given by β = D₂^ψ / d, where D₂^ψ is the correlation dimension of the multifractal eigenstates and d is the system dimension, with β < D₂^ψ / d when motion is non-ballistic in the effective D₂^ψ-dimensional space.

ABSTRACT

We study the intricate relationships between the dynamical scaling properties of electron wave packets and the multifractality of the eigenstates in quantum systems. Numerical simulations for the Harper model and the Fibonacci chain indicate that the root mean square displacement displays the scaling behavior $r(t)\sim t^β$ with $β=D_2^ψ$, where $D_2^ψ$ is the correlation dimension of the multifractal eigenstates. The equality can be generalized to $d$-dimensional systems as $β=D_2^ψ/d$, as long as the electron motion is ballistic in the effective $D_2^ψ$-dimensional space. This equality should be replaced by $β

Motivation & Objective

• To clarify the fundamental relationship between multifractal eigenstates and electron dynamics in quantum systems.
• To resolve inconsistencies in prior attempts linking spectral multifractal dimensions D_q^μ to dynamical scaling exponent β.
• To introduce a new framework where the effective dimension D₂^ψ governs electron motion, independent of energy spectrum singularities.
• To generalize the relation β = D₂^ψ for 1D systems to d-dimensional systems via effective dimensionality.
• To establish a universal criterion β ≤ D₂^ψ / d, with equality only for ballistic motion in the D₂^ψ-dimensional space.

Proposed method

• Numerical simulations of the 1D Harper model and Fibonacci chain to compute r(t) and extract β.
• Application of box-counting method to compute the spatial correlation function R(w) and determine D₂^ψ from R(w) ∼ w^{D₂^ψ}.
• Spectral averaging of R(w,E_m) over eigenstates to define a robust measure of multifractality in real space.
• Extension to 2D and 3D quasiperiodic systems (e.g., 3D Fibonacci lattice) with separable Hamiltonians to test β = D₂^ψ / d.
• Use of exactly solvable models to verify that β = D₂^ψ / d holds when motion in D₂^ψ-dimensional space is ballistic.
• Comparison of results with known disordered systems to validate β < D₂^ψ / d in non-ballistic regimes.

Experimental results

Research questions

• RQ1 How does the multifractal dimension D₂^ψ of eigenstates relate to the dynamical scaling exponent β of wave packet spreading?
• RQ2 Can the relation β = D₂^ψ / d be generalized to d-dimensional systems, and under what conditions does it hold?
• RQ3 What determines whether β = D₂^ψ / d or β < D₂^ψ / d in higher-dimensional systems?
• RQ4 Why do previous approaches based on energy spectrum multifractality fail to predict dynamics in quasiperiodic systems?
• RQ5 How does the effective dimension D₂^ψ of the multifractal space influence electron transport in disordered and quasiperiodic systems?

Key findings

• For 1D systems like the Harper model and Fibonacci chain, β = D₂^ψ, with β ≈ 0.67 for the Fibonacci chain.
• In d-dimensional systems, β = D₂^ψ / d holds when electron motion is ballistic in the D₂^ψ-dimensional effective space, as confirmed in 2D and 3D quasiperiodic lattices.
• For 3D Fibonacci lattices, D₂^ψ = 3 × D₂^ψ_x, and β = D₂^ψ / 3 reproduces the known β ≈ 0.67 from 1D chains.
• In disordered systems at the metal-insulator transition, β < D₂^ψ / d holds universally, with β = 0.33 and D₂^ψ / d ≈ 0.57–0.60 in 3D systems.
• In 2D systems with strong magnetic fields, β = 0.5 and D₂^ψ / d ≈ 0.76, confirming β < D₂^ψ / d.
• At the 2D metal-insulator transition with spin-orbit coupling, β = 0.5 and D₂^ψ / d ≈ 0.84, again satisfying β < D₂^ψ / d.