[Paper Review] Fractal Spectrum of the Aubry-Andre Model

Ang-Kun Wu|arXiv (Cornell University)|Sep 15, 2021

Cellular Automata and Applications4 citations

TL;DR

This paper uncovers the self-similar fractal structure of the energy spectrum in the Aubry-André model at the localization transition point by separating fractal filling of gaps from power-law scaling of gap sizes. It shows that fractal filling arises from specific irrational frequencies (e.g., the golden ratio), while power-law gap scaling—critical for fractal manifestation—occurs universally at the critical point, enabling self-similar spectral reconstruction in the thermodynamic limit.

ABSTRACT

The Aubry-Andre model is a one-dimensional lattice model for quasicrystals with localized and delocalized phases. At the localization transition point, the system displays fractal spectrum, which relates to the Hofstadter butterfly. In this work, we uncover the exact self-similarity structures in the energy spectrum. We separate the fractal structures into two parts: the fractal filling positions of gaps and the scaling of gap sizes. We show that the fractal fillings emerge for a certain type of incommensurate periodicity regardless of potential strength. However, the power-law scaling of gap sizes is characteristic for general incommensurability at the critical point of the model.

Motivation & Objective

• To identify the origin of fractal energy spectra in the Aubry-André model at the localization transition point.
• To separate the fractal structure into two components: fractal filling of gap positions and scaling of gap sizes.
• To determine under which conditions self-similar spectral patterns emerge in one-dimensional quasicrystals.
• To establish self-contained rules for reconstructing the fractal spectrum in the thermodynamic limit.
• To clarify the distinction between universal power-law gap scaling at criticality and non-universal fractal filling dependent on specific irrational frequencies.

Proposed method

• Analytically solving the Aubry-André model with rational periodicity (α = p/q) to derive multi-band structure and organizing rules.
• Mapping rational periodicity to multi-atom unit cells with translational symmetry to model band splitting.
• Using the self-duality of the model to identify the critical point at λ = 2t, where localization transitions occur.
• Applying iterative transformation rules (L, R, C types) to generate hierarchical gap structures from parent to child energy gaps.
• Deriving scaling factors f_L/R = 0.0729 and f_C = 0.1392 for gap size reduction across successive layers.
• Analyzing sorted gap sizes as a function of index n to extract power-law scaling δ_n ∼ n^−1.98 at criticality, with deviation due to finite system size and layer count.

Experimental results

Research questions

• RQ1 What determines the emergence of self-similar fractal filling in the energy spectrum of the Aubry-André model?
• RQ2 How does the power-law scaling of gap sizes relate to the critical point of the model?
• RQ3 Is the fractal spectrum universal across all irrational frequencies, or restricted to specific types?
• RQ4 Can the fractal spectrum be reconstructed in the thermodynamic limit using self-consistent scaling rules?
• RQ5 What role does electron filling play in the stability of the spectrum and correlation effects in quasicrystals?

Key findings

• Fractal filling of gap positions emerges only for specific irrational frequencies, such as α = (√5 − 1)/2, and is independent of potential strength.
• The power-law scaling of gap sizes, δ_n ∼ n^−1.98, is characteristic of the critical point (λ = 2t) and persists across different irrational frequencies.
• The scaling behavior is governed by the product (f_L/R²f_C)^k/3, leading to δ_n ∼ n^−2.19 in the large-n limit, with deviations due to finite system size.
• Self-similar transformations between scales are enabled by consistent multiplication factors derived from the L/R and C rules.
• The fractal spectrum requires both fractal filling (from specific irrationality) and power-law gap scaling (from criticality), with the latter being universal at λ = 2t.
• Energy resolution δ = 1/L limits observable gap scaling in localized and delocalized phases, but the power-law decay persists below this scale at criticality.