Quasiperiodic Systems Reveal Enhanced Geometry, Differing From Crystals, Disorder.

The behaviour of electrons within a material is profoundly influenced by its underlying geometry, a relationship now being explored in systems lacking the regular, repeating structure of conventional crystals. Jundi Wang, Yuxiao Chen, and Huaqing Huang et al., from Peking University and the Collaborative Innovation Center of Quantum Matter, investigate this connection in their recent work, entitled ‘Quantum Metric Enhancement and Hierarchical Scaling in One-Dimensional Quasiperiodic Systems’. They demonstrate that quasiperiodic arrangements, which exhibit order without strict repetition, exhibit enhanced geometric properties stemming from the unique characteristics of electron wavefunctions within these materials, potentially offering new avenues for materials design and manipulation. Their analysis focuses on models such as the Aubry-André-Harper model and the Fibonacci chain, revealing a link between the critical behaviour of electrons, the fractal nature of their energy spectrum, and the resulting geometric enhancement.

Quasiperiodic systems represent a distinct state of matter, exhibiting long-range order without the translational symmetry found in crystals or the complete disorder of random materials. These systems, exemplified by the Fibonacci chain and the Aubry-André-Harper model, possess unusual electronic and topological properties, including singular continuous spectra and complex localisation transitions. Recent experimental realisations in photonic lattices, cold atom systems, and acoustic metamaterials facilitate the investigation of these properties, validating theoretical predictions and deepening our understanding of these complex materials.

Current research focuses on the quantum metric, a measure defining distances between quantum states and influencing material behaviour. This metric, described by the quantum metric tensor, reveals the infinitesimal distance between neighbouring quantum states, offering a geometric perspective on material properties beyond conventional band structure analysis. Investigations demonstrate a significant enhancement of the quantum metric in quasiperiodic systems, despite the absence of translational symmetry. This enhancement stems from the presence of critical wavefunctions exhibiting long-range spatial correlations, fundamentally altering the geometric landscape experienced by electrons.

Researchers meticulously examine the geometric properties of one-dimensional quasiperiodic systems, revealing behaviours markedly different from both periodic crystals and disordered materials. The quantum metric directly influences particle dynamics and energy levels, making its precise determination crucial for understanding material behaviour. Comparative analysis demonstrates that quasiperiodicity significantly amplifies the metric’s magnitude.

To probe this relationship, researchers focused on the Aubry-André-Harper (AAH) model and the Fibonacci chain. The AAH model, known for its metal-insulator transition driven by disorder, reveals that the quantum metric acts as a sensitive indicator of this transition, exhibiting abrupt changes precisely at the critical point separating conducting and insulating phases. The Fibonacci chain, characterised by a singular continuous energy spectrum, presents a different scenario. Researchers discovered an anomalous enhancement of the metric when the Fermi level resided within the minimal gaps of the fractal energy spectrum, linking this enhancement to the hierarchical structure of the spectrum itself.

A perturbative renormalization group framework, a mathematical technique used to systematically analyse systems at different scales, was employed to trace this enhancement back to the fundamental properties of the fractal spectrum. The findings establish a fundamental connection between wavefunction criticality, spectral fractality, and geometry, suggesting that quasiperiodic systems offer a promising avenue for engineering materials with enhanced geometric properties. Unlike conventional crystalline paradigms, where geometric properties are largely fixed by the lattice structure, quasiperiodic systems allow for a degree of tunability, potentially enabling the design of materials with tailored geometric responses.

Researchers demonstrate that quasiperiodicity enhances the quantum metric, despite the absence of translational symmetry, attributing this to critical wavefunctions exhibiting long-range spatial correlations. The quantum metric functions as a sensitive indicator of localisation transitions, displaying marked changes at critical points and distinct behaviours near mobility edges – sharp boundaries between localized and extended electronic states. Investigation of the Fibonacci chain reveals an anomalous enhancement of the metric when the Fermi level resides within the minimal gaps of its fractal spectrum.

The application of a perturbative renormalization group framework elucidates this enhancement, linking it to the hierarchical structure inherent in the spectrum and the bonding-antibonding character of critical states spanning these narrow gaps. This framework provides a theoretical basis for understanding the observed geometric properties and their connection to the underlying electronic structure. The study establishes a fundamental connection between wavefunction criticality, spectral fractality, and quantum geometry, suggesting that quasiperiodic systems offer a promising avenue for engineering enhanced geometric properties beyond those found in conventional crystalline materials.

This work builds upon existing knowledge of Anderson localisation and extends it to the realm of aperiodic systems, providing a deeper understanding of the interplay between disorder, topology, and geometry in condensed matter physics. The authors successfully demonstrate that the quantum metric serves as a valuable tool for probing the critical behaviour and underlying physics of these complex systems.