On the origin of energy gaps in quasicrystalline potentials

TL;DR

The paper addresses the lack of a rigorous infinite-size theory for energy gaps in quasicrystalline potentials by introducing a configuration-space framework. It shows that energy gaps arise from resonant hybridisation of neighboring sites and that the integrated densities of states below gaps correspond to irrational areas in configuration space, verified by large-scale Wannier-based tight-binding simulations. The work provides a quantitative mechanism for gap formation, a hierarchical gap structure with irrational area ratios, and an almost flat eight-site-ring band; evidence that the 8QC lacks weakly modulated lines, making it a strong candidate for 2D MBL. Overall, it offers a generalizable analytical toolkit for infinite-size QC potentials and bridges the gap between finite-size numerics and the thermodynamic limit.

Abstract

Quasicrystals, structures that are ordered yet aperiodic, defy conventional band theory, confining most studies to finite-size real-space numerics. We overcome this limitation with a configuration-space framework that predicts and explains the positions and origins of energy gaps in quasicrystalline potentials. We find that a hierarchy of gaps stems from resonant hybridization between increasingly distant neighboring sites, pinning the integrated density of states below these gaps to specific irrational areas in configuration space. Large-scale simulations of a lowest-band tight-binding model built from localized Wannier functions show excellent agreement with these predictions. By moving beyond finite-size numerics, this study advances the understanding of quasicrystalline potentials, paving the way for new explorations of their quantum properties in the infinite-size limit.

Figures (11)

• Figure 1: Eight-fold quasicrystalline potential: (a) The eight-fold potential is formed by superimposing two square periodic lattices ($\mathbf{k}_x, \mathbf{k}_y$ and $\mathbf{k}_+, \mathbf{k}_-$) with a $45^\circ$ angle between them, see \ref{['eq:QCpotential']}. (b) Resulting potential.
• Figure 2: Tight-binding Hamiltonian of the 8QC lowest band: (a) Optical potential in the continuum, with the Amman-Beenker (AB) tiling overlaid in black. (b) Corresponding tight-binding Hamiltonian. Each dot represents the centre of mass of a numerically computed Wannier function, color-coded by onsite energy. Tunnelling amplitudes are shown in encoded in color of the line connecting sites. The lattice sites closely follow the AB tiling, and the onsite energies and tunneling amplitudes form intricate quasiperiodic patterns. The short diagonal of the rhombuses exhibit significant tunneling amplitudes, in stark contrast with the usual AB tiling where these bonds are absent. Positive tunneling amplitudes are shown but are significantly weaker than negative tunnelings. Parameters:$V_0 = 1.5 \ E_r$.
• Figure 3: Tunneling amplitudes of the 8QC as a function of distance between sites: The tunneling amplitudes exhibit an oscillating decay as a function of distance, which is caused by the oscillating sidelobes of the Wannier functions, analogous to periodic lattices. The tunnelling range decreases rapidly with lattice depth, indicating a suppression of longer-range tunneling processes.
• Figure 4: Localisation properties of the non-interacting energy spectrum of the (lowest-band) of the 8QC: Color encodes the IPR of the eigenstates. (a) Energy spectrum. (b) Energy spectrum normalized by bandwidth, where the energy of the ground-state has been subtracted. The spectrum contains a hierarchy of gaps (visible as white spaces). The ground state localizes at $1.76 \ E_r$, and the majority of the spectrum localizes between $1.76 \ E_r$ and around $3 \ E_r$. Faint in-gap lines correspond to edge states caused by the open boundary conditions. Parameters: System diameter $70 \lambda$.
• Figure 5: Eigenstates around the largest energy gaps in the localized phase: selection of pairs of states surrounding two of the largest gaps. Interestingly, these states are localized around local symmetry centres with various levels of symmetries (grey lines indicate approximate local reflection symmetries of the lattice) and are composed of small groups of resonant sites. Edge states have been filtered out by ignoring eigenstates with more than $2.5 \%$ of probability within $1 \ \lambda$ from the system's edge. All eigenstates are shown centered around their center-of-mass, they are situated in different regions of the 8QC. Parameters: $V_0=2.5 \ E_r$. System diameter $25 \ \lambda$.