Strictly localized states in the octagonal Ammann-Beenker quasicrystal

Abstract

Ammann-Beenker lattice is a two-dimensional quasicrystal with eight-fold symmetry, which can be described as a projection of a cut from a four-dimensional simple cubic lattice. We consider the vertex tight-binding model on this lattice and investigate the strictly localized states at the center of the spectrum. We use a numerical method based on the generation of finite lattices around a given perpendicular space point and QR decomposition of the Hamiltonian to count the strictly localized states. We apply this method to count the frequency of localized states in lattices of up to 100 000 sites. We obtain an orthogonal set of compact localized states by diagonalizing the position operator projected onto the manifold spanned by the zero energy states. We identify twenty localized state types and calculate their exact frequencies through their perpendicular space images. Unlike the Penrose lattice, all the localized state types are eight-fold symmetric around an eight edge vertex, and all vertex types can support localized states. The total frequency of these twenty types gives a lower bound of $f_{LS}=30796-21776\sqrt{2}\simeq 0.08547$ for the fraction of strictly localized states in the spectrum. This value is in agreement with the numerical calculation and very close to the recently conjectured exact fraction of localized states $f_{Ex}=3/2-\sqrt{2}\simeq 0.08579$.

References in corpus (7)

• Imaging Quasi-Periodic Electronic States in a Synthetic Penrose Tiling
• Antiferromagnetic order in the Hubbard Model on the Penrose Lattice
• Critical eigenstates and their properties in one and two dimensional quasicrystals
• Superlattice structure in the antiferromagnetically ordered state in the Hubbard model on the Ammann-Beenker tiling
• Spin waves and local magnetizations on the Penrose tiling
• Energy levels and their correlations in quasicrystals
• Antiferromagnetically ordered state in the half-filled Hubbard model on the Socolar dodecagonal tiling

Cited by in corpus (19)

• Hubbard Models for Quasicrystalline Potentials
• Length scale formation in the Landau levels of quasicrystals
• Physical properties of an Aperiodic monotile: Graphene-like features, chirality and zero-modes
• Electronic states of a disordered 2D quasiperiodic tiling: from critical states to Anderson localization
• Properties of the Ammann-Beenker tiling and its square approximants
• Closing of gaps and gap labeling and passage from molecular states to critical states in a 2D quasicrystal
• Confined states and topological phases in two-dimensional quasicrystalline $\pi$-flux model
• Fractal Surface States in Three-Dimensional Topological Quasicrystals
• Localized States in Local Isomorphism Classes of Pentagonal Quasicrystals
• Ferromagnetically ordered states in the Hubbard model on the $H_{00}$ hexagonal golden-mean tiling
• Confined states in the tight-binding model on the hexagonal golden-mean tiling
• Patch frequencies in Penrose rhombic tilings
• Gapless superconductivity and its real-space topology in quasicrystals
• Mott transition and correlation effects on strictly localized states in an octagonal quasicrystal
• Modulated honeycomb lattices and their magnetic properties
• On the origin of energy gaps in quasicrystalline potentials
• Emergence of quasiperiodic behavior in transport and hybridization properties of clean lattice systems
• Localization and topological signatures under periodic twisting
• Raman Characterization of Two-Dimensional Quasiperiodic Antiferromagnets on Various Lattices: Spin-Orbit Mechanism