Emergent quasiparticles in Euclidean tilings.

Area of Science:

• Condensed Matter Physics
• Materials Science
• Crystallography

Background:

• Geometric structure fundamentally dictates material properties, as seen in graphene's honeycomb lattice and its Dirac cone.
• Lattices like kagome and Lieb exhibit flat bands and pseudospin-1 Dirac dispersion, but these are considered rare phenomena in 2D systems.
• Exploring novel geometric arrangements is crucial for uncovering new quantum states and physical phenomena.

Purpose of the Study:

• To systematically characterize a large set of k-uniform tilings of the Euclidean plane.
• To identify common intrinsic properties associated with these geometric structures.
• To provide a framework for experimental prediction and interpretation of new 2D material systems.

Main Methods:

• Mathematical characterization of 1255 k-uniform tilings based on Kepler's tiling theory.
• Analysis of the electronic band structures and topological properties of these lattices.
• Identification of high-degeneracy points, flat bands, and exotic quasiparticles.

Main Results:

• The study identified 1255 distinct k-uniform lattices.
• Common features across these lattices include high-degeneracy points, flat bands, and the emergence of exotic quasiparticles.
• These findings suggest that such properties are more prevalent than previously assumed in 2D systems.

Conclusions:

• K-uniform tilings offer a rich platform for discovering new 2D materials with desirable electronic and topological properties.
• The identified common features provide a roadmap for targeted material design and experimental synthesis.
• This work facilitates the exploration of novel quantum phenomena in geometrically structured materials.