Macroscopically degenerate localized zero-energy states of quasicrystalline bilayer systems in strong coupling limit

TL;DR

This paper reveals that strongly coupled bilayer quasicrystals, formed by stacking identical 2D lattices with a twist, develop macroscopically degenerate localized zero-energy states due to emergent chiral symmetry and aperiodicity.

Contribution

It introduces the quasiband model and a geometric scheme to analyze and count zero-energy states in bilayer quasicrystals, advancing understanding of flat band physics in aperiodic systems.

Findings

Degenerate zero-energy states emerge in strong coupling bilayer quasicrystals.

The quasiband model accurately describes low energy properties and counts ZESs.

Localized ZESs have a divergent density of states similar to flat bands.

Abstract

When two identical two-dimensional (2D) periodic lattices are stacked in parallel after rotating one layer by a certain angle relative to the other layer, the resulting bilayer system can lose lattice periodicity completely and become a 2D quasicrystal. Twisted bilayer graphene with 30-degree rotation is a representative example. We show that such quasicrystalline bilayer systems generally develop macroscopically degenerate localized zero-energy states (ZESs) in strong coupling limit where the interlayer couplings are overwhelmingly larger than the intralayer couplings. The emergent chiral symmetry in strong coupling limit and aperiodicity of bilayer quasicrystals guarantee the existence of the ZESs. The macroscopically degenerate ZESs are analogous to the flat bands of periodic systems, in that both are composed of localized eigenstates, which give divergent density of states. For monolayers, we consider the triangular, square, and honeycomb lattices, comprised of homogenous tiling of three possible planar regular polygons: the equilateral triangle, square, and regular hexagon. We construct a compact theoretical framework, which we call the quasiband model, that describes the low energy properties of bilayer quasicrystals and counts the number of ZESs using a subset of Bloch states of monolayers. We also propose a simple geometric scheme in real space which can show the spatial localization of ZESs and count their number. Our work clearly demonstrates that bilayer quasicrystals in strong coupling limit are an ideal playground to study the intriguing interplay of flat band physics and the aperiodicity of quasicrystals.