Exact Eigenstates of Tight-Binding Hamiltonians on the Penrose Tiling

TL;DR

This paper constructs exact eigenstates for tight-binding models on the Penrose tiling, revealing infinitely degenerate eigenenergies and utilizing an arrow decoration ansatz based on matching rules and inflation symmetry.

Contribution

It introduces a method to find exact eigenstates on the Penrose tiling using a generalized ansatz that exploits the tiling's inflation symmetry.

Findings

Exact eigenstates are constructed for specific hopping parameters.

Eigenenergies are shown to be infinitely degenerate.

The approach can be generalized to other quasicrystalline systems.

Abstract

We investigate exact eigenstates of tight-binding models on the planar rhombic Penrose tiling. We consider a vertex model with hopping along the edges and the diagonals of the rhombi. For the wave functions, we employ an ansatz, first introduced by Sutherland, which is based on the arrow decoration that encodes the matching rules of the tiling. Exact eigenstates are constructed for particular values of the hopping parameters and the eigenenergy. By a generalized ansatz that exploits the inflation symmetry of the tiling, we show that the corresponding eigenenergies are infinitely degenerate. Generalizations and applications to other systems are outlined.