# Critical eigenstates and their properties in one and two dimensional quasicrystals

## TL;DR

This paper provides exact solutions for critical eigenstates in 1D and 2D quasiperiodic tilings, revealing their multifractal nature and properties of ground states in specific 2D quasicrystals.

## Contribution

It offers explicit solutions and multifractal spectra for eigenstates in quasiperiodic models, advancing understanding of their critical properties.

## Key findings

- Eigenstates are critical with multifractal spectra
- Critical states are common in 1D quasiperiodic chains
- Ground state properties in 2D Penrose and Ammann-Beenker tilings

## Abstract

We present exact solutions for some eigenstates of hopping models on one and two dimensional quasiperiodic tilings and show that they are "critical" states, by explicitly computing their multifractal spectra. These eigenstates are shown to be generically present in 1D quasiperiodic chains, of which the Fibonacci chain is a special case. We then describe properties of the ground states for a class of tight-binding Hamiltonians on the 2D Penrose and Ammann-Beenker tilings. Exact and numerical solutions are seen to be in good agreement.