Confined states in the tight-binding model on the hexagonal golden-mean tiling

TL;DR

This study investigates confined zero-energy states in a tight-binding model on a hexagonal golden-mean tiling, revealing how state degeneracy and eigenstate properties change with different hopping parameters.

Contribution

It identifies exact confined states in the model and explores their persistence or alteration when hopping parameters vary, highlighting degeneracy and discontinuity phenomena.

Findings

Confined states are exact eigenstates when $t_L = t_S$.

Some confined states remain eigenstates even when $t_L eq t_S$ with smooth amplitude changes.

A discontinuity in the number of confined states appears in the thermodynamic limit.

Abstract

We study the tight-binding model with two distinct hoppings $(t_L, t_S)$ on the two-dimensional hexagonal golden-mean tiling and examine the confined states with $E=0$, where $E$ is the eigenenergy. Some confined states found in the case $t_L=t_S$ are exact eigenstates even for the system with $t_L \neq t_S$, where their amplitudes are smoothly changed. By contrast, the other states are no longer eigenstates of the system with $t_L \neq t_S$. This may imply the existence of macroscopically degenerate states which are characteristic of the system with $t_L=t_S$, and that a discontinuity appears in the number of the confined states in the thermodynamic limit.