Infinite bound states and $1/n$ energy spectrum induced by a Coulomb-like potential of type III in a flat band system

TL;DR

This paper explores the unique bound states in a one-dimensional flat band system with a Coulomb-like potential, revealing infinite bound states, a $1/n$ energy spectrum, and conditions for bound states in the continuum.

Contribution

It introduces the existence of infinite bound states and a $1/n$ energy spectrum induced by a Coulomb-like potential in a flat band system, with analysis using quasi-classical approximation.

Findings

Infinite bound states exist in the system.

Bound state energies follow a $1/n$ pattern near the flat band.

Critical potential strength $\alpha_c$ determines the bound state threshold.

Abstract

In this work, we investigate the bound states in a one-dimensional spin-1 flat band system with a Coulomb-like potential of type III, which has a unique non-vanishing matrix element in basis $|1\rangle$. It is found that, for such a kind of potential, there exists infinite bound states. Near the threshold of continuous spectrum, the bound state energy is consistent with the ordinary hydrogen-like atom energy level formula with Rydberg correction. In addition, the flat band has significant effects on the bound states. For example, there are infinite bound states which are generated from the flat band. Furthermore, when the potential is weak, the bound state energy is proportional to the Coulomb-like potential strength $\alpha$. When the bound state energies are very near the flat band, they are inversely proportional to the natural number $n$ (e.g., $E_n\propto 1/n, n=1,2,3,...$). Further we find that the energy spectrum can be well described by quasi-classical approximation (WKB method). Finally, we give a critical potential strength $\alpha_c$ at which the bound state energy reaches the threshold of continuous spectrum. \textbf{After crossing the threshold, the bound states in the continuum (BIC) may exist in such a flat band system.