Infinite bound states and hydrogen atom-like energy spectrum induced by a flat band

Abstract

In this work, we investigate the bound-state problem in a one-dimensional spin-1 Dirac Hamiltonian with a flat band. It is found that the flat band has significant effects on the bound states. For example, for Dirac delta potential gδ(x), there exists one bound state for both the positive and negative potential strength g. Furthermore, when the potential is weak, the bound-state energy is proportional to the potential strength g. For square well potential, the flat band results in the existence of infinite bound states for arbitrarily weak potential. In addition, when the bound-state energy is very near the flat band, the energy displays a hydrogen atom-like spectrum, i.e. the bound-state energies are inversely proportional to the square of the natural number n (e.g., E n ∝ 1/n 2, n = 1, 2, 3, …). Most of the above nontrivial behaviors can be attributed to the infinitely large density of states of the flat band and its ensuing 1/z singularity of the Green function. The combination of a short-ranged potential and flat band provides a new possibility to get an infinite number of bound states and a hydrogen atom-like energy spectrum. In addition, our findings provide some useful insights and further our understanding of the many-body physics of the flat band.

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

“…Such a spindependent potential is a bit similar to the magnetic impurity potential in Kondo model, which may be realized in flat band materials of solid physics. The similar bound state problems with potential of type I and II have been investigated by Zhang and Zhu [28,32]. Now the Schrödinger equation with three component wave functions can be written as…”

“…It is well known that the behaviors of density of states near the threshold of a continuous energy spectrum play crucial roles in the formations of bound states [27]. In a spin-1 flat band system, due to the peculiar density of states and its ensuing 1/z singularity of Green function, a short-ranged potential, e.g., square well potential, can result in infinite bound states, even a hydrogen atom-like energy spectrum, i.e., E n ∝ 1/n 2 , n = 1, 2, 3, ... [28].…”

“…For a potential of type II, which has a unique nonvanishing matrix element in basis |2 [28], a short-ranged potential, e.g., square well potential, can cause an infinite number of bound states, even a hydrogen atom-like en-ergy spectrum. In addition, such a kind of Coulomb-like potential can result in a 1/n energy spectrum [32].…”

Proposed realization of critical regions in a one-dimensional flat band lattice with a quasi-periodic potential

“…The transition points (|α| = 1) E = E0 ≡ ±V0/2 = ±2.5m between bounded and unbounded cases for V0 = 5m are also indicated by red arrows. In the whole manuscript, we take t = m and phase φ = 0. state problems of the continuous version for above Hamiltonian with various types of potentials have been investigated by Zhang and Zhu [23,[27][28][29].…”

“…A lot of novel physics, for example, existences of localized flat band states [1][2][3], ferro-magnetism transition [4,5], super-Klein tunneling [6][7][8][9], preformed pairs [10], strange metal [11], high T c superconductivity/superfluidity [12][13][14][15][16][17][18][19][20][21][22], etc., can appear in a flat band system. Due to infinitely large density of states of flat band, a short-ranged potential can result in an infinite number of bound states, even a hydrogen atom-like energy spectrum, i.e., E n ∝ 1/n 2 , n = 1, 2, 3, ... [23]. Furthermore, a long ranged Coulomb potential can destroy completely the flat band [24,25].…”