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arXiv:2205.10477 · Physica Scripta

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

Yi‐Cai Zhang

Published Dec 29, 2021• 8 citations
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Abstract

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〉. 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 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 potential strength α. When the bound state energies are very near the flat band, they are inversely proportional to the natural number n (e.g., E n ∝ 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 α c at which the bound state energy reaches the threshold of continuous spectrum. After crossing the threshold, the bound states in the continuum (BIC) would exist in such a flat band system.

DOI
10.1088/1402-4896/ac46f4

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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〉. 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 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 potential strength α. When the bound state energies are very near the flat band, they are inversely proportional to the natural number n (e.g., E n ∝ 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 α c at which the bound state energy reaches the threshold of continuous spectrum. After crossing the threshold, the bound states in the continuum (BIC) would exist in such a flat band system.
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.
Abstract Bound states in the continuum (BICs), referring to spatially localized bound states with energies falling within the range of extended modes, have been extensively investigated in single-particle systems, leading to diverse applications in photonics, acoustics, and other classical-wave systems. Recently, there has been theoretical interest in exploring many-body BICs in interacting quantum systems, which necessitate the careful design of impurity potentials or spatial profiles of interaction. Here, we propose a type of many-body BICs localized at boundaries, which can be purely induced by the uniform onsite interaction without requiring any specific design of impurity potential or nonlocal interaction. We numerically show that three or more interacting bosons can concentrate on the boundary of a homogeneous one-dimensional lattice, which is absent at single- and two-particle counterparts. Moreover, the eigenenergy of multi-boson bound states can embed within the continuous energy spectra of extended scattering states, thereby giving rise to interaction-induced boundary many-body BICs. Furthermore, by mapping Fock states of three and four bosons to nonlinear circuit networks, we experimentally simulate boundary many-body BICs. Our findings enrich the comprehension of correlated BICs beyond the single-particle level, and have the potential to inspire future investigations on exploring many-body BICs.
Motivated by the possibility of the second energy level (0 2 + ) of 12C (in a three-alpha model) to turn into an Efimov state, we study a simple non-realistic toy model formed by three bosons interacting by the phenomenological s-wave Ali-Bodmer potential plus a Coulomb interaction. An artificial three-body potential was used to create a resonance with energy close to the energy of the 0 2 + of 12C, 0.38 MeV. The strength of the Coulomb potential is decreased until the energies of the two alpha pairs are zero. The system was placed inside a harmonic trap and a stabilization method has been used to calculate the energies of the resonances. We found that the shielded-Coulomb potential, which keeps the long tail, is not able to produce the Efimov effect. The energy of the three alphas decreases only to 0.19 MeV when the two-body energy crosses the threshold to become bound.
Motivated by the possibility of the second energy level (0 + 2 ) of 12 C (in a three-alpha model) to turn into an Efimov state, we study a simple non-realistic toy model formed by three bosons interacting by the phenomenological s-wave Ali-Bodmer potential plus a Coulomb interaction.An artificial three-body potential was used to create a resonance with energy close to the energy of the 0 + 2 of 12 C, 0.38 MeV.The strength of the Coulomb potential is decreased until the energies of the two alpha pairs are zero.The system was placed inside a harmonic trap and a stabilization method has been used to calculate the energies of the resonances.We found that the shielded-Coulomb potential, which keeps the long tail, is not able to produce the Efimov effect.The energy of the three alphas decreases only to 0.19 MeV when the two-body energy crosses the threshold to become bound.
A bound state in the continuum (BIC) is a spatially localized energy eigenstate embedded in a continuous spectrum of extended eigenstates. While diverse types of single-particle BICs have been reported in the literature, understanding how such localization persists in interacting many-body systems remains a fundamental problem. Here, we present numerical evidence and a perturbative analysis supporting the existence of many-body BICs in a one-dimensional Bose-Hubbard chain with an attractive impurity potential, a system previously known to host a BIC in the two-particle sector. Furthermore, we demonstrate that the Bethe-type construction of the known two-particle BIC breaks down already for three particles even for arbitrary finite superpositions of Bethe waves. The resulting BICs violate the eigenstate thermalization hypothesis for several local one- and two-body observables, leading to nonthermal dynamics from experimentally accessible initial states.
Abstract Historically, most of the quantum mechanical results have originated in one dimensional model potentials. However, Von-Neumann’s Bound states in the Continuum (BIC) originated in specially constructed, three dimensional, oscillatory, central potentials. One dimensional version of BIC has long been attempted, where only quasi-exactly-solvable models have succeeded but not without instigating degeneracy in one dimension. Here, we present an exactly solvable bottomless exponential potential barrier V ( x ) = − V 0 [ exp ( 2 ∣ x ∣ / a ) − 1 ] which for E < V 0 has a continuum of non-square-integrable, definite-parity, degenerate states. In this continuum, we show a surprising presence of discrete energy, square-integrable, definite-parity, non-degenerate states. For E > V 0 , there is again a continuum of complex scattering solutions ψ ( x ) whose real and imaginary parts though solutions of Schrödinger equation yet their parities cannot be ascertained as Cψ ( x ) is also a solution where C is an arbitrary complex non-real number.