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Few-Body Systems

Aharonov–Bohm Effect from Exponentially Decreasing Energy Levels of a Non-Hermitian System

K. Bakke

Published Aug 4, 2026• 0 citations
Publisher

Abstract

Abstract We introduce a non-Hermitian operator which has exponentially decreasing energy levels and study the Aharonov–Bohm effect for bound states. Further, we include the two-dimensional harmonic oscillator in this non-Hermitian quantum system and obtain real energy levels with exponentially decrease. We also show that the Aharonov–Bohm effect for bound states exists in this second case. In both cases, we go further by showing that non-null persistent currents at zero temperature and non-null revival times exist.

DOI
10.1007/s00601-026-02073-4

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Abstract We introduce a non-Hermitian operator which has exponentially decreasing energy levels and study the Aharonov–Bohm effect for bound states. Further, we include the two-dimensional harmonic oscillator in this non-Hermitian quantum system and obtain real energy levels with exponentially decrease. We also show that the Aharonov–Bohm effect for bound states exists in this second case. In both cases, we go further by showing that non-null persistent currents at zero temperature and non-null revival times exist.
The quantum states of the 2D time-dependent harmonic oscillator with the Aharonov–Bohm (AB) effect are investigated by employing the invariant operator method and the unitary transformation approach. The time-dependent radius case of a solenoid is treated. The qualitative properties of the invariant operator spectrum are described separately for the different values of the parameter C appearing in the nonlinear auxiliary equation, i.e. C>0, C=0 and C 0) or continuous (C≤0). The time-dependent cross section for a free particle scattered by the AB potential is obtained.
We study the analogue of the Aharonov–Bohm effect for bound states for a neutral particle with a permanent magnetic dipole moment interacting with an external field. We consider a neutral particle confined to moving between two coaxial cylinders and show the dependence of the energy levels on the Aharonov-Casher quantum flux. Moreover, we show that the same flux dependence of the bound states can be found when the neutral particle is confined to a one-dimensional quantum ring and a quantum dot, and we also calculate the persistent currents in each case.
A version of the electric Aharonov-Bohm effect is proposed where the quantum system which picks up the Aharonov-Bohm phase is confined to a Faraday cage with a time-varying spatially uniform scalar potential. The electric and magnetic fields in this region are effectively zero for the entire period of the experiment. The observable consequence of this version of the electric Aharonov-Bohmn effect is to shift the energy levels of the quantum system rather than shift the fringes of the two-slit interference pattern. We show a strong mathematical connection between this version of the scalar electric ABABAB effect and the ac Stark effect.
We introduce a two-dimensional non-Hermitian lattice model with an imaginary magnetic field and elucidate various unique features which are absent in Hermitian lattice models with real magnetic fields. To describe the imaginary magnetic field, we consider both the Landau gauge and the symmetric gauge, which are related by a generalized gauge transformation, changing not only the phase but also the amplitude of the wave function. We discuss the complex energy spectrum and the non-Hermitian Aharonov-Bohm effect as examples of properties which are due to the imaginary magnetic field independent of the generalized gauge transformation. We show that the energy spectrum does not converge as the lattice size is made larger, which comes from the intrinsic nonperiodicity of the model. However, we have found that the energy spectrum does converge if one fixes the length of one side and makes the other side longer; this asymptotic behavior can be understood in the framework of the non-Bloch band theory. We also find an analog of the Aharonov-Bohm effect; the net change of the norm of the wave function upon adiabatically forming a closed path is determined by the imaginary magnetic flux enclosed by the path, which provides an experimentally observable feature of the imaginary magnetic field.
The behavior of a neutral particle with magnetic quadrupole moment that interacts with axial magnetic and electric fields is analyzed. From the interaction of the magnetic quadrupole moment with the axial magnetic field, a spectrum of energy analogous to a Coulomb potential in two dimensions is obtained. Furthermore, the presence of an axial electric field is also considered. From the interaction of the magnetic quadrupole moment with this electric field, an analog of the Aharonov-Bohm effect is obtained. Finally, the Aharonov-Bohm effect for bound states is analyzed when this neutral particle system is subject to the two-dimensional harmonic oscillator.