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Europhysics Letters (EPL)

Aharonov-Bohm quantum phase influence on a non-Hermitian quantum system

K. Bakke

Published Mar 30, 2026• 1 citation
Publisher

Abstract

Abstract We study the influence of the Aharonov-Bohm quantum phase on a quantum system described by a non-Hermitian Hamiltonian. We show that a non-Hermitian Hamiltonian has real energy eigenvalues which are influenced by the Aharonov-Bohm quantum phase. In addition, we show that persistent currents can arise. Finally, we calculate the revival times in this two-dimensional system and show that they are influenced by Aharonov-Bohm quantum phase.

DOI
10.1209/0295-5075/ae58cc

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Abstract We study the influence of the Aharonov-Bohm quantum phase on a quantum system described by a non-Hermitian Hamiltonian. We show that a non-Hermitian Hamiltonian has real energy eigenvalues which are influenced by the Aharonov-Bohm quantum phase. In addition, we show that persistent currents can arise. Finally, we calculate the revival times in this two-dimensional system and show that they are influenced by Aharonov-Bohm quantum phase.
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.
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 relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of H is obtained using pseudo-bosonic operators.
In this paper, we study the quantum dynamics of an electron/hole in a two-dimensional quantum ring within a spherical space. For this geometry, we consider a harmonic confining potential. Suggesting that the quantum ring is affected by the presence of an Aharonov–Bohm flux and a uniform magnetic field, we solve the Schrödinger equation for this problem and obtain exactly the eigenvalues of energy and corresponding eigenfunctions for this nanometric quantum system. Afterwards, we calculate the magnetization and persistent current are calculated, and discuss influence of curvature of space on these values.
The spectrum of non-Hermitian Hamiltonians is studied using the point-group symmetries of the Hermitian and non-Hermitian parts of the Hamiltonian. We show that, in principle, the symmetry properties of the Hamiltonian are responsible for the appearance of real eigenvalues in the spectrum of the non-Hermitian Hamiltonian. In practice, however, one must take into account the ``strength'' of the non-Hermitian part of the Hamiltonian. An effective energy-dependent Hermitian Hamiltonian is constructed that has the same real spectrum as the non-Hermitian Hamiltonian. Several properties, including the self-orthogonality of the states at branch points of the spectrum, are proven using this effective representation. An interesting possibility of complex eigenvalues returning to the real axis is exemplified along with several examples of non-Hermitian Hamiltonians with real eigenvalues.
The exact energy spectrum and wave function of a charged particle moving in the Coulomb field and Aharonov-Bohm's magnetic flux are solved by the nonintegrable phase factor. The universal formula for the matrix elements of the radial operator ${r}^{\ensuremath{\alpha}}$ of arbitrary power $\ensuremath{\alpha}$ is given by an analytical solution. The difference between the classical limit of matrix elements of inverse radius in quantum mechanics and the Fourier components of the corresponding quantity for the pure Coulomb system in classical mechanics is examined in reference to the correspondence principle of Heisenberg. Explicit calculation shows that the influence of nonlocal Aharonov-Bohm effect exists even in the classical limit. The semiclassical quantization rule for systems containing the topological effect is presented in the light of Heisenberg's corresponding principle.