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arXiv:2610.08521

Universal critical $g$ factor for spin-1/2 Aharonov-Bohm bound states

Vinícius Salem, Edilberto O. Silva, Fabiano M. Andrade

quant-phhep-thmath-phSubmitted Oct 6, 2026• Announced Oct 7, 2026

Abstract

Whether the spin-1/2 Aharonov-Bohm Hamiltonian supports bound states has remained controversial since Hagen concluded that no such states exist for a Dirac particle. Here, we show that Hagen's solution corresponds to a particular member of the one-parameter family of self-adjoint extensions that describe the singular Zeeman interaction, and we identify which member the microscopic physics selects. Matching the extension parameter to a finite-radius flux tube yields $ν=0$ precisely at $g=2$, and it does so in every flux sector. The critical $g$ factor is therefore universal, $g_c=2$, and depends on neither the flux sector nor the regularization radius. The flux sector controls the depth of the bound state energy, and we obtain a closed-form expression showing that binding deepens markedly with the integer part of the flux. Hagen's conclusion is thus confirmed and sharpened: it is not a statement about one arbitrary member of a family of extensions, but about the member that the physics selects when the magnetic moment takes its Dirac value, and an anomalous moment is a necessary and sufficient condition for binding in the singular channel.

Comments: 8 pages, 2 figures, comments are welcome

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Universal critical

g

factor for spin-1/2 Aharonov-Bohm bound states

Whether the spin-1/2 Aharonov-Bohm Hamiltonian supports bound states has remained controversial since Hagen concluded that no such states exist for a Dirac particle. Here, we show that Hagen's solution corresponds to a particular member of the one-parameter family of self-adjoint extensions that describe the singular Zeeman interaction, and we identify which member the microscopic physics selects. Matching the extension parameter to a finite-radius flux tube yields

ν=0

precisely at

g=2

, and it does so in every flux sector. The critical

g

factor is therefore universal,

g_c=2

, and depends on neither the flux sector nor the regularization radius. The flux sector controls the depth of the bound state energy, and we obtain a closed-form expression showing that binding deepens markedly with the integer part of the flux. Hagen's conclusion is thus confirmed and sharpened: it is not a statement about one arbitrary member of a family of extensions, but about the member that the physics selects when the magnetic moment takes its Dirac value, and an anomalous moment is a necessary and sufficient condition for binding in the singular channel.

It is well known that the formal Aharonov–Bohm Hamiltonian operator, describing the interaction of a charged particle with a magnetic vortex, has a four-parameter family of self-adjoint extensions, which reduces to a two-parameter family if one requires that the Hamiltonian commutes with the angular momentum operator. The question we study here is which of these self-adjoint extensions can considered as limits of regularized Aharonov–Bohm Hamiltonians, that is Pauli Hamiltonians in which the magnetic field corresponds to a flux tube of nonzero diameter. We show that not all the self-adjoint extensions in this two-parameter family can be obtained by these approximations, but only two one-parameter subfamilies. In these two cases we can choose the gyromagnetic ratio in the approximating Pauli Hamiltonian in such a way that we get convergence in the norm resolvent sense to the corresponding self-adjoint extension.

In this paper, we study the effects of rotation in the spin-1/2 non-relativistic Aharonov-Bohm problem for bound states. We use a technique based on the self-adjoint extension method and determine an expression for the energies of the bound states. The inclusion of the spin element in the Hamiltonian guarantees the existence of bound state solutions. We perform a numerical analysis of the energies and verify that both rotation and the spin degree of freedom affect the energies of the particle. The main effect we observe in this analysis is a cutoff value manifested in the Aharonov-Bohm flux parameter that delimits the values for the positive and negative energies.

We show that the Hamiltonian H and the helicity operator Λ of a Dirac particle moving in two dimensions in the presence of an infinitely thin magnetic flux tube each admit a four-parameter family of self-adjoint extensions. Each extension is in one-to-one correspondence with the boundary conditions (BCs) to be satisfied by the eigenfunctions at the origin. Although the actions of these two operators commute before specification of BCs, to ensure helicity conservation it is not sufficient to take the same BCs for both operators. We show that, given certain relations between the parameters of the extensions, it is possible to write down the most general domain where both operators H and Λ are self-adjoint with helicity conservation and also Aharonov-Bohm symmetry (ϕ→ϕ + 1) preserved, where ϕ is the magnetic flux in natural units. The continuity of the dynamics is also obtained. Our results imply that neither helicity conservation nor Aharonov-Bohm symmetry by itself solves the problem of choosing the `physical' BCs for this system.

The ((2+1)-dimensional) Aharonov–Bohm effect is analyzed for a spin-1/2 particle in the case that a 1/rpotential is present. Scalar and vector couplings are each considered. It is found that the approach in which the flux tube is given a finite radius that is taken to zero only after a matching of boundary conditions does not give physically meaningful results. Specifically, the operations of taking the limit of zero flux tube radius and the Galilean limit do not commute. Thus there appears to be no satisfactory solution of the relativistic Aharonov–Bohm–Coulomb problem using the finite radius flux tube method.

In this work, we review two methods used to approach singular Hamiltonians in (2+1) dimensions. Both methods are based on the self-adjoint extension approach. It is very common to find singular Hamiltonians in quantum mechanics, especially in quantum systems in the presence of topological defects, which are usually modelled by point interactions. In general, it is possible to apply some kind of regularization procedure, as the vanishing of the wave function at the location of the singularity, ensuring that the wave function is square integrable and then can be associated with a physical state. However, an analysis based on the self-adjoint extension approach can lead to more general boundary conditions that still gives acceptable physical states. We exemplify the methods by exploring the bound and scattering scenarios of a spin 1/2 charged particle with an anomalous magnetic moment in the Aharonov-Bohm potential in the conical space.