The Aharonov-Bohm-Casher problem is examined for a charged particle describing a circular path in the presence of a Lorentz-violating background that is nonminimally coupled to a spinor and a gauge field. The particle eigenenergies were evaluated, showing that the LV background is able to lift the original degenerescence in the absence of magnetic field even for a neutral particle. The Aharonov-Casher phase is used to impose an upper bound on the background magnitude. A similar analysis is accomplished in a space endowed with a topological defect, revealing that both the disclination parameter and the LV background are able to modify the particle eigenenergies. We also analyze a particular case where the particles interact harmonically with the topological defect and the LV background, with similar results.

In this work, we propose a $CPT$-even and Lorentz-violating dimension-five nonminimal coupling between fermionic and gauge fields, involving the $CPT$-even and Lorentz-violating gauge tensor of the Standard Model extension. This nonminimal coupling modifies the Dirac equation, whose nonrelativistic regime is governed by a Hamiltonian which induces new effects, such as an electric-Zeeman-like spectrum splitting and an anomalouslike contribution to the electron magnetic moment, among others. Some of these new effects allow one to constrain the magnitude of this nonminimal coupling in 1 part in ${10}^{16}(\mathrm{eV}{)}^{-1}$.

In this work the dynamics of a 2D Dirac oscillator in the spacetime of a magnetic cosmic string is considered. It is shown that earlier approaches to this problem have neglected a \(\delta \) function contribution to the full Hamiltonian, which comes from the Zeeman interaction. The inclusion of spin effects leads to results which confirm a modified dynamics. Based on the self-adjoint extension method, we determined the most relevant physical quantities, such as energy spectrum, wave functions and the self-adjoint extension parameter by applying boundary conditions allowed by the system.

We examine the bound state and scattering problem of a spin-one-half particle undergone to an Aharonov-Bohm potential in a conical space in the nonrelativistic limit. The crucial problem of the $\delta$-function singularity coming from the Zeeman spin interaction with the magnetic flux tube is solved through the self-adjoint extension method. Using two different approaches already known in the literature, both based on the self-adjoint extension method, we obtain the self-adjoint extension parameter to the bound state and scattering scenarios in terms of the physics of the problem. It is shown that such a parameter is the same for both situations. The method is general and is suitable for any quantum system with a singular Hamiltonian that has bound and scattering states.

Based on the discussions about the Aharonov-Casher effect in the Lorentz symmetry violation background, we show that the analogue of the relativistic Landau quantization in the Aharonov-Casher setup can be achieved in the Lorentz-symmetry violation background.