Bounds on Lorentz-violating parameters in magnetically confined 2D systems: A phenomenological approach

We present a unified, SI-consistent framework to constrain minimal SME effective coefficients a μ ( eff ) and b μ ( eff ) using magnetically confined two-dimensional electron systems under a uniform magnetic field. Working in the nonrelativistic (Schrödinger–Pauli) limit with effective mass, we derive the radial problem for cylindrical geometries and identify how spatial components ( a ( eff ) , b ( eff ) ) reshape the effective potential, via 1 / r and r terms or spin-selective offsets, while timelike components ( a 0 ( eff ) , b 0 ( eff ) ) act through a global energy shift and a spin-momentum coupling. We caution that the timelike labels refer to the transformation properties of these backgrounds under the observer Lorentz group, not to Lorentz-scalar quantities. Phenomenological upper bounds follow from requiring LV-induced shifts to lie below typical spectroscopic resolutions: | a 0 ( eff ) | ≲ δ E , | b z ( eff ) | ≲ δ E / ħ , and compact expressions for | a φ ( eff ) | and | b 0 ( eff ) | that expose their dependence on device scales ( r 0 , B 0 , μ , m ). Dimensional analysis clarifies that, in this regime, spatial a i ( eff ) carry momentum dimension and b i ( eff ) carry inverse-time/length dimensions, ensuring unit-consistent reporting. Finite-difference eigenvalue calculations validate the scaling laws and illustrate spectral signatures across realistic parameter sets. The results show that timelike sectors (notably a 0 ( eff ) ) are tightly constrained by state-of-the-art μ eV-resolution probes, while spatial and axial sectors benefit from spin- and m -resolved spectroscopy and geometry-dependent sensitivity, providing a reproducible pathway to test Lorentz symmetry in condensed-matter platforms. We emphasize that the quantities constrained here should be interpreted as material- and device-dependent effective coefficients in an effective-mass description; connecting them to vacuum SME coefficients requires a microscopic band-structure derivation beyond our phenomenological scope.