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

We present a unified, SI-consistent framework to constrain minimal SME coefficients $a_{\mu }$ and $b_{\mu }$ 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 ($\mathbf{a},\mathbf{b}$) reshape the effective potential, via $1\mathrm{/}r$ and $r$ terms or spin-selective offsets, while scalar components ($a_0,b_0$) act through a global energy shift and a spin-momentum coupling. Phenomenological upper bounds follow from requiring LV-induced shifts to lie below typical spectroscopic resolutions: $\mathrm{\mid }{a}_0\mathrm{\mid }\lesssim \delta E$, $\mathrm{\mid }{b}_z\mathrm{\mid }\lesssim \delta E\mathrm{/}\mathrm{\hslash }$, and compact expressions for $\mathrm{\mid }{a}_{\phi }\mathrm{\mid }$ and $\mathrm{\mid }{b}_0\mathrm{\mid }$ that expose their dependence on device scales ($r_0$, $B_0$, $\mu$, $m$). Dimensional analysis clarifies that, in this regime, spatial $a_i$ carry momentum dimension and $b_i$ carry inverse-time/length dimensions, ensuring gauge-independent, unit-consistent reporting. Finite-difference eigenvalue calculations validate the scaling laws and illustrate spectral signatures across realistic parameter sets. The results show that scalar sectors (notably $a_0$) are tightly constrained by state-of-the-art $\mu$eV-resolution probes, while spatial and axial sectors benefit from spin- and $m$-resolved spectroscopy and geometric leverage, providing a reproducible pathway to test Lorentz symmetry in condensed-matter platforms.