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

TL;DR

This work develops a SI-consistent, nonrelativistic SME framework for magnetically confined 2D electrons, deriving a cylindrical radial reduction that maps the minimal SME coefficients $a_\mu$ and $b_\mu$ onto measurable spectral features. It dissects how scalar components $(a_0,b_0)$ produce uniform shifts and spin–momentum couplings, while spatial components $(a_\varphi, b_z)$ deform the radial confinement and induce spin-dependent offsets; the bounds are translated into SI units via spectroscopic resolution $\delta E$ and device scales $(r_0,B_0,\mu,m)$. The paper provides explicit phenomenological limits: $|a_0| \lesssim \delta E$, $|b_z| \lesssim \delta E/\hbar$, $|a_\varphi| \lesssim \mu\delta E / |\hbar m/r_0 - eB_0 r_0/2|$, and $|b_0| \lesssim \mu\delta E / (\hbar^2| m/r_0 - eB_0 r_0/(2\hbar)|)$, with finite-difference numerics validating the scaling and spectral fingerprints. It further develops SI reporting conventions, introduces dimensionless cross-platform diagnostics, and presents a concrete experimental roadmap to exploit μeV-scale spectroscopy in quantum dots and rings to probe Lorentz symmetry, complementing high-energy tests with tabletop precision measurements.

Abstract

We present a unified, SI-consistent framework to constrain minimal SME coefficients $a_μ$ and $b_μ$ 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 $1/r$ 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: $|a_0|\lesssimδE$, $|b_z|\lesssimδE/\hbar$, and compact expressions for $|a_\varphi|$ and $|b_0|$ that expose their dependence on device scales ($r_0$, $B_0$, $μ$, $m$). 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 $μ$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.

Figures (4)

• Figure 1: Eigenvalues $\varepsilon_n$ (eV) as a function of $a_0$ (eV) for $m=1$, $s=1$. Reference set: $\mu=0.067\,m_e$, $r_0=10$ nm, $B_0=1$ T. The range shown corresponds to meV-scale resolutions ($a_0 \sim 10^{-20}$ eV). The dependence is strictly linear: $a_0$ acts as a uniform vertical shift and does not change level spacings.
• Figure 2: Same calculation as Fig. \ref{['fig:En_vs_a0_case1']}, now zoomed into the $\mu$eV-resolution regime ($a_0 \sim 10^{-25}$ eV). The curves remain parallel and offset, confirming that $a_0$ leaves relative splittings essentially unchanged even at the $\mu$eV scale.
• Figure 3: Eigenvalues $\varepsilon_n$ plotted versus $b_0$ (m$^{-1}$) for $m=1$, $s=1$, using the parameter range relevant for the bounds at the $10^9$ m$^{-1}$ level (Table \ref{['tab:bounds_a0_b0']}). The vertical axis is rescaled (meV $\times 10^3$) for readability. Within this realistic range, the curves are essentially flat: the $b_0$-induced spin-momentum correction is below current meV-scale resolution. Parameters as in Fig. \ref{['fig:En_vs_a0_case1']}.
• Figure 4: Same calculation as Fig. \ref{['fig:Energy_vs_b0_case1']}, but zoomed into an order-of-magnitude tighter window ($b_0 \sim 10^4$ m$^{-1}$), representative of $\mu$eV-level metrology. A weak but visible $n$-dependence now emerges because $b_0$ couples spin to the azimuthal kinetic momentum. Higher-$n$ states, being more spatially extended, are slightly more sensitive.