Effects of Lorentz Symmetry Violation in the Spectra of Rare-Earth Ions in a Crystal Field

We demonstrate that experiments measuring the transition energies of rare-earth ions doped in crystalline lattices are sensitive to violations of Local Lorentz Invariance and Einstein's Equivalence Principle. Using the crystal field of LaCl\(_{3}\) as an example, we calculate the frame-dependent energy shifts of the transition frequencies between low-lying states of Ce\(^{3+}\), Nd\(^{3+}\), and Er\(^{3+}\) dopants in the context of the Standard Model Extension, and show that they have high sensitivity to electron anomalies that break rotational invariance.