Galilei invariance and geometric oscillations about the half-filled Landau level.*

Geometric oscillation measurements by Kamburov et al. [Phys. Rev. Lett. 113, 196801 (2014)] called into question the long held belief that the half-filled lowest Landau level of electrons is a Fermi liquid-metallic state of nonrelativistic composite fermions, finding the oscillation minima to be determined by the electron density for ν 1/2. These (and subsequent) measurements highlight the incompleteness of our understanding of the half-filled Landau level: mean-field treatments of the Dirac (Son) and nonrelativistic (Halperin-Lee-Read) composite fermion theories give identical predictions for the locations of oscillation minima; if the effects of Chern-Simons gauge field fluctuations (ignored in any mean-field treatment) are included, the comparison between theory and experiment improves dramatically. Here we study the extent to which the disagreement between composite fermion mean-field theory and experiment might instead be attributed to an approximate Galilei symmetry using the Dirac composite fermion mean-field theory. We further comment on the relevance of an approximate Galilei symmetry to metallic states found at other even-denominator filling fractions.