Broken reflection in fourfold Landau fields: amplitude drift, exact witnesses, and LiErF4

A fourfold order parameter ψ with the action ψ → iψ is often called Z4 order, but its Landau energy can keep internal reflections that the crystal does not require. For one complex scalar every C4-invariant potential of degree ≤ 4 is reflective after a rephasing. The first obstruction appears at sixth order, through the invariant J6 = Im(c11 c10*), the Landau form of a known relative crystal-field phase at S4 sites. When the angular energy is a single harmonic, J6 ≠ 0 is equivalent to a drift of the preferred angle with the order-parameter amplitude, Lorentzian in ρ². Vacuum amplitude–phase mixing and barrier-saddle displacement are exact readings of one Bezoutian kernel and, at sixth order, vanish only if J6 = 0. The phase offset inside a domain wall is a nonlocal linear response and can have the opposite sign. A second harmonic, allowed at eighth order, permits reflection breaking without drift. Whether J6 is allowed is decided by the image of the space group on the order parameter. In scheelite LiErF4 the uniform magnetization has a reflection-free image, and a single-ion model with four published crystal fields predicts that the in-plane easy axis turns by 1.9–4.1° between 0.05 and 4 T; the ordered antiferromagnetic order parameter has a reflective image. Code, data and preregistration files: https://doi.org/10.5281/zenodo.24034636