The Jahn-Teller theorem states that any non-linear molecule or crystal site with electronic degeneracy will spontaneously distort to break the symmetry and lower its energy. When distortions couple across a lattice, cooperative Jahn-Teller ordering emerges.
Jahn-Teller Physics: At each site, the degenerate e_g orbital states (d_{z²}, d_{x²−y²}) couple linearly to the octahedral Q₂ stretching mode (q) via H_JT = −λ·q·σ_z. The adiabatic potential E±(q) = K·q²/2 ∓ λ|q| is a Mexican hat / double well. Below the cooperative ordering temperature T_JT ∝ λ²/K, all sites distort coherently (orbital ordering). The lattice shows site distortions: elongated (green) = positive Q₂, compressed (gold) = negative Q₂. The alternating checkerboard pattern is the orbital-ordered ground state of manganites.