Charge and orbital order due to cooperative Jahn-Teller effect in   manganite chains
  We derive an effective Hamiltonian that takes into account the quantum nature of phonons and models cooperative Jahn-Teller effect in the adiabatic regime and at strong electron-phonon coupling in one dimension. Our approach involves mapping a strong-coupling problem to a weak-coupling one by using a duality transformation. Subsequently, a sixth-order perturbation theory is employed in the polaronic frame of reference where the small parameter is inversely (directly) proportional to the coupling (adiabaticity). We study charge and orbital order in ferromagnetic manganite chains and address the pronounced electron-hole asymmetry in the observed phase diagram. In particular, at strong coupling, we offer an explanation for the observed density dependence of the wavevector of charge modulation, i.e., wavevector is proportional to (independent of) electron density on the electron-doped (hole-doped) side of the phase diagram of manganites. We also provide a picture for the charge and orbital order at special fillings $ \frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, and $ \frac{1}{5}$; while focusing on the ordering controversy at fillings $\frac{1}{3}$ and $\frac{1}{4}$, we find that Wigner-crystal arrangement is preferred over bi-stripe order. 
Ravindra Pankaj and Sudhakar Yarlagadda
cond-mat.str-el