Orbital Order, Stripe Phases and Mott Transition in a Planar Model for Manganites

Understanding orbital ordered (OO) Mott insulating states lies at the heart of a consistent resolution of the colossal magneto-resistance (CMR) observed in manganites, where its melting induces a low-$T$ insulator-metal transition for $0.25 \le x\le 0.45$. Motivated thereby, we study the OO states in a planar model for bilayer manganites using DMFT and finite-size diagonalisation methods. We derive the correct OO ground states observed in manganites for $x=0,{1/2},{2/3},{3/4}$ in exact agreement with observations, including the charge-orbital-magnetic ordered stripe phases for $x>{1/2}$. These OO states are {\it exactly} shown to be associated with an ``alloy'' ordering of the $d_{3x^{2}-r^{2}}/d_{3y^{2}-r^{2}}$ orbitals on each $Mn^{3+}$ site.