Curie temperature of the two band double exchange model for manganites

We consider two-band double exchange model and calculate the critical temperature in ferromagnetic regime (Curie temperature). The localized spins are represented in terms of the Schwinger-bosons, and two spin-singlet Fermion operators are introduced. In terms of the new Fermi fields the on-site Hund's interactions are in a diagonal form and one accounts for them exactly. Integrating out the spin-singlet fermions we derive an effective Heisenberg type model for a vector which describes the local orientations of the total magnetization. The transversal fluctuations of the vector are the true magnons in the theory, which is a base for Curie temperature calculation. The critical temperature is calculated employing the Schwinger-bosons mean-field theory. While approximate, this technic of calculation captures the essentials of the magnon fluctuations in the theory, and for 2D systems one obtains zero Curie temperature, in accordance with Mermin-Wagner theorem.