  We have analyzed different correlation functions in a realistic spin-orbital
model for half-doped manganites. Using a finite-temperature diagonalization
technique the CE phase was found in the charge-ordered phase in the case of
small antiferromagnetic interactions between $t_{2g}$ electrons. It is shown
that a key ingredient responsible for stabilization of the CE-type spin and
orbital-ordered state is the cooperative Jahn-Teller (JT) interaction between
next-nearest Mn$^{+3}$ neighbors mediated by the breathing mode distortion of
Mn$^{+4}$ octahedra and displacements of Mn$^{+4}$ ions. The topological phase
factor in the Mn-Mn hopping leading to gap formation in one-dimensional models
for the CE phase as well as the nearest neighbor JT coupling are not able to
produce the zigzag chains typical for the CE phase in our model.
