  At half doping, the ground state of three-dimensional manganite perovskite
oxides like R$_{1-x}$Ca$_x$MnO$_3$, where R is a trivalent ion such as La, Pr,
etc, is still unclear. Many experimental findings agree better with the
combined magnetic, charge, and orbital order characteristic of the
"intermediate phase", introduced by Efremov et al. in 2004 [Nature Mats. 3,
853]. This phase consists of spin dimers (thus incorporating aspects of the
Zener polaron phase (ZP) proposed in 2002 by Daoud-Aladine et al. [Phys. Rev.
Lett. 89, 097205]), though formed by a pair of parallel Mn spins of different
magnitude, in principle (thereby allowing for a degree of Mn charge
disproportionation: not necessarily as large as that of Mn$^{3+}$-Mn$^{4+}$ in
Goodenough's original CE phase [Phys. Rev. 100, 564 (1955)]). In the
intermediate phase, consecutive spin dimers localed along the planar zig-zag
chains are oriented at a constant relative angle $\Phi$ between them. Varying
Mn-charge disproportionation and $\Phi$, the intermediate phase should allow to
continuously interpolate between the two limiting cases of the CE phase and the
dimer phase denoted as "orthogonal intermediate $\pi/2-$phase". It is not easy
to find a microscopic model able to describe the phenomenological intermediate
phase adequately for the spin, charge, and orbital degrees of freedom
simultaneously. Here, we study the quantum spin excitations of a planar model
of interacting localized spins, which we found can stabilize the intermediate
phase classically. We compare the quantum magnons of the intermediate phase
with those of the CE and orthogonal $\pi/2$ phases, in the context of recent
experimental results.
