The phase diagram of the spin-orbital (SO) Kugel-Khomskii ($d^9$) model
posed a challenging theoretical problem [1], yet it is still unknown
[1]. Here we investigate the phase diagrams of the $d^9$ model,
depending on Hund’s exchange $J_H$ and the $e_g$ orbital splitting $E_z$,
for a bilayer and a monolayer square lattice using
Bethe-Peierls-Weiss method
with exact diagonalization of a cubic or square cluster
coupled to its neighbors in $ab$ planes by the mean-field (MF) terms.
The cluster MF method
confirms existence of singlet phases similar to those obtained by
variational wave functions [2], and enables finite SO
order parameter independent of spin and orbital ordering. For a bilayer
we obtain phases with interlayer spin singlets stabilized by holes in
$3z^2-r^2$ orbitals and with alternating
plaquette valence-bond (PVB) as well as two new phases with SO
entanglement, in addition to the antiferromagnetic
($G$-AF, $A$-AF) and ferromagnetic (FM) order.
For a monolayer we obtained at temperature $T=0$:
(i) the PVB phase, (ii) two AF phases with either $3z^2-r^2$
or $x^2-y^2$ orbitals occupied, and (iii) a FM phase.
However, after including thermal fluctuations ($T>0$) we found the
same entangled SO phases as for a bilayer at $T=0$.
This shows that both quantum and thermal fluctuations can stabilize
phases with exotic SO order while the classical
spin order is destroyed.
[1] L. F. Feiner, A. M. Ole\’s, and J. Zaanen,
Phys. Rev. Lett. \textbf{78}, 2799 (1997).
[2] A. M. Ole\’s, Acta Phys. Polon. A \textbf{115}, 36 (2009).
