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Spin-orbital order in undoped manganite LaMnO3 at finite temperature

We investigate the evolution of spin and orbital order in undoped LaMnO$_3$ under increasing temperature with a model including both superexchange and Jahn-Teller interactions. We used several cluster mean field calculation schemes and find coexisting $A$-type antiferromagnetic ($A$-AF) and $C$-type alternating orbital order at low temperature. The value of the Jahn-Teller coupling between strongly correlated $e_g$ orbitals is estimated from the orbital transition temperature at $T_{\rm OO}\simeq 780$ K. By a careful analysis of on-site and on-bond correlations we demonstrate that spin-orbital entanglement is rather weak. We have verified that the magnetic transition temperature is influenced by entangled spin-orbital operators as well as by entangled orbital operators on the bonds but the errors introduced by decoupling such operators partly compensate each other. Altogether, these results justify why the commonly used disentangled spin-orbital model is so successful in describing the magnetic properties and the temperature dependence of the optical spectral weights for LaMnO$_3$.