Biquadratic and ring exchange interactions in orthorhombic perovskite manganites

We use ab initio electronic structure calculations within the GGA+U approximation to density functional theory (DFT) to determine the microscopic exchange interactions in the series of orthorhombic rare-earth manganites (o-RMnO$_3$). Our motivation is to construct a model Hamiltonian (excluding effects due to spin-orbit coupling), which can provide an accurate description of the magnetism in these materials. First we map the exchange couplings for several representatives of o-RMnO$_3$ series onto a Heisenberg Hamiltonian and find a clear deviation from the Heisenberg-like behavior. We demonstrate that this deviation can be explained only by the presence of relatively strong higher order exchange interactions (biquadratic and four-spin ring couplings) and show that they have the strongest effect in compounds, where nearest-neighbor exchange interactions are weakened due to the presence of large GdFeO$_3$-type distortion. Finally we discuss how these higher order terms determine magnetic ground states, influence magnetic excitations and define the multiferroic properties of o-RMnO$_3$.