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We use ab initio electronic structure calculations within the generalizedgradient approximation (GGA+U) to density functional theory (DFT) to determinethe microscopic exchange interactions in the series of orthorhombic rare-earthmanganites, o-$R$MnO$_3$. Our motivation is to construct a model Hamiltonian(excluding effects due to spin-orbit coupling), which can provide an accuratedescription of the magnetism in these materials. First, we consider TbMnO$_3$,which exhibits a spiral magnetic order at low temperatures. We map the exchangecouplings in this compound onto a Heisenberg Hamiltonian and observe a cleardeviation from the Heisenberg-like behavior. We consider first the couplingbetween magnetic and orbital degrees of freedom as a potential source ofnon-Heisenberg behavior in TbMnO$_3$, but conclude that it does not explain theobserved deviation. We find that higher order magnetic interactions(biquadratic and four-spin ring couplings) should be taken into account for aproper treatment of the magnetism in TbMnO$_3$ as well as in the otherrepresentatives of the o-$R$MnO$_3$ series with small radii of the $R$ cation.

Biquadratic and ring exchange interactions in orthorhombic perovskite manganites