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Magnetic exchange interactions in SrMnO$_3$

We calculate Heisenberg-type magnetic exchange interactions for SrMnO$_3$ under isotropic volume expansion using an approach that is based on total energy variations due to infinitesimal spin rotations around a given reference state. Our total energy calculations using density functional theory (DFT) indicate a transition from antiferromagnetic to ferromagnetic coupling for increasing interatomic distances, corresponding to a sign change of the nearest neighbor exchange interaction. This sign change cannot easily be understood from a standard superexchange mechanism. Furthermore, the exchange interaction strongly depends on the corresponding reference state. This "non-Heisenberg" behavior increases with increasing volume and is also confirmed through non-collinear DFT calculations. An orbital- and energy-resolved decomposition of the exchange coupling suggests that an increased partial occupancy of $e_g$ orbitals near the Fermi level is crucial both for the sign change and the non-Heisenberg behavior of the nearest neighbor interaction. Furthermore, even though both $e_g$ and $t_{2g}$ contributions to the exchange interactions decay exponentially for large inter-atomic distances, the $e_g$ contribution remains surprisingly strong over relatively large distances along the crystal axes.