We study magnetism on the fcc lattice with tetragonal distortions, withgeneral exotic directional magnetic interactions allowed by symmetry. Weconsider two models, corresponding to a uniform tetragonal distortion, or atwo-sublattice model with a tetragonal screw axis. We establish their lowtemperature phase diagrams in the semi-classical limit using classicaloptimization and consideration of fluctuations both analytically and by MonteCarlo simulation. Both order by disorder and exchange anisotropy mechanismsfavor a $\langle 110\rangle$ easy axis for magnetization. We also show thatspin-lattice coupling can give rise to an intermediate temperature paramagnetcnematic/orthorhombic phase, and discuss the transitions to/from this state.These results are relevant to a family of insulating magnetic doubleperovskites, and find immediate application to the ferromagnet Ba$_2$NaOsO$_6$.
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