Magnetic and orbital correlations in multiferroic CaMn$_7$O$_{12}$ probed by x-ray resonant elastic scattering

The quadruple perovskite CaMn$_7$O$_{12}$ is a topical multiferroic, in which the hierarchy of electronic correlations driving structural distortions, modulated magnetism, and orbital order is not well known and may vary with temperature. x-ray resonant elastic scattering (XRES) provides a momentum-resolved tool to study these phenomena, even in very small single crystals, with valuable information encoded in its polarization- and energy-dependence. We present an application of this technique to CaMn$_7$O$_{12}$. By polarization analysis, it is possible to distinguish superstructure reflections associated with magnetic order and orbital order. Given the high momentum resolution, we resolve a previously unknown splitting of an orbital order superstructure peak, associated with a distinct textit{locked-in} phase at low temperatures. A second set of orbital order superstructure peaks can then be interpreted as a second-harmonic orbital signal. Surprisingly, the intensities of the first- and second-harmonic orbital signal show disparate temperature and polarization dependence. This orbital re-ordering may be driven by an exchange mechanism, that becomes dominant over the Jahn-Teller instability at low temperature.