Spectroscopic signatures and origin of hidden order in Ba<inf>2</inf>MgReO<inf>6</inf>

Clarifying the underlying mechanisms that govern ordering transitions in condensed matter systems is crucial for comprehending emergent properties and phenomena. While transitions are often classified as electronically driven or lattice-driven, we present a departure from this conventional picture in the case of the double perovskite Ba2MgReO6. Leveraging resonant and non-resonant elastic x-ray scattering techniques, we unveil the simultaneous ordering of structural distortions and charge quadrupoles at a critical temperature of Tq ~ 33 K. Using a variety of complementary first-principles-based computational techniques, we demonstrate that, while electronic interactions drive the ordering at Tq, it is ultimately the lattice distortions that dictate the specific ground state that emerges. Our findings highlight the crucial interplay between electronic and lattice degrees of freedom, providing a unified framework to understand and predict unconventional emergent phenomena in quantum materials.