Dipolar and quadrupolar correlations in the $5d^2$ Re-based double perovskites Ba$_2$YReO$_6$ and Ba$_2$ScReO$_6$

Double perovskites containing heavy transition metal ions are an important family of compounds for the study of the interplay between electron correlation and spin-orbit coupling. Here, by combining magnetic susceptibility, heat capacity, and neutron scattering measurements, we investigate the dipolar and quadrupolar correlations in two prototype rhenium-based double perovskite compounds, Ba$_2$YReO$_6$ and Ba$_2$ScReO$_6$. A type-I dipolar antiferromagnetic ground state with a propagation vector $\mathbf{q} = (0, 0, 1)$ is observed in both compounds. At temperatures above the magnetic transitions, a quadrupolar ordered phase is identified. Weak spin excitations, which are gapped at low temperatures and softened in the correlated paramagnetic phase, are explained using a minimal model that considers both the dipolar and quadrupolar interactions. At larger wavevectors, we observe dominant phonon excitations that are well described by density functional calculations.