Suppression of effective spin-orbit coupling by thermal fluctuations in spin-orbit coupled antiferromagnets

We apply the finite-temperature variational cluster approach to a strongly correlated and spin-orbit coupled model for four electrons (i.e., two holes) in ${t}_{2g}$ subshell. focus on parameters suitable antiferromagnetic Mott insulators, particular, ${\mathrm{Ca}}_{2}{\mathrm{RuO}}_{4}$, identify crossover from low-temperature regime, where coupling is essential, high-temperature regime it leaves few signatures. The seen one-particle spectra, $xz$ $yz$ spectra are almost one dimensional (as expected weak coupling) at high temperature. At lower temperature, mixes all three orbitals, they become more dimensional. However, stronger effects two-particle observables like weight states with definite on-site angular momentum. thus enigmatic intermediate-temperature ``orbital-order phase transition,'' which has been reported various x-ray diffraction absorption experiments $T\ensuremath{\approx}260$ K, as signature of onset spin-orbital correlations.