Magnetization dynamics fingerprints of an excitonic condensate $t_{2g}^{4}$ magnet

The competition between spin-orbit coupling $\lambda$ and electron-electron interaction $U$ leads to a plethora of novel states of matter, extensively studied in the context of $t_{2g}^4$ and $t_{2g}^5$ materials, such as ruthenates and iridates. Excitonic magnets -- the antiferromagnetic state of bounded electron-hole pairs -- is a prominent example of phenomena driven by those competing energy scales. Interestingly, recent theoretical studies predicted that excitonic magnets can be found in the ground-state of spin-orbit-coupled $t_{2g}^4$ Hubbard models. Here, we present a detailed computational study of the magnetic excitations in that excitonic magnet, employing one-dimensional chains (via density matrix renormalization group) and small two-dimensional clusters (via Lanczos). Specifically, first we show that the low-energy spectrum is dominated by a dispersive (acoustic) magnonic mode, with extra features arising from the $\lambda=0$ state in the phase diagram. Second, and more importantly, we found a novel magnetic excitation forming a high-energy optical mode with the highest intensity at wavevector $q\to 0$. In the excitonic condensation regime at large $U$, we also have found a novel high-energy $\pi$-mode composed solely of orbital excitations. These unique fingerprints of the $t_{2g}^4$ excitonic magnet are important in the analysis of neutron and RIXS experiments.