  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.
