Van Vleck excitons in Ca$_{2}$RuO$_{4}$

A framework is presented for modeling and understanding magnetic excitations in localized, intermediate coupling magnets where the interplay between spin-orbit coupling, magnetic exchange, and crystal field effects are known to create a complex landscape of unconventional magnetic behaviors and ground states. A spin-orbit exciton approach for modeling these excitations is developed based upon a Hamiltonian which explicitly incorporates single-ion crystalline electric field and spin exchange terms. This framework is then leveraged to understand a canonical Van Vleck $j\rm{_{eff}}=0$ singlet ground state whose excitations are coupled spin and crystalline electric field levels. Specifically, the anomalous Higgs mode [Jain et al. Nat. Phys. 13, 633 (2017)], spin-waves [S. Kunkem\"{o}ller et al. Phys. Rev. Lett. 115, 247201 (2015)], and orbital excitations [L. Das et al. Phys. Rev. X 8, 011048 (2018)] in the multiorbital Mott insulator Ca$_2$RuO$_4$ are captured and good agreement is found with previous neutron and inelastic x-ray spectroscopic measurements. Furthermore, our results illustrate how a crystalline electric field-induced singlet ground state can support coherent longitudinal, or amplitude excitations, and transverse wavelike dynamics. We use this description to discuss mechanisms for accessing a nearby critical point.