Fractionalized excitations and resonant inelastic x-ray spectra in frustrated spin-1/2 trimer chains

The interacting spin-1/2 trimer system hosts a range of exotic quasiparticles, which have been observed through inelastic neutron scattering (INS) and this system has attracted significant attention in recent studies. We theoretically investigate the resonant inelastic x-ray scattering (RIXS) spectra in a frustrated spin-1/2 trimer chain, as realized in ${\mathrm{Na}}_{2}{\mathrm{Cu}}_{3}{\mathrm{Ge}}_{4}{\mathrm{O}}_{12}$, with Cu ${d}^{9}$ 1/2 spins. We compute multispin correlations contributing to spin-conserving (SC) and non-spin-conserving (NSC) RIXS cross sections using ultrashort core-hole lifetime expansion within the Kramer-Heisenberg formalism. These excitations involve flipping spins of up to three spin-1/2 trimers and include the INS single spin-flip excitations in the lowest order of the NSC channel. We identify the fractionalization of two coupled frustrated trimers in terms of spinons, doublons, and quartons in the spectra evaluated using exact diagonalization, complementing prior studies single spin-spin flip excitation in inelastic neutron scattering. Specifically, we uncover two new high-energy modes at $\ensuremath{\omega}\ensuremath{\approx}2.4{J}_{1}$ and $3.0{J}_{1}$ in the NSC and SC channels that are accessible at the Cu $K$-edge and $L$-edge RIXS spectra, which were missing in the INS study. This, therefore, provides pathways to uncover all the possible excitations in coupled trimers. Our work opens new opportunities for understanding the nature of fractionalization and RIXS spectra of frustrated, low-dimensional spin chains.