High-energy magnetic excitations in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$: Towards a unified description of the electronic and magnetic degrees of freedom in the cuprates

We investigate the high-energy magnetic excitation spectrum of the high-$T_c$ cuprate superconductor Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ (Bi-2212) using Cu $L_3$ edge resonant inelastic x-ray scattering (RIXS). Broad, dispersive magnetic excitations are observed, with a zone boundary energy of $\sim$300 meV and a weak dependence on doping. These excitations are strikingly similar to the bosons proposed to explain the high-energy `kink’ observed in photoemission. A calculation of the spin-response based on the ARPES-derived electronic structure and YRZ-quasi-particles reproduces the key features of the observed magnetic dispersions with no adjustable parameters. These results show that it is possible to reconcile the magnetic and electronic properties of the cuprates within a unified framework.