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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, dispersivemagnetic excitations are observed, with a zone boundary energy of $\sim$300 meVand a weak dependence on doping. These excitations are strikingly similar tothe bosons proposed to explain the high-energy `kink' observed inphotoemission. A calculation of the spin-response based on the ARPES-derivedelectronic structure and YRZ-quasi-particles reproduces the key features of theobserved magnetic dispersions with no adjustable parameters. These results showthat it is possible to reconcile the magnetic and electronic properties of thecuprates within a unified framework.

physical review letters

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