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 ar
e 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.
The dynamics of S=1/2 quantum spins on a 2D square lattice lie at the heart of the mystery of the cuprates cite{Hayden2004,Vignolle2007,Li2010,LeTacon2011,Coldea2001,Headings2010,Braicovich2010}. In bulk cuprates such as LCO{}, the presence of a weak
interlayer coupling stabilizes 3D N{e}el order up to high temperatures. In a truly 2D system however, thermal spin fluctuations melt long range order at any finite temperature cite{Mermin1966}. Further, quantum spin fluctuations transfer magnetic spectral weight out of a well-defined magnon excitation into a magnetic continuum, the nature of which remains controversial cite{Sandvik2001,Ho2001,Christensen2007,Headings2010}. Here, we measure the spin response of emph{isolated one-unit-cell thick layers} of LCO{}. We show that coherent magnons persist even in a single layer of LCO{} despite the loss of magnetic order, with no evidence for resonating valence bond (RVB)-like spin correlations cite{Anderson1987,Hsu1990,Christensen2007}. Thus these excitations are well described by linear spin wave theory (LSWT). We also observe a high-energy magnetic continuum in the isotropic magnetic response. This high-energy continuum is not well described by 2 magnon LSWT, or indeed any existing theories.
