The dynamics of a doped hole in cuprates is not controlled by spin fluctuations

Twenty seven years after the discovery of high-temperature superconductivity \cite{BedMu}, consensus on its theoretical explanation is still absent. To a good extent, this is due to the difficulty of studying strongly correlated systems near half-filling, needed to understand the behaviour of one or few holes doped into a CuO$_2$ layer. To simplify this task it is customary to replace three-band models \cite{Emery} describing the doping holes as entering the O $2p$ orbitals of these charge-transfer insulators \cite{ZSA} with much simpler one-band Hubbard or $tJ$ models \cite{rev1,rev2}. Here we challenge this approach, showing that not only is the dynamics of a doped hole easier to understand in models that explicitly include the O orbitals, but also that our solution contradicts the long-held belief that the quantum spin fluctuations of the antiferromagnetic (AFM) background play a key role in determining this dynamics. Indeed, we show that the correct, experimentally observed dispersion is generically obtained for a hole moving on the O sublattice, and coupled to a N\'eel lattice of spins without spin fluctuations. This marks a significant conceptual change in our understanding of the relevant phenomenology and opens the way to studying few-holes dynamics without finite-size effect issues \cite{BayoB}, to understand the actual strength of the "magnetic glue".