Calculation of the four-spin cyclic exchange in cuprates

Starting from the three-band Hubbard model for the cuprates, we calculate analytically the four-spin cyclic exchange in the limit of infinite on-site Coulomb repulsion and zero O-O hopping $t_{pp}$ using two methods: i) perturbation theory in $t_{pd}\mathrm{/}\mathrm{\Delta }$, where $t_{pd}$ is the Cu-O hopping and $\mathrm{\Delta }$ the Cu-O charge transfer energy and ii) exact solution of a Cu${}_4$O${}_4$ plaquette. The latter method coincides with the first to order eight in $t_{pd}$ and permits to extend the results to $t_{pd}\mathrm{/}\mathrm{\Delta }$ of order one. The results are relevant to recent experimental and theoretical research that relate the splitting of certain spin excitations with $\mathrm{\Delta }$ and the superconducting critical temperature.