Abstract
The evolution of the so-called ``41 meV resonance'' in the magnetic response of YBCO cuprates is studied with slave boson theory for the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}J$ model. The resonance appears as a collective spin fluctuation in the $d$-wave superconducting (SC) state. It is undamped at optimal doping due to a threshold in the excitation energies of particle-hole pairs with relative wave vector $(\ensuremath{\pi},\ensuremath{\pi})$. When hole filling is reduced, the resonance moves to lower energies and broadens. Below the resonance energy we find a crossover to an incommensurate response in agreement with a recent experiment on ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{6.6}$. We show that dynamic nesting in the $d$-wave SC state causes this effect.
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