Based on recently proposed antiferromagnetic spin fluctuation exchange models for $d_{x^2-y^2}$ superconductors, we show that coupling to harmonic phonons cannot account for the observed isotope effect in the cuprate high-$T_c$ materials, whereas coupling to strong anharmonic multiple-well lattice tunneling modes can. Our results thus point towards a strongly enhanced effective electron-phonon coupling and a possible breakdown of Migdal-Eliashberg theory in the cuprates.

• Received 27 December 1994

DOI:https://doi.org/10.1103/PhysRevLett.75.4504