Isotope effect on exchange in antiferromagnetic insulators

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Motivated by the recent proposal of a nonvanishing isotope effect \ensuremath{\alpha} on the Heisenberg exchange integral J in ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$, we have constructed a spin-phonon Hamiltonian as an effective model for strongly interacting electrons at half filling, coupling to local anharmonic phonons. We have studied the model in the limit when the typical phonon energy is much larger than J. Applying our results to ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$ and using realistic parameters, we can explain the order of magnitude of \ensuremath{\alpha}.