Anharmonic phonon frequency shift in MgB2

We compute the anharmonic shift of the phonon frequencies in MgB2, using density functional theory. We explicitly take into account the scattering between different phonon modes at different q points in the Brillouin zone. The shift of the E-2g mode at the Gamma point is only +5% of the harmonic frequency, therefore questioning the accepted explanation of the anomalous isotope effect on the critical temperature. This result comes from the cancellation between the contributions of the four- and three-phonon scattering, respectively, +10% and -5%. A similar shift is predicted at the A point, in agreement with inelastic x-ray scattering phonon-dispersion measurements.