Correlation effects on electron-phonon coupling in semiconductors: many-body theory along thermal lines

A method is proposed for the inclusion of electron correlation in the calculation of the temperature dependence of band structures arising from electron-phonon coupling. It relies on an efficient exploration of the vibrational phase space along the recently introduced thermal lines. Using the $G_0W_0$ approximation, the temperature dependence of the direct gaps of diamond, silicon, lithium fluoride, magnesium oxide, and titanium dioxide is calculated. Within the proposed formalism, a single calculation at each temperature of interest is sufficient to obtain results of the same accuracy as in alternative, more expensive methods. It is shown that many-body contributions beyond semi-local density functional theory modify the electron-phonon coupling strength by almost $50$% in diamond, silicon, and titanium dioxide, but by less than $5$% in lithium flouride and magnesium oxide. The results reveal a complex picture regarding the validity of semi-local functionals for the description of electron-phonon coupling.