  Elucidating the details of the electron-phonon coupling in semiconductors and
insulators is a topic of pivotal interest, as it governs the transport
mechanisms and is responsible for various phenomena such as spectral-weight
transfers to phonon sidebands and self-trapping. Here, we investigate the
influence of the electron-phonon interaction on the excitonic peaks of rutile
TiO$_2$, revealing a strong anisotropic polarization dependence with increasing
temperature, namely an anomalous blueshift for light polarized along the a-axis
and a conventional redshift for light polarized along the c-axis. By employing
many-body perturbation theory, we identify two terms in the electron-phonon
interaction Hamiltonian that contribute to the anomalous blueshift of the
a-axis exciton. Our approach paves the way to a complete ab initio treatment of
the electron-phonon interaction and of its influence on the optical spectra of
polar materials.
