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.
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