  The Allen-Heine-Cardona theory allows us to calculate phonon-induced electron
self-energies from first principles without resorting to the adiabatic
approximation. However, this theory has not been able to account for the change
of the electron wave function, which is crucial if interband energy differences
are comparable to the phonon-induced electron self-energy as in
temperature-driven topological transitions. Furthermore, for materials without
inversion symmetry, even the existence of such topological transitions cannot
be investigated using the Allen-Heine-Cardona theory. Here, we generalize this
theory to the renormalization of both the electron energies and wave functions.
Our theory can describe both the diagonal and off-diagonal components of the
Debye-Waller self-energy in a simple, unified framework. For demonstration, we
calculate the electron-phonon coupling contribution to the
temperature-dependent band structure and hidden spin polarization of BiTlSe2
across a topological transition. These quantities can be directly measured. Our
theory opens a door for studying temperature-induced topological phase
transitions in materials both with and without inversion symmetry.
