The principle of detailed balance between electrons and phonons in presence of excitonic effects

Based on a many-body formulation, we derive an electron-phonon coupling including excitonic effects that preserves thermodynamic detailed balance between electronic and phononic scattering processes. We start from the microscopic electron-nucleus Hamiltonian, expand around the Born-Oppenheimer equilibrium geometry, and construct an effective action for the electronic and phononic propagators. From the same effective action, we derive both electronic and phononic self-energies in terms of a nonlocal vertex $\mathcal G^{\textrm{s}}$ including excitonic effects, which generalizes the usual local interaction vertex $g^{\textrm{s}}$. When electrons and phonons are well-defined quasiparticles in the screened-exchange approximation and $\mathcal G^{\textrm{s}}$ is taken in its static limit, both self-energies reduce to Fermi-golden-rule expressions containing the same $\mathcal G^{\textrm{s}}$, thereby ensuring detailed balance. As an application, we compute electronic and phononic linewidths in graphene and illustrate this common-vertex construction. We analyze the competition between the reduced scattering phase space induced by the screened-exchange band structure and the enhancement of the electron-phonon vertex due to excitonic effects, finding that the vertex enhancement can compensate for and overcome the phase-space reduction in both electronic and phononic linewidths.