  In solid state physics, the electron-phonon interaction (EPI) is central to many phenomena. The theory of the renormalization of electronic properties due to EPIs became well established with the theory of Allen-Heine-Cardona, usually applied to second order in perturbation theory (P2). However, this is only valid in the weak coupling regime, while strong EPIs have been reported in many materials. Although non-perturbative (NP) methods have started to arise in the last years, they are usually not well justified, and it is not clear to what degree they reproduce the exact theory. To address this issue, we present a stochastic approach for the evaluation of the non-perturbative interacting Green's function in the adiabatic limit, and show it is equivalent to the Feynman expansion to all orders in the perturbation. Also, by defining a self-energy, we can reduce the effect of broadening needed in numerical calculations, improving convergence in the supercell size. In addition, we clarify whether it is better to average the Green's function or self-energy. Then we apply the method to a graphene tight-binding model, and obtain several interesting results: (i) The Debye-Waller term, which is normally neglected, does affect the change of the Fermi velocity. (ii) The P2 and NP self-energies differ even at room temperature for some k-points, raising the question of how well P2 works in other materials. (iii) Close to the Dirac point, positive and negative energy peaks merge. (iv) In the strong coupling regime, a peak appears at energy E=0, which is consistent with previous works on disorder and localization in graphene. (v) The spectral function becomes more asymmetric at stronger coupling and higher temperatures. Finally, in the Appendix we show that the method has better convergent properties when the coupling is strong relative to when it is weak, and discuss other technical aspects. 
Non-perturbative Green's function method to determine the electronic   spectral function due to electron-phonon interactions: Application to a   graphene model from weak to strong coupling