In and Out-of-Equilibrium Ab Initio Theory of Electrons and Phonons

In this work, we lay down the ab initio many-body quantum theory of electrons and phonons in equilibrium as well as in steady-state or time-varying settings. Our focus is on the harmonic approximation, but the developed tools can readily incorporate anharmonic effects. We begin by showing the necessity of determining the ab initio Hamiltonian in a self-consistent manner to ensure the existence of an equilibrium state. We then identify the correct partitioning into a “noninteracting” and an “interacting” part to carry out diagrammatic expansions in terms of dressed propagators and screened interactions. The final outcome is the finite-temperature nonequilibrium extension of the Hedin equations, showcasing the emergence of the coupling between electrons and coherent phonons through the time-local Ehrenfest diagram. The Hedin equations have limited practical utility for real-time simulations of systems driven out of equilibrium by external fields. To overcome this limitation, we leverage the versatility of the diagrammatic approach to generate a closed system of differential equations for the dressed propagators and nuclear displacements. These are the Kadanoff-Baym equations for electrons and phonons. The formalism naturally merges with the theory of conserving approximations, which guarantee the satisfaction of the continuity equation and the conservation of total energy during time evolution. As an example, we show that the popular Born-Oppenheimer approximation is not conserving whereas its dynamical extension is conserving, provided that the electrons are treated in the Fan-Migdal approximation with a dynamically screened electron-phonon coupling. We also derive the formal solution of the Kadanoff-Baym equations for nonequilibrium steady states, which is useful for studies in photovoltaics and optoelectronics. Interestingly, the expansion of the phononic Green’s function around the quasiphonon energies points to a correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.

Popular Summary

The interaction between electrons and phonons—sound and optical waves propagating in a crystal—plays a vital role in determining all of a crystal’s properties. To miniaturize or enhance any optoelectronic device, it is imperative to comprehend the fundamental laws governing these interactions. Yet, despite the concepts of electrons and phonons being around for more than a century, how to deal with electrons and phonons away from equilibrium and how to refine the approximations underlying state-of-the-art formulas have remained crucial and unresolved issues. Here, we provide a definitive solution to these long-standing problems.

We lay down the quantum theory of electrons and phonons, applicable at any temperature, both in and out of equilibrium. An essential merit of the theory is the possibility of systematically improving the accuracy of a calculation by incrementally incorporating controlled corrections.

Our contribution will guide the development of more accurate equations and will aid the implementation of computer programs to predict and explain the unique properties exhibited by newly discovered and yet-to-come crystals.

Figure 1

$\mathsf{L}$-shaped Konstantinov-Perel’ contour.

Figure 2

Low-order diagrams in the expansion of the interacting electronic Green’s function $G$.

Figure 3

Low-order diagrams in the expansion of the interacting phononic Green’s function $\tilde{D}$.

Figure 4

Resummation of the tadpole diagrams.

Figure 5

Expansion of the phononic (top) and electronic (middle) self-energies and of the polarization (bottom) in $G$-skeleton diagrams and $D$-skeleton diagrams.

Figure 6

Expansion of the phononic (top) and electronic (bottom) self-energy in $G$-skeleton diagrams, $D$-skeleton diagrams, and $W$-skeleton diagrams. To facilitate the discussion in the main text, we label the $\mathrm{\Sigma }$ diagrams.

Figure 7

Expansion of the ${\mathrm{\Phi }}_{\mathrm{xc}}$ functional in $G$-skeleton diagrams and $D$-skeleton diagrams.

Figure 8

Diagrams of the ${\mathrm{\Phi }}_{\mathrm{xc}}$ functional leading to the self-energies Eqs. (108) and (111) through the functional derivatives in Eqs. (127).