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.


I. INTRODUCTION
The concept of phonons as quasi-particles describing independent excitations of the nuclear lattice dates back to almost a century ago [1,2].Nonetheless, the first rigorous theory of electrons and phonons saw the light of day in 1961 [3].In a seminal paper Baym showed how to map the original electronnuclear Hamiltonian onto a low-energy or equivalently electronphonon (ℎ) Hamiltonian, and derived a set of equations for the electronic and phononic Green's functions (GF) and .The Baym equation for the electronic GF was rather implicit though.In the mid-sixties Hedin used the same technique as Baym, the so called source-field method or field-theoretic approach, to generate a more explicit set of equations for the electronic GF at clamped nuclei [4].The contributions by Baym and Hedin have been largely ignored by the electron-phonon community (including ourselves) in favor of semi-empirical Hamiltonians.Only recently the works of Baym, Hedin and a few others [5][6][7] have been rigorously merged by Giustino in a unified many-body GF framework [8], which we keep naming the Hedin-Baym equations (instead of "Hedin equations") after Giustino.Despite these recent notable advances the formal theory of electrons and phonons is still not complete.We stress here the word "formal" as this paper does not address specific aspects or computational strategies related to theℎ interaction, for which we refer the reader to excellent textbooks [9][10][11][12][13][14][15] and modern comprehensive reviews [8], rather it establishes a mathematically rigorous apparatus for the quantum treatment of electrons and nuclei in the so called harmonic approximation.
Three pivotal issues are still waiting to be clarified and solved.
The first issue has to do with the ab initioℎ Hamiltonian, often replaced by semi-empirical Hamiltonians or left unspecified as unnecessary for the implementation of approximate formulas for the phononic dispersions, life-times, etc..The ab initioℎ Hamiltonian is, however, of paramount relevance.
It is necessary for assessing the validity of semi-empirical approaches, for improving approximations based on perturbation theory, for making fair comparisons between different methods and between different approximations within the same method as well as for benchmarking the harmonic approximation against other methods like, e.g., the surface hopping approach [16] or the exact-factorization scheme [17,18].A plausible explanation for the scarce attention given to the ab initioℎ Hamiltonian is that the minimal sensible approximation is nonperturbative.In fact, the ab initioℎ Hamiltonian evaluated at zeroℎ coupling does not even contain phonons.
A clean derivation of theℎ Hamiltonian can be found in the Baym's work [3].However, Baym's original expression as well as equivalent expressions designed for having a good starting point for perturbative expansions [7] necessitate the knowledge of the exact equilibrium electronic density 0 .This means that the ab initioℎ Hamiltonian is unknown unless 0 is calculated by other means, e.g., by solving the original electron-nuclear problem.Even assuming that we could find 0 somehow, we would still have to face a practical problem.All many-body techniques (including those based on GF) can only be implemented in some approximation, for the exchangecorrelation (xc) potential in Density Functional Theory (DFT), for the self-energy in GF theory, for the configuration state functions in the Multi-Configurational Hartree-Fock method, for the intermediate states in the Algebraic Diagrammatic Con-struction scheme, etc.An approximation in any of the available many-body methods inevitably leads to an inconsistency if the ab initioℎ Hamiltonian is evaluated at the exact 0 .The inconsistency lies in the fact that the forces acting on the nuclei would not vanish in thermal equilibrium.As we shall see, the ab initioℎ Hamiltonian must be determined self-consistently for it to be used in practice.Such self-consistent concept is completely general, i.e., it is not limited to GF approaches.Of course if an exact method is used than the self-consistent density 0 coincides with the exact one.
The second issue is the extension of the Hedin-Baym equations at finite temperature and out-of-equilibrium.This is especially relevant in the light of the overwhelming and steadily increasing number of time-resolved spectroscopy experiments.We mention that at zero-temperature a nonequilibrium Green's function (NEGF) formulation has been put forward in terms of the nuclear-density correlation function [6,19,20].We here present the finite-temperature and nonequilibrium extension of the Hedin-Baym equations for the electronic GF and the displacement-displacement correlation function (or phononic GF) [8].We anticipate that the main differences are: (i) in all internal vertices the time arguments must be integrated over the -shaped Konstantinov-Perel's contour [21][22][23][24][25]; (ii) the electronic GF satisfies a Dyson equation with an extra timelocal self-energy, whose inclusion is fundamental to recover the Ehrenfest (mixed quantum-classical) dynamics [26][27][28][29][30][31][32][33][34][35][36].In the context of material science the Ehrenfest self-energy plays a key role in the description of polarons [31,37] and it is expected to be fundamental to detect the phonon-induced coherent modulation of the excitonic resonances [38].
The Hedin-Baym equations have limited practical utility in nonequilibrium problems.Furthermore, the original derivation based on the source-field method allows for solving these equations only iteratively, starting from an approximation to the vertex.The question of whether the iterative procedure converges towards the exact solution is still open.In most applications only one iteration step is performed since the equations resulting from the second interaction are already too complex.All these considerations bring us to the third issue, i.e., how to systematically improve the approximations possibly preserving all fundamental conservation laws.We here present a diagrammatic derivation of the Hedin-Baym equations based on the skeletonic expansion of the electronic and phononic selfenergies in terms of interacting electronic and phononic GF and the screened interaction.We highlight three essential merits of the diagrammatic derivation: (i) the possibility of including relevant scattering mechanisms through a proper selection of Feynman diagrams (to be converted into mathematical expressions through the Feynman rules which we provide); (ii) the possibility of using the Φ-derivable criterion [39] to have a fully conserving dynamics; (iii) the possibility of closing the Kadanoff-Baym equations (KBE) [40] through a skeletonic expansion of the self-energies in terms of only interacting GF.The KBE are integro-differential equations for the electronic and phononic NEGF [25,41], and they are definitely more practical for investigating the real-time evolution of systems driven out of equilibrium.Furthermore, through the so called Generalized Kadanoff-Baym Ansatz for fermions [42] and bosons [43] the KBE can be mapped onto a much simpler system of ordinary differential equations for a large number of self-energy approximations [43][44][45][46][47][48][49].
The paper is organized as follows.In Section II we derive the low-energy Hamiltonian for any system of electrons and nuclei, highlighting its dependence on the equilibrium electronic density and pointing out the necessity of a self-consistent procedure for its determination.In Section III we specialize the discussion to lattice periodic systems, introduce general time-dependent external perturbations and derive theℎ Hamiltonian on the -shaped contour.The equations of motion for the electronic and phononic field operators are derived in Section IV.In Section V we define the many-particle electronic and phononic GF on the contour and construct the Martin-Schwinger hierarchy that these GF satisfy.In Section VI we present the Wick's theorem as the solution of the noninteracting Martin-Schwinger hierarchy and in Section VII we provide the exact formula of the many-body expansion of the interacting GF in terms of the noninteracting ones.The many-body expansion is mapped onto a diagrammatic theory in Section VIII where we also introduce the notion of self-energies and skeleton diagrams.The skeletonic expansion of the self-energies in terms of the interacting GF and screened Coulomb interaction is shown to lead to the Hedin-Baym equations in Section IX.The Hedin-Baym equations are applicable to systems in-and out-of equilibrium as well as at zero and finite temperature.To study the system evolution or the finite-temperature spectral properties the equations of motion for the GF are more convenient than the Hedin-Baym equations.These equations of motion are derived in Section X.In Section XI we discuss the so called Φ-derivable approximations to the self-energies.The GF satisfying the equations of motion with Φ-derivable self-energies guarantee the fulfillment of all fundamental conservation laws.In Section XII we convert the equations of motion into a coupled system of integro-differential equations for the Keldysh components of the GF; these are the KBE.We first discuss the self-consistent solution of the equilibrium problem and then derive the realtime equations of motion to study the system evolution.We also present the formal solution of the KBE in the long-time limit and for steady-state conditions.The expansion of the phononic GF around the quasi-phonon energies reveals the possibility of a correlation-induced splitting of the phonon dispersion in materials with no time-reversal invariance.The presented formalism can be extended to deal with much more general Hamiltonians than theℎ Hamiltonian.In Section XIII we provide a summary of the main results, illustrate possible extensions and discuss their physical relevance.

II. QUANTUM SYSTEMS OF ELECTRONS AND NUCLEI
In this section we lay down a quantum theory of electrons and nuclei which is suited whenever the average nuclear positions remain close to the equilibrium values.Under this "nearequilibrium hypothesis" the nuclei stay away from each other and they can be treated as quantum distinguishable particles.In fact, we can distinguish them by means of techniques like scanning tunneling microscopy or electron diffraction [8].
Let ̂ ( = ) be the field operators that annihilate an electron in position with spin , hence they satisfy the anticommutation relations { ̂ ( ), ̂ † ( ′ )} = ′ ( − ′ ) ≡ ( − ′ ), and ̂ , ̂ be the position and momentum operators of the -th nucleus, = 1, … , satisfying the standard commutation relations [ ̂ , , ̂ , ] − = , with and running over the three components of the vectors.The Hamiltonian describing an unperturbed system of electrons interacting with nuclei of charge and mass reads (we use atomic units throughout the paper) where is the electronic Hamiltonian, is the nuclear Hamiltonian (with ̂ 2 = ̂ ⋅ ̂ ) and is the electron-nucleus interaction.In Eqs.(2)(3)(4) the integral ∫ ≡ ∫ ∑ signifies a spatial integral and a sum over spin, ( , ′ ) = 1∕| − ′ | is the Coulomb interaction and ̂ ( ) ≡ ̂ † ( ) ̂ ( ) is the density operator in for particles of spin .For later purposes we find it convenient to collect all nuclear position and momentum operators into the vectors ̂ = ( ̂ 1 , … , ̂ ) and ̂ = ( ̂ 1 , … , ̂ ).We also find it useful to define the nuclear potential energy appearing in Eq. ( 3), and the electron-nuclear potential appearing in Eq. ( 4).The operators of ̂ act on the directproduct space ⊗ where is the electronic Fock space and is the Hilbert space of the distinguishable nuclei.Expansion around thermal equilibrium.-Considerthe interacting system of electrons and nuclei in thermal equilibrium at a certain temperature.Under the "near-equilibrium hypothesis" we can approximate the full Hamiltonian by its second-order Taylor expansion around the equilibrium values of the nuclear positions, which we name 0 = ( 0 1 , … , 0 ), and around the equilibrium value of the electronic density, which we name 0 ( ).In fact, also the electronic density must stay close to 0 ( ) for otherwise the forces acting on the nuclei would be strong enough to drive the nuclei away from 0 .Notice that 0 and 0 ( ) do in general depend on the temperature.We also observe that the existence of an inertial reference frame for the coordinates 0 is supported by the macroscopic size of the system, i.e., → ∞.For finite systems, e.g., molecules or molecular aggregates, the choice of a suitable reference frame is more subtle, see Refs [50][51][52][53][54][55].
We introduce the displacement (or position fluctuation) operators ̂ and the density fluctuation operator Δ ̂ according to In the following we refer to these operators as the fluctuation operators.Formally, the "near-equilibrium hypothesis" is equivalent to restrict full the space ⊗ to the subspace of states giving a small average of ̂ and Δ ̂ .
The expansion of the nuclear potential energy around the equilibrium nuclear positions yields to second order In the first term we recognize the electrostatic energy of a nuclear geometry 0 .As this term is only responsible for an overall energy shift we do not include it in the following discussion.Similarly, the expansion of the electron-nuclear potential yields to second order where we define , ( ) Inserting Eq. ( 9) into Eq.( 4) we see that the first term gives rise to a purely electronic operator; it is the potential energy operator for electrons in the classical field generated by a nuclear geometry 0 .The second and third terms emerge when relaxing the infinite-mass approximation for the nuclei.The third term of Eq. ( 9) is already quadratic in the displacements and it can therefore be multiplied by the equilibrium density, i.e., ̂ → 0 [3].Going beyond the quadratic (or harmonic) approximation the replacement ̂ → 0 is no longer justified; in this case the third term gives rise to the so called Debye-Waller (DW) interaction [56].
The low-energy Hamiltonian.-Inserting the expansion of − and into Eqs.( 3) and ( 4) the total Hamiltonian becomes The first two terms describe an uncoupled system of noninteracting electrons and interacting nuclei in the electric field generated by a frozen electronic density 0 ( ) where we define the elastic tensor which is real and symmetric under the exchange ( , ) ↔ ( , ).
Already at this stage of the presentation a remark is due.The eigenvalues 2 of the tensor , ; , ≡ , ; , ∕ √ are not physical and can even be negative such that ̂ 0, ℎ does not have a proper ground state.It is therefore generally not possible to define annihilation and creation operators ̂ and ̂ † to rewrite Eq. (13) in the form ̂ 0, ℎ = ∑ ( ̂ † ̂ + 1 2 ).The ab initio low-energy Hamiltonian evaluated at vanishing coupling , [hence ̂ ′ = 0, see Eq. (21)] does not contain physical quanta of vibrations (phonons in solids).These excitations can only emerge from a proper nonperturbative treatment, see Section XII A.
The Hamiltonian in Eq. ( 11) is the low-energy approximation of the full Hamiltonian in Eq. (1).The expansion around the equilibrium nuclear geometry and around the equilibrium density has inevitably made Eq. (11) to depend on these quantities.The scalar potential and the electron-nuclear coupling are determined from the sole knowledge of the equilibrium positions 0 whereas the elastic tensor depends on both 0 and 0 , see Eq. (14).Notice that the dependence of ̂ on 0 is through as well as Δ ̂ , see Eq. (21).In the following we assume that 0 is known and therefore that and are given.Strategies to obtain good approximations to the equilibrium nuclear geometry are indeed available, e.g., the Born-Oppenheimer approximation, see also discussion in Ref. [20].Alternatively, 0 can be taken from X-ray crystallographic measurements.The equilibrium density 0 must instead be determined, and the proper way of doing it is self-consistently.Let us expand on this point.
We write the dependence of ̂ on 0 explicitly: ̂ = ̂ [ 0 ].For any given many-body treatment (whether exact or approximate) a possible self-consistent strategy to obtain 0 is: (i) Make an initial guess 0  1 and (ii) Use the chosen many-body treatment to calculate the equilibrium density 0 2 of ̂ [ 0 1 ], then the equilibrium density 0  3 of ̂ [ 0 2 ] and so on and so forth until convergence.If the initial guess 0  1 does already produce a good approximation to then a partial self-consistent scheme in which is not updated is also conceivable.Self-consistency is however unavoidable to determine 0 in Δ ̂ .In fact, it is only at self-consistency that the equilibrium value Δ = 0, an essential requirement for the r.h.s. of Eq. ( 22) to vanish and hence for the nuclear geometry to remain stationary.In Section XII A we discuss how to implement the self-consistent strategy using NEGF.In particular we show that 0 can be obtained from the self-consistent solution of the Dyson equation for the Matsubara GF.
For any given equilibrium geometry 0 different scenarios are possible.If 0 is too off target the self-consistent scheme may not converge, indicating that the nuclear geometry must be improved.If convergence is achieved then the self-consistent equilibrium state can be either stable or unstable.In the unstable scenario an infinitesimally small perturbation brings the nuclei away from 0 , indicating again that the nuclear geometry must be improved.Let us finally consider the stable scenario.The nuclear geometry can in this case be further optimized by minimizing the total energy (at zero temperature) or the grand-potential (at finite temperature) with respect to 0 .At the minimum the equilibrium geometry is the exact one only if an exact many-body treatment is used.Needless to say that the minimum is defined up to arbitrary overall shifts and rotations of the nuclear coordinates.

III. INTERACTING HAMILTONIAN FOR ELECTRONS AND PHONONS IN-AND OUT-OF-EQUILIBRIUM
Independently of the method chosen to find 0 and of the self-consistent many-body treatment chosen to determine 0 the low-energy Hamiltonian of a system of electrons and nuclei is given by Eq. (11).Let us discuss in detail the case of a crystal and introduce some notations.
In a crystal we can label the position of every nucleus with the vector (of integers) of the unit cell it belongs to and with the position relative to some point of the unit cell, i.e., 0 = , = 0 + 0 .If the unit cell contains nuclei then = 1, … , .By definition the vector 0 for the -th nucleus is the same in all unit cells, and the mass = , = and charge = , = of the -th nucleus in cell is independent of .The invariance of the crystal under discrete translations implies that the elastic tensor depends only on the difference between unit cell vectors, i.e., , ; ′ ′ , ′ = , ; ′ , ′ ( − ′ ).(23) The periodicity of the crystal also implies an important property for the electron-nuclear coupling .According to the definition in Eq. (10b) we have , ( ) = − ( ∕ ) ( , 0 ).The Coulomb interaction depends only on the relative coordinate and therefore ( , 0 ) = ( + 0 ′ , 0 + 0 ′ ) = ( + 0 ′ , 0 + ′ ) for all vectors 0 ′ .This implies that , ( ) = + ′ , ( + 0 ′ ).
Equilibrium Hamiltonian.-Weconsider a finite piece of the crystal with cells along direction = , , and impose the Born-von Karman boundary conditions.The total number of cells is therefore = .Accordingly the displacement and momentum operators can be expanded as where the sum over = ( , , ) runs over all vectors satisfying the property = 2 with integers, and ∈ (− , ] for = , , .In Eqs.(25) the vectors ( ) with components , ( ) form an orthonormal basis for each , i.e., ( ) † ⋅ ′ ( ) = ′ .In three dimensions the set of all vectors ( ) spans a 3 dimensional space for each .We refer to these vectors as the normal modes.As we shall see the most convenient choice of normal modes depends on the approximation made to treat the problem.A typical choice is the eigenbasis of the Hessian of the Born-Oppenheimer energy.At this stage of the presentation the set of normal modes is just a basis to expand the displacement and momentum operators.
The hermiticity of the operators ̂ , and ̂ , imposes the following constraints on the operators ̂ and ̂ and on the normal modes: Inserting the expansions Eqs.(25) into Eqs.( 13) and ( 21) we obtain where < l a t e x i t s h a 1 _ b a s e 6 4 = " 0 f 8 9 Y V J 5 8 5 g j 9 w P n 8 A r R O P v g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l 6 8 j 6 i 8 M J f 3 / V 7 n e / 6 h K O z F R w S E = " > A A A B 9 X i c b V B N S w M x E M 3 W r 1 q / q h 6 9 B I v g q e y K q M e i F y 9 C B f s B 3 W 2 Z T b N t a J J d k q x S l v 4 P L x 4 U 8 e p / 8 e a / M W 3 and For crystals the Hamiltonian of Eq. ( 11) is known as the electron-phonon (ℎ) Hamiltonian.
Nonequilibrium Hamiltonian.-Weare interested in formulating a NEGF approach to deal with systems described by the ℎ Hamiltonian of Eq. ( 11) possibly driven out of equilibrium by external driving fields.Of course the external fields must be such that the nonequilibrium density and displacements are small enough to justify the harmonic approximation.As we shall see the developed formalism can accommodate many different kinds of drivings.Without any loss of generality we take the system in thermal equilibrium for times < 0 and then perturb it by letting The timedependence of the Coulomb interaction andℎ coupling may be due to, e.g., an adiabatic switching protocol or a sudden quench of the interaction, whereas the time dependence of the one-particle Hamiltonian ℎ and elastic tensor may be due to laser fields, phonon drivings, etc.
One remark about the dependence of ̂ ( ) on .If the operator does not depend on we can safely write ̂ ( ) = ̂ in the first line of Eq. (32).However, if we do so in the second line then it is not clear where to place the operator ̂ after the contour ordering.The reason to keep the contour argument even for operators that do not have an explicit time dependence (like the operators ̂ , ̂ and ̂ ) stems from the need of specifying their position along the contour, thus rendering unambiguous the action of  .Once the operators are ordered we can omit the time arguments if there is no time dependence.
To shorten the equations we gather the displacement and momentum operators into a two-dimensional vector of operators having components The commutation relations for the ̂ -operators follow from the commutation relations [ ̂ , , ̂ ′ ′ , ′ ] = ′ ′ ′ after inserting the expansions in Eqs.(25) and using the properties in Eq. (26).We find where Let us express the contour Hamiltonian ̂ ( ), see Eqs. ( 11) and (31), in terms of the ̂ -operators.We first write the final result and then prove its correctness.We have where and In Eqs.(37a) and (37d) we introduce a (two-dimensional vector) shift ( ) with the purpose of simplifying the NEGF treatment, see below.The exact form of the shift is irrelevant for the time being as the terms containing ( ) cancel out in the sum of Eq. ( 36), thereby ̂ ( ) is independent of this quantity.The shifted electronic Hamiltonian ̂ 0, ( ) evaluated at zero shift, i.e., ( ) = 0, is the same as ̂ 0, ( ), see Eqs. (12) and (31a).The first term of the shifted phononic Hamiltonian ̂ 0, ℎ ( ) is the same as ̂ 0, ℎ ( ), see Eqs. (27) and (31b), whereas the second term coincides with the contribution to ̂ ′ coming from 0 in Δ ̂ , see Eqs. (28) and (31d).The -interaction Hamiltonian ̂ − ( ) has not changed, see Eqs. (11) and (31c).Finally, theℎ interaction Hamiltonian ̂ − ℎ ( ) evaluated at zero shift coincides with the contribution to ̂ ′ coming from ̂ in Δ ̂ .We conclude that Eq. ( 36) is identical to the contour Hamiltonian for any two-dimensional vector ( ).

IV. EQUATIONS OF MOTION FOR OPERATORS IN THE CONTOUR HEISENBERG PICTURE
The contour Heisenberg picture is an extremely useful concept to develop the NEGF formalism.We define the contour evolution operators according to where  is the anti-contour-ordering operator.These operators are unitary for = ± and hermitian for = 0 − .For all ∈ we have the property  ( 0 , )  ( , 0 ) = 1.We define an operator in the contour Heisenberg picture as It is straightforward to verify that ( ± ) = ( ).The equation of motion for operators in the contour Heisenberg picture follows directly from the definitions in Eqs. ( 39) As the Hamiltonian is written in terms of ̂ ( ) and ̂ it is clear that the equation of motion for these operators plays a crucial role in the following derivation.Equations of motion.-UsingEq. ( 41) the equation of motion for the ̂ -operators reads To remove the inhomogeneous term with 0 we define the 2 × 2 GF 0, ′ as the solution of and satisfying the periodic Kubo-Martin-Schwinger (KMS) boundary conditions along the contour .We use 0 to define the shift in Eqs. ( 37) Then the equation of motion for the (time-dependent) shifted displacement reads which is a homogeneous equation in the field operators.Notice that the shift is proportional to the identity operator and therefore , ( ) = ( ).Such proportionality also implies that the operators ̂ ( ) and ̂ ( ′ ) [these operators are not in the contour-Heisenberg picture and in particular ̂ ( ′ ) does not depend on ′ ] satisfy the same commutation relations for any and ′ : From the commutator [ ̂ ( ), ̂ ( )] we can easily derive the equation of motion for the electronic field operators where we define the shifted one-particle Hamiltonian see Eq. (37a), and in the last term of the r.h.s.we recognize that the density operator in Eq. (37d) multiples the shifted displacement ̂ defined in Eq. ( 45).

V. GREEN'S FUNCTIONS AND MARTIN-SCHWINGER HIERARCHY
The building blocks of the NEGF formalism are the electronic and phononic GF.Let us introduce a notation which is used throughout the reminder of the paper.We denote the position, spin and contour-time coordinates of an electronic field operator with the collective indices etc.. Thus, for instance, ).We recall that the electronic field operators have no explicit dependence on the contour-time, i.e., ̂ ( , ) = ̂ ( ).Without any risk of ambiguity we use the same notation to denote the momentum, branch, component and contour-time coordinates of a phononic field operator etc.. Thus, for instance, . We also use the superscript star " * " to denote the composite index with reversed momentum, e.g., * = − , , , .Of course starring twice is the same as no starring, i.e., * * = .We then define where ( 1 , 2 ) is the Dirac-delta on the contour [25].Accordingly, the equations of motion for the field operators, i.e., Eqs. ( 46) and (48), are shortened as To distinguish the integration variables of the electronic operators ̂ from those of the phononic operators ̂ we use a bar for the former and a tilde for the latter, thus Setting = = 0 in the r.h.s. of Eqs.(53) we obtain the equations of motion of the field operators governed by the Hamiltonian We shall use this observation in the next section.
Green's functions.-Thephononic and electronic GF are contour-ordered correlators of strings of phononic and electronic field operators.The -particle phononic GF is defined as In this definition can also be a half-integer; in particular ̃ 1∕2 (1) = (1)∕ 1∕2 .The phononic GF ̃ differs from that in Refs.[57,58] as it is defined in terms of the shifted operators ̂ instead of ̂ .We observe that the operators appearing in the second equality are not in the contour Heisenberg picture [compare with Eq. ( 32)].The phononic GF is totally symmetric under an arbitrary permutation of its arguments 1, 2, ..., 2 .Similarly, we define the -particle electronic GF as The electronic GF is totally antisymmetric under an arbitrary permutation of the arguments 1, 2, ..., and 1 ′ , 2 ′ , ..., ′ .Martin-Schwinger hierarchy.-Takinginto account that the commutation relations for the operators ̂ are identical to the commutation relations for the operators ̂ , see Eq. ( 47), the equations of motion for ̃ with = 1∕2, 1, 3∕2, ... read where the variable 1 in the l.h.s. is at place and we define ̃ 0 = 1 and ̃ −1∕2 = 0.The first term in the r.h.s.originates from the equation of motion Eq. (53a).The last term in the r.h.s.originates from the derivative of the Heaviside step-functions implicit in the contour ordering [25].These derivatives generate quantities like ( , )[ ̂ , ( ), ̂ , ( )] = ( , ) ,− , see Eq. (34), which multiplied by lead to ( , * ) ≡ ( , ) ,− .To shorten the equations we have also introduced the symbol " ⊓ " above an index to indicate that the index is missing from the list.In a similar way we can derive the equations of motion for the -particle GF and find for = 1, 2, 3, … where the variable 1 in the l.h.s. is at place and we define 0 = 1.The first two terms in the r.h.s.originate from the equation of motion Eq. (53b).The last term in the r.h.s.originates from the the derivative of the Heaviside stepfunctions implicit in the contour ordering.In the electronic case ( ; ) ≡ ( − ) ( , ).We further notice that the last argument of the GF +1 is 1+ .We use the superscript " + " to indicate that the contour time is infinitesimally later than ̄ 1 .This infinitesimal shift guarantees that the creation operator ̂ † ( 1) in +1 ends up to the left of the annihilation operator ̂ ( 1) when the operators are contour ordered.The equation of motion for with derivative with respect to the primed arguments can be worked out similarly.All equations of motion must be solved with KMS boundary conditions, i.e., ̃ must be periodic on the contour with respect to all times 1 , … , 2 and must be antiperiodic on the contour with respect to all times 1 , … , , ′  1 , … , ′ .If theℎ coupling = 0 then ̃ couples only to ̃ −1 .In the electronic sector things are different.For = 0 the equations of motion reduce to the Martin-Schwinger hierarchy for a system of only electrons, and couples to −1 and +1 through the Coulomb interaction .For to couple only to −1 the Coulomb interaction has to vanish too.
When both -andℎ interactions are present, couples to −1 , +1 as well as to mixed GF consisting of a mix string of ̂ , ̂ † , and ̂ operators, see second line in the equations of motion.Likewise, ̃ couples to ̃ −1 but also to mixed GF, see first term in the r.h.s. of the equations of motion.The equations of motion for the mixed GF can be derived in precisely the same way, see also Ref. [59].We refer to the full set of equations as the Martin-Schwinger hierarchy for electron-phonon systems.In the next sections we lay down a perturbative method to calculate all GF.
The Wick theorem provides the solution of the Martin-Schwinger hierarchy with r.h.s evaluated at = = 0.This is the same as solving the Martin-Schwinger hierarchy for a system of electrons and phonons governed by the Hamiltonian ̂ 0 , see comment above Eq.(54).As ̂ 0 depends on explicitly, setting = = 0 in the r.h.s. of Eqs. ( 57) and ( 58) is not the same as solving the Martin-Schwinger hierarchy with ̂ | = =0 (noninteracting hierarchy).Henceforth we name the GF governed by ̂ 0 as the independent GF and we denote them by 0, and 0, .We then have and the like with time derivatives with respect to the primed arguments.The independent GF satisfy two independent hierarchies.Despite the similarities the phononic and electronic hierarchies present important differences.The sum in the r.h.s.runs over all arguments of 0, in Eq. (59a) whereas it runs only over the unprimed arguments of 0, in Eq. ( 58).Moreover, in the phononic case can also be a half-integer.From Eq. (59a) we see that the integer connects with the integer −1 and the half-integer connects with the half-integer −1.Therefore, we have two separate hierarchies of equations for the phononic GF.
Wick's theorem for phonons.-Theproof of Wick's theorem for 0, goes along the same lines as in Ref. [58].The phononic GF 0, = 0 for all half-integers .This can easily be proven by considering the average of the equation of motion Eq. ( 46) with = 0.By definition this average is proportional to 0,1∕2 and the only solution satisfying the KMS boundary conditions is 0,1∕2 = 0. Consider now = 3∕2: and the like for the variables 2 and 3. We see that 0,3∕2 = 0 is a solution satisfying the KMS boundary conditions.By induction, 0, = 0 for half-integers.Henceforth we only consider integers in the noninteracting case.For = 1 we have the equation of motion for 0,1 : Comparing with Eq. ( 43) we realize that Without any risk of ambiguity we denote 0,1 (1, 2) simply by 0 (1,2) in the remainder of the paper.For 0, with > 1 the solution of Eq. (59a) is given by the so-called hafnian [59][60][61].
The hafnian can be defined recursively starting from any of the arguments in 0, .Choosing for instance the argument we have Using again Eq. ( 63) for 0, −1 and then for 0, −2 and so on and so forth we obtain an expansion of 0, in terms of products of 0 's.A compact way to write this expansion is where the sum runs over all permutations of the indices 1, 2, … , 2 .The recursive form of Eq. ( 63) makes it clear that 0, satisfies Eq. (59a) and the KMS boundary conditions.
Wick's theorem for electrons.-Thesolution of Eq. ( 58) has been discussed at length in Refs.[25,62].We here write the final result for completeness.For = 1 Eq.( 58) yields to be solved with KMS boundary conditions.Like for the phononic case we shorten the notation and write 0,1 (1; 2) = 0 (1; 2).The superscript " " reminds us that this GF depends on through ℎ .For 0, with > 1 the solution of Eq. ( 58) is again defined recursively choosing either an unprimed or a primed argument 0, ( Using again Eq. ( 66) for 0, −1 and then for 0, −2 and so on and so forth we obtain an expansion of 0, in terms of products of 0 's.A compact way to write this expansion is the determinant where the sum runs over all permutations of the indices 1, 2, ..., and (−) is the sign of the permutation.The recursive form of the Wick's theorem highlights the differences between the phononic and the electronic case.For 0, we need to connect unprimed arguments to primed arguments in all possible ways.For 0, there is no such distinction, and we need to connect all arguments in all possible ways.

VII. EXACT GREEN'S FUNCTIONS FROM WICK'S THEOREM
The interacting GF ̃ and can be expanded in powers of the -interaction andℎ coupling .In this section we focus on the one-particle electronic GF ≡ 1 , the oneparticle phononic GF ̃ ≡ ̃ 1 and the half-particle phononic GF ̃ 1∕2 .The final results are Eqs.(75), (77) and (78).In Section IX we show that the perturbative expansion leads to a closed system of equations for these quantities.Higher order Green's functions as well as mixed Green's functions (relevant for linear response theory) can be investigated along the same lines, see Ref. [25].
The starting point is the Hamiltonian written in the form of Eq. ( 36), i.e., ̂ = ̂ 0 + ̂ − + ̂ − ℎ with ̂ 0 defined in Eq. (54).Inside the contour-ordering the Hamiltonians ̂ 0 , ̂ − and ̂ − ℎ can be treated as commuting operators and hence the exponential of their sum can be separated into the product of three exponentials.It is then natural to define the independent averages as We emphasize again that the independent averages are not the same as the noninteracting averages, i.e., the averages with = = 0, since both ̂ 0, and ̂ 0, ℎ depend on , see Eqs.(37a) and (37b).
One-particle electronic Green's function.-TheGF ≡ 1 is defined in Eq. (56).We have where = , and = , in accordance with the notation of Eq. (50).The denominator in this equation is the interacting partition function  = Tr[ − ( ̂ − ̂ ) ]. Expanding the exponentials containing ̂ − and ̂ − ℎ we find To facilitate the identification of the expansion terms we use a tilde for the contour times of theℎ interaction Hamiltonian.Let us write the integrated Hamiltonians in terms of ( ; ) and ( ; ), see Eqs. (52c) and (52d).We have The infinitesimal shift in the contour-times of the electronic creation operators guarantees that these operators end up to the left of the annihilation operators calculated at the same contour-times after the contour reordering.Inserting Eqs.(71) into Eq.( 70) we are left with the evaluation of contour-ordered strings like ⟨ { ̂ … ̂ ̂ † … ̂ † ̂ … ̂ }⟩ 0 with an arbitrary number of operators.We observe that ̂ 0, acts on the Fock space of the electrons and ̂ 0, ℎ acts on the Hilbert space of distinguishable nuclei.As such, the eigenkets of ̂ 0 = ̂ 0, + ̂ 0, ℎ factorize into tensor products of kets in and kets in .Therefore, the partition function for independent electrons and phonons factorizes into electron and phonon contributions.The same type of factorization allows us to simplify the independent average of any string of operators as where the average ⟨…⟩ 0, is performed with ̂ 0, and the average ⟨…⟩ 0, ℎ is performed with ̂ 0, ℎ .To order in and to order in the numerator of the GF contains the independent average of the following string With similar manipulations we can work out the expansion of the partition function.Taking into account that 0, ∕2 is nonvanishing only for even integers the expansion of the interacting GF reads with The zero-th order term in the expansion of Eq. ( 75) ( = = 0) is the GF 0,1 = 0 calculated from Eq. (65) where ⃖⃖ ⃗ −1 0 is defined in Eq. (52b).Using Wick's theorem for 0, and 0, Eqs. ( 75) and ( 76) provide an exact expansion in terms of the one-particle electronic GF 0 and phononic GF 0 .
Half-particle phononic Green's function.-Theinteracting GF ̃ 1∕2 is proportional to the time-dependent average of the field operator ̂ , i.e., ̃ 1∕2 ( ) = 1 1∕2 ( ).Proceeding along the same lines as for the derivation of the expansion Eq. (77) we find where we take into account that 0, vanishes for half-integers .The average vanishes for = 0, in agreement with the equation of motion Eq. (46).

VIII. DIAGRAMMATIC THEORY
The expansions of , ̃ and contain the GF 0 and 0 , the Coulomb interaction and theℎ coupling .Let us assign a graphical object to these quantities.We use an oriented line from 2 to 1 to represent 0 (1; 2), and a wiggly line between 1 and 2 to represent (1; 2) = (2; 1).For the noninteracting phononic GF 0 (1, 2) = 0 (2, 1) we use a spring from 1 to 2. Theℎ coupling (1; 2 * ) is instead represented by a square, half black and half white, where 1 is attached to the black vertex and 2 is attached to the white vertex.To summarize < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 / a c g w We can now represent every term of the expansions with diagrams.The diagrams for are either connected or products of a connected diagram and a vacuum diagram.In a connected diagram for ( ; ) all internal vertices are connected to both and through 0 , 0 , and .Thus a disconnected -diagram is characterized by a subset of internal vertices that are not connected to either or and hence they form a vacuum diagram.
Similarly, the diagrams for ( ) fall into two main classes: those with all internal vertices connected to and those where a subset of internal vertices is disconnected, thus forming a vacuum diagram.The diagrams for ̃ ( , ) can instead be grouped into three different classes: (c1) all internal vertices connected to both and (c2) a subset of internal vertices connected only to and the complementary set connected only to and (c3) diagrams where a subset of internal vertices is not connected to either or , thus forming a vacuum diagram.In all cases the contributions containing vacuum diagrams factorize and cancel with the expansion of the partition function , see Eq. ( 76).Furthermore, many connected diagrams are topologically equivalent and it is therefore enough to consider only the topologically inequivalent diagrams.The number of topologically equivalent diagrams cancel the combinatorial factor 2 !(2 )! in Eqs. ( 75) and ( 77) and 2 !(2 + 1)! in Eq. ( 78).The proof of these statements goes along the same lines as the proof for only electrons and we refer to Refs.[25,59] for more details.The resulting formulas for , ̃ and become and where the labels " " and " . ." indicate that when expanding 0, in determinants and 0, in hafnians only connected and topologically inequivalent diagrams are retained.In particular the expansion Eq. ( 81) for ̃ contains all diagrams in classes (c1) and (c2).
Diagrammatic expansion for .-InFig. 2 we show a few low-order Feynman diagrams for .The Feynman rules to convert the diagrams into a mathematical expression are • Number all vertices and assign an interaction ( ; ) to a < l a t e x i t s h a 1 _ b a s e 6 4 = " k t K 5 Low-order diagrams in the expansion of the interacting electronic Green's function .
wiggly line connecting and , anℎ coupling ( ; * ) to a square with white vertex in and black vertex , a GF 0 ( ; + ) to an oriented line from to , a GF 0 ( , ) to a spring connecting and .
• Integrate over all internal vertices and multiply by + (−) where is the number of electronic loops, is the number of wiggly lines and 2 is the number of squares.
There are diagrams (5-th and 7-th diagrams in the second row of Fig. 2) that are one-0 -line reducible, i.e., they can be disconnected into two pieces by cutting an internal 0 -line.We define the irreducible self-energy Σ as the set of all one-0 -line irreducible diagrams with the external (ingoing and outgoing) 0 -line removed.Then can be written as a geometric series Each product in this formula stand for a space-spin-time convolution.The self-energy Σ = Σ[ 0 , 0 , , ] is an infinite sum of irreducible diagrams with 0 -lines and 0 -lines connected through and .
Among the self-energy diagrams there are some with selfenergy insertions, i.e., diagrams that can be disconnected into two pieces by cutting two 0 -lines.We say that a diagram is -skeletonic if it does not contain self-energy insertions.Then the full set of Σ-diagrams is obtained by dressing the 0 -lines of the skeleton diagrams with all possible self-energy insertions.This amounts to evaluate the skeleton diagrams with the interacting GF instead of 0 [25].Denoting by Σ 1skel the sum of only -skeleton diagrams we can write Diagrammatic expansion for ̃ .-Letus now consider the one-particle phononic GF ̃ .Expanding 0, and 0, in Eq. ( 81) according to Wick's theorem and representing every term of the expansion with a diagram we obtain the diagrammatic expansion of ̃ .The Feynman rules for the ̃ -diagrams are the same as for the diagrams.In Fig. 3 we illustrate a few low-order diagrams.
We can clearly distinguish the diagrams belonging to class (c2).These are the double tadpole diagrams (see 2-nd, 4-th, 5-th and 6-th diagrams of Fig. 3).The single tadpole diagrams constitute the diagrammatic expansion of the half-particle phononic GF ̃ 1∕2 = ∕ 1∕2 .In Fig. 4 we show the diagrammatic expansion of .The full set of diagrams can be easily summed up, see r.h.s. in Fig. 4 where the oriented double line denotes the interacting GF .The Feynman rules for the -diagrams are the same as for the diagrams and diagrams except that the prefactor is + +1 (−) +1 where is the number of electronic loops, is the number of wiggly lines and + 1 is the number of squares, see Eq. ( 82).We thus have a result that could alternatively be found by direct integration of the equation of motion Eq. ( 46) after taking into account that the electronic density (1) = − (1; 1 + ).We can therefore write the expansion of ̃ as in which the products in this formula stand for momentummode-component-time convolutions.The phononic self-energy Π = Π[ 0 , 0 , , ] is the set of all diagrams that, after the removal of the ingoing and outgoing 0 -lines, cannot be cut into two pieces by cutting an internal -line (one-0 -line irreducible diagrams).
The phononic Green's function ̃ ( , ) does not fulfill a Dyson equation but the fully connected phononic GF does.The GF ( , ) can alternatively be written in terms of the fluctuation operators Δ ̂ ( ) 1 w e p / 7 3 S e q N I t l y 8 w S G g g 8 l i x i B J t c u n J d d 1 i t e a 6 3 3. Low-order diagrams in the expansion of the interacting phononic Green's function ̃ .
< l a t e x i t s h a 1 _ b a s e 6 4 = " .. < l a t e x i t s h a 1 _ b a s e 6 4 = " F J P 8 T 2 f A P f 5 / y F J 2 U i 9 X 1 r l r u c E = " e V 1 0 r o q e 9 f l S r 1 S q j a y O P J w B u d w C R 7 c Q B U e o A Z N Y I D w D K / w 5 j w 6 L 8 6 7 8 7 F s z T n Z z C n 8 g f P 5 A 5 U l j N w = < / l a t e x i t > = < l a t e x i t s h a 1 _ b a s e 6 4 = " 8 r J b w z v w u U l 2 P y S L v 0 a y r U 6 N h Δ ̂ ( ).We have The GF ( , ) fulfills a Dyson equation since Like for the electronic Green's function we can express the phononic self-energy in terms of -skeleton diagrams.We remove all diagrams with Σ-insertions inside the Π-diagrams and then replace 0 by the full GF , thus obtaining the functional Π 1skel : The topological idea of the skeletonic expansion in is completely general and it can be extended to the phononic GF and the screened Coulomb interaction .This is done in the next section.

IX. FROM THE SKELETONIC EXPANSION IN AND TO THE HEDIN-BAYM EQUATIONS
Skeletonic expansion in .-Ifwe remove all phononic selfenergy insertions inside the Π 1skel -diagrams and then replace 0 with we can write Here Π 2skel contains all doubly skeletonic self-energy diagrams, i.e., all those diagrams that do not contain either Σ-insertions or Π-insertions.Examples of Π 2skel -diagrams are shown in Fig. 5 (top) where the double spring represents .A similar       procedure can be applied to the electronic self-energy except that for Σ we must exclude the time-local diagrams.In fact, a Π-insertion is here equivalent to a Σ-insertion and it would therefore lead to a double counting.Therefore where Σ Eh is the self-energy in the second diagram of Fig. 2 with 0 → .Using the Feynman rules where in the last equality we use Eq. ( 85).The self-energy Σ Eh is known as the Ehrenfest self-energy.The Ehrenfest approximation consists in including the phononic feedback on the electrons through Σ Eh , the electronic feedback on the phonons through the density, see Eq. ( 85), and in setting Π = 0.In this approximation the nuclei are therefore treated as classical particles since they are described in terms of displacements and momenta only, i.e., the components of [26][27][28][29][30][31][32][33][34][35][36].The importance of the Ehrenfest diagram in the description of polarons has been pointed out in Refs.[31,37].We expect that the Ehrenfest diagram is also crucial to capture the phononinduced coherent modulation of the excitonic resonances [38].
Examples of Σ 2skel diagrams are shown in Fig. 5 (middle).Skeletonic expansion in .-Like in the case of only electrons [4] we can further reduce the number of diagrams by removing all those diagrams containing a polarization insertion, which we here define as a piece that can be cut away by cutting two -lines and at the same time it does not break into two disjoint pieces by cutting one -line or one -line, an example is the 5-th diagram in the Σ-expansion of Fig. 5 (middle).The polarization diagrams are therefore one--line irreducible and one--line irreducible.To the best of our knowledge the diagrammatic definition of the polarization in systems of electrons and phonons is given here for the first time.In Fig. 5 (bottom) we show a few low-order diagrams for = 2skel [ , , , ] which are both -skeletonic and -skeletonic.The polarization contains both electronic and phononic contributions.For later purposes we also define the one--line irreducible, one--line irreducible and two--lines reducible kernel ̃ from the polarization according to where the dark-grey bubble represents and the square grid represents ̃ ( 1, 2; 3, 4) = ̃ ( 2, 1; 4, 3).This is the same definition used in the case of systems of only electrons.In fact, the so called vertex function Γ = − ̃ relates to through the well known formula = − Γ (the factor of " " in the second diagram of Eq. ( 94) comes from the Feynman rules for the polarization diagrams).Following the same strategy as in Ref. [25] one can show that the kernel ̃ satisfies the Bethe-Salpeter equation < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D 4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q p x o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 z m F P 3 A + f w B t c 5 B L < / l a t e x i t > <l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D 4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q p x o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 ˜< l a t e x i t s h a 1 _ b a s e 6 4 = " w P

R X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t >
+ < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D 4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q p x o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 ˜< l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D 4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q p x o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 where the kernel ̃ , represented by the square with only vertical lines, is one--line irreducible, one--line irreducible and two--lines irreducible.
From the polarization diagrams we can construct the dynamically screened interaction in the usual manner  We say that a diagram is -skeletonic if it does not containinsertions.Then the desired expression for Π and Σ is obtained by discarding all those diagrams which are not -skeletonic, and then replacing with .Phononic self-energy.-Forthe phononic self-energy we get where Π 3skel is the sum of all the triply skeletonic self-energy diagrams, i.e., all those diagrams that do not contain either Σinsertions, Π-insertions or -insertions.In Fig. 6 (top) we show a few low-order diagrams of the triply skeletonic expansion for Π 3skel .They are one--line irreducible, one--line irreducible and two--lines irreducible, i.e., they cannot break into disjoint pieces by cutting two -lines for otherwise they would contain a polarization insertion.We then have either one-line reducible diagrams or one--line irreducible diagrams.We can write Π in a compact form using the polarization .We define the dressed (or screened)ℎ coupling as < l a t e x i t s h a 1 _ b a s e 6 4 = " w P

+
< l a t e x i t s h a 1 _ b a s e 6 4 = " K h H 3 q 6 S x 9 I 3 n Y v t h D q 6 + V S n j m In terms of the dressedℎ coupling the phononic self-energy can be represented as < l a t e x i t s h a 1 _ b a s e 6 4 = " e D d 5 f 5 W Z L u c E X D u y q s 6 2 6 C T e e 4 w = " > in agreement with the field-theoretic approach [8].We see from this expression that oneℎ coupling is bare whereas the other is dressed.As pointed out in Refs.[8,63] this structure has to be properly taken into account for the calculation of phonons, although a symmetrized form where bothℎ couplings are dressed has been shown to give satisfactory results [64], which has been attributed to error cancellations [65].Electronic self-energy.-Forthe electronic self-energy Σ the only diagram for which we should not proceed with the < l a t e x i t s h a 1 _ b a s e 6 4 = " c p

R X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 r S H v 8 c 4 8 z x s z A m T 0 O j R 4 + s m B l 8 = " >
r e 2 d 8 q 7 l b 3 9 g 8 O j 6 v F J R 8 e p I r R N Y h 6 r X o g 1 5 U z S t m G G 0 1 6 i K B Y h p 9 1 w e p / 7 3 S e q N I t l y 8 w S G g g 8 l i x i B J t c u n J d d 1 i t e a 6 3 A < l a t e x i t s h a 1 _ b a s e 6 4 = " q V F n U N e s W W w C h l d q q M 7 x S / L 7 W 8 x w B W j R k w s I V R x e y u m I 6 I I N T a d k g 3 B X X 5 5 l b Q v a u 5 l r X 5 f r z T u 8 j i K c A K n U A U X r q A B t 9 C E F l A Y w j O 8 w h s S 6 A W 9 o 4 9 F a w H l M 8 f w B + j z B 5 M q j W A = < / l a t e x i t > (d) < l a t e x i t s h a 1 _ b a s e 6 4 = " h I x w U 5 5 Q w < l a t e x i t s h a 1 _ b a s e 6 4 = " D 9 A 7 H a i + S p P 7 A x 3 s g / q F X 1 / + o 5 w = " > replacement is the Hartree diagram [first diagram in Fig. 5 (middle)] since here every polarization insertion is equivalent to a self-energy insertion, hence → would lead to a double counting.Therefore where Σ H is the Hartree diagram.The self-energy Σ 3skel is also called the exchange-correlation (xc) self-energy.Henceforth we shall equivalently write Σ 3skel or Σ xc .The diagrammatic expansion of the xc self-energy plays a crucial role in diagrammatic theory since the Bethe-Salpeter kernel ̃ (1, 2; 3, 4) = − Σ xc (1; 3)∕ (4; 2), see Eq. ( 95).This statement can be proven along the same lines as in Ref. [25].We illustrate in Fig. 6 (bottom) a few low-order diagrams of the triply skeletonic expansion of Σ 3skel .Diagrams like the last two in the figure must be included since by cutting the two -lines we get a piece that is neither a polarization insertion nor a Π-insertion.
Like the phononic self-energy also the electronic self-energy Σ can be written in a compact form using the polarization , or more precisely the kernel ̃ .Let us consider the admissible "effective" interactions that can sprout from, e.g., the left vertex of Σ xc .Keeping an eye on Fig. 6  ] con-tain a Π-insertion whereas [ ] contains a polarization insertion.We conclude that the total "effective" interaction sprouting from the left vertex is To make these graphical considerations rigorous we define the phonon-mediated -interaction from the dressedℎ coupling according to < l a t e x i t s h a 1 _ b a s e 6 4 = " + 0 r 4

2
< l a t e x i t s h a 1 _ b a s e 6 4 = " T P 3 S 5 3 T 2 X 3 y s T j w m n q r t I or in formulas and the total screened interaction ̃ ≡ + ℎ according to < l a t e x i t s h a 1 _ b a s e 6 4 = " F J P 8 T 2 or in formulas The total electronic self-energy can then be written as + < l a t e x i t s h a 1 _ b a s e 6 4 = " m p I q g t w + k 8 9 C H F K s j Y 9 F 0 W G c e G I = " > A A A B 7 n i c b V B N S w M x E J 3 U r 1 q / q h 6 9 B I v g q e y K q B e h o A e P F e 0 H t E v J p t k 2 N M k u S V Y o S 3 + E F w + K e P X 3 e P P f m L Z 7 0 N Y H A 4 / 3 Z p i Z F y a C G + t 5 3 6 i w s r q o k m l D r E i q 5 E P z F l 5 d J 8 6 z q X 1 T P 7 8 8 r t d s 8 j i I c w T G c g g + X U I M 7 q E M D K I z g G V 7 h D S X o B b 2 j j 3 l r A e U z h / A H 6 P M H 8 p a P V Q = = < / l a t e x i t >

⌃ =
< l a t e x i t s h a 1 _ b a s e 6 4 = " c p Hedin-Baym equations.-Wesummarize in Table I the fundamental equations that relate the various many-body quantities, i.e., , , Σ, Π, ̃ , , , and ̃ .We here align with Ref. [8] and call to the full set of equations in Table I the Hedin-Baym equations for electrons and phonons.The Hedin-Baym equations provide a closed system of equations for any diagrammatic approximation to the xc self-energy through the irreducible kernel ̃ = Σ xc ∕ .They are equations on the contour and can therefore be used to study systems in equilibrium at any temperature as well as systems driven away from equilibrium by external fields.For systems in equilibrium at zero temperature the adiabatic assumption in conjunction with the assumption of a nondegenerate ground state allows for deforming the contour into a single branch going from −∞ to +∞, i.e., the real axis [25].In this case the contour Green's functions become the more familiar time-ordered Green's functions and the Hedin-Baym equations reduce to those presented in Ref. [8].We emphasize that no such shortcut is possible at finite temperature.One way to avoid the use of the -shaped contour for equilibrium systems at finite temperature is the analytic continuation (from Matsubara to retarded to time-ordered) which may however be rather cumbersome in the presence of singularities or branch cuts, although notable progresses have been recently made [66][67][68].
Like for the Hedin equations for only electrons the Hedin-Baym equations can be iterated to obtain an expansion of Σ and Π in terms of , , and .If we start with ̃ = 0, hence the electronic self-energy Σ is approximated by where Σ GW = is the well known self-energy with RPA screened interaction = + 0 , and is the so called Fan-Migdal self-energy [69,70] with dressed electron-phonon coupling The phononic self-energy for ̃ = 0 is simply We remark that the response function 0 appearing in and Π cannot, in general, be built with a quasi-particle GF since both Σ GW and Σ FM are non-local in time.Although the timelocal Coulomb-hole plus Screened Exchange (COH-SEX) version of Σ GW often provides a good compromise in the trade-off between accuracy and computational cost we are not aware of a similar time-local version of Σ FM .In Section X we show that the approximation in Eqs. ( 108) and ( 111) is conserving, i.e., the resulting GF satisfy all fundamental conservation laws.
Inserting an approximation for the self-energies Σ and Π into the Dyson equations = 0 + 0 Σ and = 0 + 0 Π we obtain a closed system of equations for and for any 0 and 0 .The GF 0 depends only on the parameters of the Hamiltonian, see also Appendix A, whereas the GF 0 depends also on the nuclear displacements through ℎ .The Hedin-Baym equations must therefore be coupled to Eq. ( 85).
The Hedin-Baym equations can alternatively be derived using the field-theoretic approach [6,8,20,63,71].We emphasize that the field-theoretic approach prescinds from any diagrammatic notion, i.e., it does not tell us how to expand the various many-body quantities diagrammatically.
For simplicity we specialize the discussion to the relevant case of external perturbations that do not break the lattice periodicity of the crystal, albeit the developed formalism is far more general.Let us write the coordinates = ( , , ) of any point in space as the sum of the vector 0 of the unit cell the point belongs to and a displacement spanning the unit cell centered at the origin, i.e., = 0 + .We introduce a generic one-electron Bloch basis, e.g., the Kohn-Sham basis, where the vector takes the same values as the vector defined below Eqs.(25).The wavefunctions Ψ ( ) can be thought of as the one-electron eigenfunctions in some potential, e.g., the Kohn-Sham potential, with the periodicity of our lattice; hence the index can be thought of as a band index.The matrix element of any two-point correlator with the same periodicity, calculated by sandwiching with Ψ * < l a t e x i t s h a 1 _ b a s e 6 4 = " p I C a E T M m J 9 S y W J m P a z + c 0 5 P r f K E I e x s i U N n q u / J z I S a T 2 N A t s Z E T P W y 9 5 M / M / r p y a 8 8 T M u k < l a t e x i t s h a 1 _ b a s e 6 4 = " E Y + c J 1 r g A P 0 n X B 7 j G < l a t e x i t s h a 1 _ b a s e 6 4 = " C a E T M m J 9 S y W J m P a z + c 0 5 P r f K E I e x s i U N n q u / J z I S a T 2 N A t s Z E T P W y 9 5 M / M / r p y a 8 8 T M u k C a E T M m J 9 S y W J m P a z + c 0 5 P r f K E I e x s i U N n q u / J z I S a T 2 N A t s Z E T P W y 9 5 M / M / r p y a 8 8 T M u k < l a t e x i t s h a 1 _ b a s e 6 4 = " E Y + c J 1 r g A P 0 n X B 7 j G < l a t e x i t s h a 1 _ b a s e 6 4 = " , where is the number of equivalent lines yielding the same self-energy diagram by their respective removal.The lowest order diagrams of the expansion are shown in Fig. 7.The additional minus sign is due to the fact that the removal of a -line from a vacuum diagram changes the number of electronic loops by one.By construction the Φ xc functional has the property that where the subscript " " refers to the symmetrized derivative [ ∕ (1, 2) + ∕ (2, 1)]∕2.The Hartree self-energy is obtained from the functional derivative of the Hartree functional < l a t e x i t s h a 1 _ b a s e 6 4 = " 9 r 3 7 N K Y Z q K 6 U p y L t k U i r 4 J V z V Y Y = " > A A A B 9 X i c b V B N S w M x E J 3 1 s 9 a v q k c v w S J 4 s e x K U S 9 C w U u P F e w H d N e S T b N t a J J d k q x S l v 4 P L x 4 U 8 e p / 8 e a / M W 3 3 o K 0 P B h 7 v z T A z L 0 w 4 0 8 Therefore the full self-energy Σ is the functional derivative of the Φ-functional defined as In most cases we can only deal with approximate functionals.These are obtained by selecting an appropriate subset of Φ-diagrams.We say that the self-energies are Φ-derivable whenever there exists an approximate functional Φ such that Σ xc and Π can be written as in Eqs.(127).The Φ-functional is invariant under gauge transformations and contour-time deformations, i.e., → ( ) with ( 0 ) = 0 and ( 0 − ) = 0 − , implying the fulfillment of the continuity equation and energy conservation for the GF that satisfy the equations of motion Eqs. ( 116) and (126) with Φ-derivable self-energies [39].We C a E T M m J 9 S y W J m P a z + c 0 5 P r f K E I e x s i U N n q u / J z I S a T 2 N A t s Z E T P W y 9 5 M / M / r p y a 8 8 T M u k C a E T M m J 9 S y W J m P a z + c 0 5 P r f K E I e x s i U N n q u / J z I S a T 2 N A t s Z E T P W y 9 5 M / M / r p y a 8 8 T M u k C a E T M m J 9 S y W J m P a z + c 0 5 P r f K E I e x s i U N n q u / J z I S a T 2 N A t s Z E T P W y 9 5 M / M / r p y a 8 8 T M u k mention here that the use of Φ-derivable self-energies evaluated at different input GF still guarantee the satisfaction of all conservation laws provided that they are convoluted with the same input GF [43].In other words, self-consistency is not required for having a conserving theory.We conclude by observing that the approximation to the self-energies discussed at the end of Section IX (derived by setting ̃ = 0 in the Hedin-Baym equations) is Φ-derivable.The diagrams for Φ xc are those of the GW approximation plus a second infinite sum of ring diagrams in which one -line is replaced by ( ), see Fig. 8.

XII. KADANOFF-BAYM EQUATIONS
Placing the arguments on different branches of the contour and using the Langreth rules [25,74] we can convert the equations of motion Eqs. ( 116) and (126) into a coupled system of equations for the Keldysh components of and .These are the Kadanoff-Baym equations (KBE) for systems of electrons and phonons to be solved with KMS boundary conditions.As for the case of only electrons the equations for the Matsubara components decouple.The Matsubara GF M and M allow for calculating the initial thermal average of any onebody operator for electrons and of any quadratic operator in the displacements and momenta for the nuclei.The KBE for the GF with times on the horizontal branches (hence the left/right and lesser/greater components) allow for monitoring the system evolution as well as for calculating electronic and phononic spectral functions of the system in any stationary state.As already pointed out below Eqs.(31) we can study how the system responds to different kind of external perturbations, e.g., interaction quenches, laser fields, phonon drivings, etc..

A. Self-consistent Matsubara equations
The preliminary step to solve the equations of motion for the GF consists in solving the Matsubara problem.The Matsubara self-energies do indeed depend only on the Matsubara GF [25] and therefore the equations for the Matsubara components are closed.The Matsubara component of any correlator ( 1 , 2 ) with arguments on is defined as M ( 1 , 2 ) ≡ ( 1 = 0 − expanding the Matsubara GF and self-energies according where the Matsubara frequencies = 2 ∕ for periodic functions like and Π and = (2 +1) ∕ for antiperiodic functions like and Σ c .Taking into account that along the vertical track = = 0 since Δ = 0, see Eq. ( 121), the equations of motion Eqs. ( 116) and (126) yield where ( ) = ( ) 0 0 1 , see Eq. (38a).For any approximation to Σ c [ , , , ] and Π[ , , , ] these equations can be solved self-consistently.We recall that = [ 0 ] depends on the equilibrium density see Eq. ( 14).Thus, the Matsubara equations are coupled even if we set Σ c = Π = 0.It is only in the partial self-consistent scheme discussed at the end of Section II that the Matsubara equations with Σ c = Π = 0 decouple (the elastic tensor is not updated in this case).Phononic self-energy in the clamped+static approximation.-Inall physical situations of relevance setting Σ c = Π = 0 is a very poor approximation.For the electronic self-energy the approximation is a "gold standard" for obtaining accurate or at least reasonable results [75][76][77][78][79][80].What about the phononic self-energy?Let us explore the physics of Eq. (123) when is calculated by summing all diagrams withoutℎ coupling, i.e., ≃ clamp .This approximation for corresponds to the response function of a system of only electrons interacting through the Coulomb repulsion and feeling the potential generated by clamped nuclei in 0 [8].Using the definition in Eq. (124) the Matsubara phononic self-energy in the clamped approximation reads ) where we use that along the vertical track (̄ 1 , 0 − ) = (̄ 1 ) is independent of .Let us further approximate the r.h.s. with its value at = 0 [81].This is the static approximation and it is similar in spirit to the statically screened approximation of .The response function M ( ̄ 1 , ̄ 2 ; 0) calculated at a zero frequency is identical to the retarded or advanced response function R∕A ( ̄ 1 , ̄ 2 ; 0) also calculated at zero frequency [25].Therefore the phononic self-energy in the clamped+static approximation can be written as with Inserting this approximation into Eq.(132b) we see that the effect of Π is to renormalize the (1, 1) block of .In other words the interacting in the clamped+static approximation has the same form as the noninteracting 0 with a renormalized elastic tensor 1 2 ( ) → renorm .(137) Connection with the Born-Oppenheimer approximation.-Ifwe use the Born-Oppenheimer (BO) approximation to evaluate the equilibrium nuclear positions and electronic density then 0 ≃ 0,BO and 0 ≃ 0,BO .Correspondingly we have an approximation to theℎ coupling ≃ BO and to the elastic tensor ≃ BO .In the BO approximation the renormalized elastic tensor is exactly the Hessian  of the BO energy calculated in 0,BO , i.e., renorm 1 2 ( ) =  1 2 ( ), see Ref. [8] or Appendix B. The Hessian has positive eigenvalues 2 since 0,BO is the global minimum of the BO energy.The frequencies ≡ √ 2 ≥ 0 are called the phonon frequencies and they provide an excellent starting point already for a clamped response function clamp evaluated at the RPA level.Nonetheless, the importance of going beyond the static approximation (especially for metallic systems) has been reported in the literature [82][83][84].These considerations make it clear that in order to extract physical phonons from the ab initioℎ Hamiltonian, it is necessary to treat theℎ coupling in a nonperturbative manner.
In the basis of the normal modes of the Hessian we have  1 2 ( ) = .Thus the (1, 1) block of the interacting phononic GF in the clamped+static approximation for Π is simply, see Eq. (A13), As the phonon frequencies are all positive we can use them to construct the phononic annihilation operators and creation operators ̂ † with commutation relations [ ̂ , ̂ † ′ ′ ] = , ′ ′ .These results has led several authors to partition the low-energy Hamiltonian in Eq. ( 11) in a slightly different way, see for instance Refs.[5,7,8,63,85].The main difference consists in using the Hessian to define the phononic Hamiltonian In such alternative partitioning the remainder (a quadratic form in the nuclear displacements) must be treated somehow.As we have shown the NEGF formalism and related diagrammatic expansions are most easily formulated with the partitioning of Eq. ( 36).In fact, there is no particular convenience in rewriting the full Hamiltonian in terms of the operators ̂ and ̂ † since ̂ 0, ℎ is not diagonal.
There is instead a convenience in using the Hessian eigenbasis to define ̂ and ̂ since the interacting phononic GF in the clamped+static approximation is diagonal, see Eq. ( 138).In this approximation ̂ ( ̂ † ) annihilates (creates) a quantum of vibration (the phonon) characterized by a well defined energy (the frequency ) and hence an infinitely long life-time.Conserving approximations.-Theclamped+static approximation is not a conserving approximation.Going beyond it the concept of phonons as infinitely long-lived lattice excitations is no longer justified.Phonons become quasi-phonons or dressed phonons by acquiring a finite life-time [8].Still, the clamped+static approximation remains an excellent starting point, often providing quantitative interpretations of Raman spectra.Accordingly, the minimal phononic self-energy which is at the same time conserving and physically sensible is Π = with the RPA = 0 + 0 , or equivalently Π = 0 with = + = + 0 and = + 0 .We emphasize that is not the RPA response function at clamped nuclei since the GF in 0 = − are evaluated with the Fan-Migdal self-energy.As pointed out in Section XI this phononic self-energy is Φ-derivable, the xc functional being the sum of all diagrams in the second row of Fig. 8.For the theory to be conserving the electronic selfenergy must be consistently derived from the same functional Φ.Therefore any calculation with Π = should be done with an electronic Fan-Migdal self-energy Σ FM = .We can add to Σ FM the self-energy and still be conserving since the diagrams in the first row of Fig. 8 do not contribute to Π.

B. Time-dependent evolution and steady-state solutions
With the Matsubara GF at our disposal we can proceed with the calculation of all other Keldysh components by timepropagation.The right component of any correlator ( 1 , 2 ) with arguments on is defined as ⌉ ( , ) ≡ ( ± , 0 − ).The equations for the right components of and follow from Eqs. (116) and (126) when setting 1 = + or 1 = − and 2 = 0 − .Using the Langreth rules [25] we find Henceforth we use the short-hand notation " ⋅ " for time convolutions between 0 and ∞ and " ⋆ " for convolutions between 0 and [24,25].Similarly, the equations for the left component, defined for any correlator as ⌈ ( , ) ≡ ( 0 − , ± ), follow when setting 1 = 0 − and 2 = − or + in the counterparts of the equations of motion Eqs. ( 116) and ( 126) with derivative with respect to 2 .We find where the left arrow over the derivative indicates that the derivative acts on the quantity to its left.At fixed these equations are first order integro-differential equations in which must be solved with initial conditions ⌉ ( 0 , ) = M (0, ), ⌈ ( , 0 ) = M ( , 0), ⌉ ( 0 , ) = M (0, ), ⌈ ( , 0 ) = M ( , 0). (144b) The dependence on time in ℎ HF ( , ) and ( , ) may be due to some external laser field and/or phonon driving.The retarded/advanced as well as the left/right components of the self-energies depend not only on the left and right GF but also on the lesser and greater GF.For any correlator the lesser component is defined as < ( , ′ ) ≡ ( − , ′ + ) whereas the greater component is defined as > ( , ′ ) ≡ ( + , ′ − ).The retarded and advanced components are not independent quantities since R∕A ( , ′ ) = ( − ′ ) ( ) ± (± ∓ ′ )[ > ( , ′ ) − < ( , ′ )], where is the weight of a possible singular part of the correlator ( 1 , 2 ) [for, e.g., and we have = = 0 while for the electronic self-energy Σ we have that Σ is the sum of the HF and Ehrenfest diagrams, see Eq. ( 113)].To close the set of equations we need the equations of motion for ≶ and ≶ .These are obtained by setting 1 = 1± and 2 = 2∓ in Eqs. ( 116) and ( 126) and in their counterparts with derivative with respect to 2 .We find which must be solved with initial conditions < ( 0 , 0 ) = M (0, 0 + ), > ( 0 , 0 ) = M (0 + , 0), (146a) < ( 0 , 0 ) = M (0, 0 + ), > ( 0 , 0 ) = M (0 + , 0). (146b) Finally, we need the equation of motion for the displacement and momentum at = 0 since ( ) appears explicitly in the equations for the electronic GF.These are given by Eqs.(120) when setting = ± and read From these equations we see that the equilibrium density 0 appearing in the full Hamiltonian ̂ must be the same as the self-consistent density in Eq. (133) for otherwise the r.h.s. of the equation of motion for does not vanish at = 0.The set of equations Eqs.(142), ( 143) and (145) are the KBE for systems of electrons and phonons.The KBE, together with the initial conditions for the Keldysh components of and , completely determine the electronic and phononic GF with one and two real times once a choice for the self-energies is made.These equations have been so far solved only in relatively simple model systems [57,59,86,87].Under certain approximations (Generalized Kadanoff-Baym Ansatz [42,43], diagonal density matrices and Markov approximation) the KBE can be shown to reduce to the well known Boltzmann equations.
In the clamped+static approximation for Π the KBE for simplify considerably since Π ≶ = Π ⌉ = Π ⌈ = 0 and Π 1 2 ,R∕A , see Eq. (136).We recall that in this approximation the resulting GF are not conserving since Π clamp+stat is not the functional derivative of any Φ functional.In particular the total energy of the unperturbed system is not constant in time.
Steady-state solution.-TheKBE for the lesser/greater GF can be formally solved.This is done in Refs.[24,25] for where it is also shown that lim , ′ →∞ ≶ ( , ′ ) = R ⋅ Σ ≶ ⋅ A ( , ′ ) provided that Σ c, vanishes when the separation between its time arguments go to infinity.Following the same mathematical steps one can show a similar relation for the phononic GF lim which is valid provided that Π vanishes when the separation between its time arguments go to infinity.Further assuming that in the long-time limit the system attains a steady-state we can Fourier transform with respect to the time-difference and find where and ( ) = lim →∞ ( , ).In most physical situations ( , ) = ( ) is independent of time.By construction, see Eq. ( 88) and Appendix C, R ( ) = A ( ) † and hence Π R ( ) = Π A ( ) † .We can then uniquely write the retarded/advanced phononic self-energy as where Λ ℎ, and Γ ℎ, are self-adjoint matrices.Accordingly, the phononic spectral function of the system at the steady state reads The phononic self-energy has only one nonvanishing block, which is the block (1, 1).If Γ 11 ℎ, ( ) ≠ 0 we can discard the positive infinitesimal in Eq. ( 152) and find for the block (1, 1) of the spectral function We observe that the retarded/advanced phononic GF differs from its more conventional form in that it is not multiplied by the BO frequency , see e.g., Refs.[8,63].This is simply due to a different definition of the correlator where √ ̂ is used in place of ̂ .We prefer to stick to our original definition as it does not contain any notion of the BO approximation.The phononic GF in Eq. ( 154) is written in an arbitrary basis of normal modes; it reduces to the BO diagonal form in the basis of the BO normal modes if Π 11,R∕A ( ) → Π clam+stat .
Among the possible steady states we have thermal equilibrium.In this case we can obtain all Keldysh components from the spectral function.Indeed the fluctuation-dissipation theorem implies that < ( ) = − ( ) ℎ, ( ), (155a) where ( ) = 1∕( − 1) is the Bose function and ̄ ( ) = ( ).Taking into account that ̄ ( ) − ( ) = 1 we also have Quasi-phonons and life-times.-FromEq. ( 154) we also see that any frequency-dependent phononic self-energy gives rise to a finite life-time (Λ ℎ and Γ ℎ are related by a Hilbert transformation) in the same way as any approximation beyond HF for the electronic self-energy gives rise to a finite life-time for the electrons.We can estimate the frequency renormalization and the phononic life-time assuming that the correction to the clamped+static Born-Oppenheimer approximation is small.Working in the basis of the normal modes of the BO Hessian we approximate [8] where The quasi-phonon weight ± ℎ, gives the probability that a lattice vibration with momentum along the normal mode excites a phonon with quantum numbers .The remaining spectral weight (1 − ± ℎ, ) is absorbed by collective phononic excitations arising from correlation effects.
If the function Γ dyn ( )∕ is small for ≃ Ω ± and slowly varying in then we can approximate it with its value in Ω ± for ≃ Ω ± and rewrite Eq. (158) as is the phononic life-time.This formula agrees with the expression in Refs.[8,10,56] for ± ℎ, = 1.We observe that, in general, Ω + ≠ −Ω − , and therefore the energy lost from the emission of a phonon is not the same as the energy gained by the absorption of the same phonon.Energy is however conserved since the same analysis carried for [

XIII. SUMMARY AND OUTLOOK
In this work we have developed the many-body theory of the electron-phonon problem.The first step was to clarify the dependence of the ab initioℎ Hamiltonian on the equilibrium electronic density.Only through a self-consistent procedure it is possible to determine theℎ Hamiltonian and hence to make fair comparisons between different many-body methods as well as between different approximations within the same method.The analysis has also highlighted an issue affecting those semi-empirical approaches that include theℎ effects through the addition of a term of the form ∑ ̂ † ̂ + ∑ , [ , ̂ ̂ † + h.c.], with ̂ purely electronic operators, to the electronic Hamiltonian at clamped nuclei.Beside being devoid of a theoretical foundation, the phonon frequencies may get significantly renormalized by the phononic self-energy, possibly becoming complex.
We identify in Eq. ( 36) the most suitable partitioning of the ab initioℎ Hamiltonian and in Eq. ( 55) the most suitable phononic GF for the NEGF formulation.After mapping the cumbersome many-body expansion onto a diagrammatic theory we have delved into the diagrammatic content of the self-energies until reaching a closed form in terms of skeleton diagrams.We have in this way provided the diagrammatic derivation of the Hedin-Baym equations for in-and out-ofequilibrium systems at any temperature.The main differences with respect to case of equilibrium at zero-temperature [8] is the domain of the time-integration for the internal vertices and the appearance of the Ehrenfest self-energy.The merits of the diagrammatic approach are (i) it provides a systematic way for improving many-body approximations through a proper selection of physically insightful Feynman diagrams (ii) it naturally combines with the Φ-derivable theory to generate fully conserving GF and (iii) it is versatile in the resummation of different diagrammatic series.The last point is especially relevant for the skeletonic expansion in terms of only and , which is crucial for closing the KBE.In fact, the Hedin-Baym equations are of scarce practical use to study the time-dependent evolution of theℎ system.On the contrary the KBE, being them integro-differential equations, are better suited for real-time simulations.Considering the fast pace of progress in time-resolved experiments we foresee a growing interest in this direction.The interest is also fueled by the possibility of solving the KBE using a time-linear scheme for a large number of self-energy approximations [43][44][45][46][47][48][49], thus making NEGF competitive with the fastest quantum methods currently available.
The NEGF formalism presented in this work is applicable to Hamiltonians that are far more general than theℎ Hamiltonian.One straightforward extension is to make the one-electron Hamiltonian nonlocal in space and nondiagonal in spin, i.e., 0, ( ) = ∫ ′ ̂ † ( )ℎ( , ′ , ) ̂ † ( ′ ), and to consider a spin-dependent -interaction ( , ′ ).This allows for including the coupling with vector potentials as well as relativistic corrections like the spin-orbit interaction, and hence to deal with magnetic systems and noncollinear spins [88].It also makes possible to use pseudo-potentials, thus eliminating the core degrees of freedom.Another extension consists in considering anℎ interaction Hamiltonian like ̂ − ℎ = ∑ , ∫ ′ − ( , ′ ) ̂ † ( ) ̂ ( ′ ) ̂ .In fact, the only property required by our derivation is − ( , ′ ) = * ( ′ , ), which is clearly satisfied by theℎ coupling, see Eq. (30).The spatial nonlocality of allows for coupling bosonic particles like photons to nonlocal electronic operators like the current, thereby providing a quantum treatment of the light-matter interaction.We can also study exotic couplings between electrons and the bosonic momentum since we do not need to assume that − = 0 for = 2.A further extension concerns 0, ℎ as the only properties required by our derivation are ( ) = † ( ) = * (− ).Thus the formalism can deal with Hamiltonians containing terms proportional to ̂ † ̂ and ̂ † ̂ .If the bosonic particles are photons these terms arise in the effective Hamiltonian for squeezed light [89].
We finally observe that all these extensions widen the class of external drivings to those that break the crystal periodicity.
Coming back to theℎ Hamiltonian, the theory presented in this work indicates the route for the treatment of anharmonic effects [90,91].To third order in the fluctuation operators we must add to the Hamiltonian in Eq. ( 36) the Debye-Waller interaction, which is proportional to ̂ 2 Δ ̂ , and the next-toquadratic term in the expansion of − , which is proportional to ̂ 3 .These two extra pieces in the Hamiltonian can be treated in precisely the same manner as we have treated ̂ − and ̂ − ℎ in Section VII.The diagrammatic theory will be enriched by two more couplings, the Debye-Waller one connecting two -lines to two -lines, and the one stemming from the cubic displacement connecting three -lines.The systematic analysis of the resulting diagrammatic expansion is a topic for future research.Another topic for future research is the generalization of the NEGF formalism for superconductors.Starting as always from the Hamiltonian Eq. ( 36) we here outline the steps for this achievement (i) add to ̂ an infinitesimally small term breaking the (1) gauge symmetry, e.g., ∫ ′ Δ( , ′ ) ̂ † ( ) ̂ † ( ′ ) + ℎ. .; (ii) derive the noninteracting Martin-Schwinger hierarchy for the many-particle GF containing annihilation operators and creation operators with ≠ ; (iii) establish the Wick's theorem as the solution of the generalized noninteracting Martin-Schwinger hierarchy and (iv) expand the GF with + = 1 (this is the so called Nambu GF) as outlined in Section VII.
We conclude with our perspective on the combination of the NEGF formalism with DFT, which we intentionally left aside in this work as it was not necessary in developing the theory.Zerotemperature [92,93] or finite-temperature [94] DFT as well as Density Functional Perturbation Theory (DFTP) [95][96][97][98][99] provide an invaluable capacity to obtain the ab initio parameters of the Hamiltonian [100][101][102][103][104][105][106].Through DFT and DFPT one can minimize the total energy with respect to the nuclear positions and find an approximation for 0 , and hence for the electronic potential and coupling through Eqs.(10a) and (10b).In the partial self-consistent scheme for the determination of the ℎ Hamiltonian, see end of Section II, the equilibrium DFT density can also be used to calculate the elastic tensor .However, the DFT density cannot be used in the term containing Δ ̂ , see Eq. ( 21) and following discussion, for otherwise the nuclei would start drifting away from 0 .DFPT can also be used to approximate the dressedℎ coupling, necessary for the calculation of the phononic self-energy and phonon-inducedinteraction, see Table I.In fact, = ( + ) = ( + ) .However, DFPT provides in the clamped+static approximation which we have already observed to be nonconserving.One may think to restore the conserving properties by resorting to Time-Dependent (TD) DFT [107][108][109][110], which gives us access to the full frequency dependence of the clamped response function.This is unfortunately not so.The problem lies in the fact that from the TDDFT expression = 0 + 0 ( + xc ) , with xc the xc kernel, it is not obvious to trace the Φ-functional fulfilling = − 1 2 Φ xc ∕ [25].One possibility would be to construct the kernel from the conserving linearized Sham-Schlüter equation [111] as discussed in Ref. [112].However, the linearized Sham-Schlüter equation is itself an approximate equation, hence the resulting theory will never be exact.We finally remark that if one is only interested in spectral properties then the use of non-conserving approximations is much less critical.In this case the fundamental requirement is to use positive-semidefinite (PSD) self-energies [113,114].PSD approximations can be evaluated with the dressed DFTPℎ coupling by simply paying attention to the double-counting problem.
We hope that our contribution to theℎ problem will stimulate further research both at a fundamental and computational level and that it will be of help in the development and in the implementation of accurate approximation schemes to face the novel challenges posed by new materials and time-resolved experiments.
m a e v F e r c + 5 q 0 5 K 5 s 5 h D + w P n 8 A / E u P x A = = < / l a t e x i t >G s 0 (1; 2) = < l a t e x i t s h a 1 _ b a s e 6 4 = " D S s / o t L G M 3 t G c b w 7 Y + b 2 o w P 1 u y 4 = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y

2 <
X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H 8 P j L 8 = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " D S s / o t L G M 3 t G c b w 7 Y + b 2 o w P 1 u y 4 = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 r S H v 8 c 4 8 z x s z A m T 0
h h P o r b l w o 4 t Y P c e f f O G 2 z 0 N Y D F w 7 n 3 M u 9 9 4 Q J o 0 o 7 z r d V 2 t j c 2 t 4 p 7 1 b 2 9 g 8 O j + z j k 5 6 K U 4 l J F 8 c s l o X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l k Q S z v O 4 + g Z t U 9 3 c L 4 k E 0 = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M e P G Y Q F 6 Q L G F 2 0 k n G z M 4 u M 7 N C W P I F X j w o 4 t V P 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e Q 2 N r e 2 d / K 7 h b 3 9 g 8 O j 4v F J S 0 e J Y t h k k Y h U J 6 A a B Z f Y N N w I 7 M Q K a R g I b A e T + 7 n f f k K l e S Q b Z h q j H 9 K R 5 E P O q L F S / a p f L L l l d w G y T r y M l C B D r V / 8 6 g 0 i l o Q o D R N U 6 6 7 n x s Z P q T K c C Z w V e o n G m L I J H W H X U k l D 1 H 6 6 O H R G L q w y I M N I 2 Z K G L N T f E y k N t Z 6 G g e 0 M q R n r V W 8 u / u d 1 E z O 8 8 1 M u 4 8 S g Z M t F w 0 Q Q E 5 H 5 1 2 T A F T I j p p Z Q p r i 9 l b A x V Z Q Z m 0 3 B h u C t v r x O W t d l 7 6 Z c q V d K 1 U Y W R x 7 O 4 B w u w Y N b q M I D 1 K A J D B C e 4 R X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l k Q S z v O 4 + g Z t U 9 3 c L 4 k E 0 = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M e P G Y Q F 6 Q L G F 2 0 k n G z M 4 u M 7 N C W P I F X j w o 4 t V P 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e Q 2 N r e 2 d / K 7 h b 3 9 g 8 O j 4 v F J S 0 e J Y t h k k Y h U J 6 A a B Z f Y N N w I 7 M Q K a R g I b A e T + 7 n f f k K l e S Q b Z h q j H 9 K R 5 E P O q L F S / a p f L L l l d w G y T r y M l C B D r V / 8 6 g 0 i l o Q o D R N U 6 6 7 n x s Z P q T K c C Z w V e o n G m L I J H W H X U k l D 1 H 6 6 O H R G L q w y I M N I 2 Z K G L N T f E y k N t Z 6 G g e 0 M q R n r V W 8 u / u d 1 E z O 8 8 1 M u 4 8 S g Z M t F w 0 Q Q E 5 H 5 1 2 T A F T I j p p Z Q p r i 9 l b A x V Z Q Z m 0 3 B h u C t v r x O W t d l 7 6 Z c q V d K 1 U Y W Rx 7 O 4 B w u w Y N b q M I D 1 K A J D B C e 4 R X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l k Q S z v O 4 + g Z t U 9 3 c L 4 k E 0 = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M e P G Y Q F 6 Q L G F 2 0 k n G z M 4 u M 7 N C W P I F X j w o 4 t V P 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e Q 2 N r e 2 d / K 7 h b 3 9 g 8 O j 4 v F J S 0 e J Y t h k k Y h U J 6 A a B Z f Y N N w I 7 M Q K a R g I b A e T + 7 n f f k K l e S Q b Z h q j H 9 K R 5 E P O q L F S / a p f L L l l d w G y T r y M l C B D r V / 8 6 g 0 i l o Q o D R N U 6 6 7 n x s Z P q T K c C Z w V e o n G m L I J H W H X U k l D 1 H 6 6 O H R G L q w y I M N I 2 Z K G L N T f E y k N t Z 6 G g e 0 M q R n r V W 8 u / u d 1 E z O 8 8 1 M u 4 8 S g Z M t F w 0 Q Q E 5 H 5 1 2 T A F T I j p p Z Q p r i 9 l b A x V Z Q Z m 0 3 B h u C t v r x O W t d l 7 6 Z c q V d K 1 U Y W R x 7 O 4 B w u w Y N b q M I D 1 K A J D B C e 4 R X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 r S H v 8 c 4 8 z x s z A m T 0 O j R 4 + s m B l 8 = " > A A A B 6 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B Z B E E I i R T 0 W v H i s 0 C 9 o Q 9 l s N + 3 S 3 U 3 Y 3 Q g l 9 C 9 4 8 a C I V / + Q N / + N m z Y H b X 0 w 8 H h v h p l 5 Y c K Z N p 7 3 7 Z Q 2 N r e 2 d 8 q 7 l b 3 9 g 8 O j 6 v F J R 8 e p I r R N Y h 6 r X o g 1 5 U z S t m G G 0 1 6 i K B Y h p 9 1 w e p / 7 3 S e q N I t l y 8 w S G g g 8 l i x i B J t c u n J d d 1 i t e a 6 3 A r L 6 6 R z 7 f o 3 b v 2 x X m u 0 i j j K c A b n c A k + 3 E I D H q A J b S A w g W d 4 h T d H O C / O u / O x b C 0 5 x c w p / I H z + Q O 6 0 Y 1 y < / l a t e x i t > +... < l a t e x i t s h a 1 _ b a s e 6 4 = " z b F b k u W e 4 W 9 k 3 + 5 H k 4 B T O 4 A I 8 u I I q 1 K A O D W C g 4 B l e 4 c 1 5 c l 6 c d + d j 0 b r m Z D M n 8 A f O 5 w / 7 A 5 I 1 < / l a t e x i t > P 2skel = < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l

FIG. 5 .
FIG.5.Expansion of the phononic (top) and electronic (middle) selfenergies and of the polarization (bottom) in -skeleton diagrams and -skeleton diagrams.
1 0 r o q e 9 f l S r 1 S q j a y O P J w B u d w C R 7 c Q B U e o A Z N Y I D w D K / w 5 j w 6 L 8 6 7 8 7 F s z T n Z z C n 8 g f P 5 A 5 U l j N w = < / l a t e x i t > = < l a t e x i t s h a 1 _ b a s e 6 4 = " c y t 8 8 9  a C k 8 8 c w h 8 4 n z 8 5 e 4 0 t < / l a t e x i t > W =.(96) X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > J s P B U b g r f 4 8 j J p n 9 W 9 i / r 5 / X m t c V P E U Y Y j O I Z T 8 O A S G n A H T W g B g R E 8 w y u 8 O c J 5 c d 6 d j 3 l r y S l m D u E P n M 8 f w w S O E Q = = < / l a t e x i t > X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 d 4 FIG.6.Expansion of the phononic (top) and electronic (bottom) self-energy in -skeleton diagrams, -skeleton diagrams and -skeleton diagrams.To facilitate the discussion in the main text we have labelled the Σ-diagrams.
1 0 r o q e 9 f l S r 1 S q j a y O P J w B u d w C R 7 c Q B U e o A Z N Y I D w D K / w 5 j w 6 L 8 6 7 8 7 F s z T n Z z C n 8 g f P 5 A 5 U l j N w = < / l a t e x i t > = < l a t e x i t s h a 1 _ b a s e 6 4 = " j / d o D w a O b C 0 7 y K S U 2 0 3 1 2 9 T 3 h / 0 X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " w P X R F x s G Y 5 n z U K X U C d a 5 S A v r q 0 g = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U C 9 C Q A 8 e E z A P S J Y w O + l N x s z O L j O z Q g j 5 A i 8 e F P H q J 3 n z b 5 w k e 9 D E g o a i q p

s h a 1 _ b a s e 6 4 =
" c p v L 1 y l k Q S z v O 4 + g Z t U 9 3 c L 4 k E 0 = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M e P G Y Q F 6 Q L G F 2 0 k n G z M 4 u M 7 N C W P I F X j w o 4 t V P 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e Q 2 N r e 2 d / K 7 h b 3 9 g 8 O j 4 v F J S 0 e J Y t h k k Y h U J 6 A a B Z f Y N N w I 7 M Q K a R g I b A e T + 7 n f f k K l e S Q b Z h q j H 9 K R 5 E P O q L F S / a p f L L l l d w G y T r y M l C B D r V / 8 6 g 0i l o Q o D R N U 6 6 7 n x s Z P q T K c C Z w V e o n G m L I J H W H X U k l D 1 H 6 6 O H R G L q w y I M N I 2 Z K G L N T f E y k N t Z 6 G g e 0 M q R n r V W 8 u / u d 1 E z O 8 8 1 M u 4 8 S g Z M t F w 0 Q Q E 5 H 5 1 2 T A F T I j p p Z Q p r i 9 l b A x V Z Q Z m 0 3 B h u C t v r x O W t d l 7 6 Z c q V d K 1 U Y W R x 7 O 4 B w u w Y N b q M I D 1 K A J D B C e 4 R X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " w P X R F x s G Y 5 n z U K X U C d a 5 S A v r q 0 g = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U C 9 C Q A 8 e E z A P S J Y w O + l N x s z O L j O z Q g j 5 A i 8 e F P H q J 3 n z b 5 w k e 9 D E g o a i q p v u r i A R X B v X / X Z y a + s b m 1 v 5 7 c L O 7 t 7 + Q f H w q K n j V D F s s F j E q h 1 Q j Y J L b B h u B L Y T h T Q K B L a C 0 e 3 M b z 2 h 0 j y W D 2 a c o B / R g e Q h Z 9 R Y q X 7 T K 5 b c s j s H W S V e R k q Q o d Y r f n X 7 M U s j l I Y J q n X H c x P j T 6 g y n A m c F r q p x o S y E R 1 g x 1 J J I 9 T + Z H 7 o l J x Z p U / C W N m S h s z V 3 x M T G m k 9 j g L b G V E z 1 M v e T P z P 6 6 Q m v P Y n X C a p Q c k W i 8 J U E B O T 2 d e k z x U y I 8 a W U K a 4 v Z W w I V W U G Z t N w Y b g L b + 8 S p o X Z e + y X K l X S t W 7 L I 4 8 n M A p n I M H V 1 C F e 6 h B A x g g P M M r v D m P z o v z 7 n w s W n N O N n M M f + B 8 / g C Q V Y z M < / l a t e x i t > = < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l k Q S z v O 4 + g Z t U 9 3 c L 4 k E 0 = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M e P G Y Q F 6 Q L G F 2 0 k n G z M 4 u M 7 N C W P I F X j w o 4 t V P 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e Q 2 N r e 2 d / K 7 h b 3 9 g 8 O j 4 v F J S 0 e J Y t h k k Y h U J 6 A a B Z f Y N N w I 7 M Q K a R g I b A e T + 7 n f f k K l e S Q b Z h q j H 9 K R 5 E P O q L F S / a p f L L l l d w G y T r y M l C B D r V / 8 6 g 0 i l o Q o D R N U 6 6 7 n x s Z P q T K c C Z w V e o n G m L I J H W H X U k l D 1 H 6 6 O H R G L q w y I M N I 2 Z K G L N T f E y k N t Z 6 G g e 0 M q R n r V W 8 u / u d 1 E z O 8 8 1 M u 4 8 S g Z M t F w 0 Q Q E 5 H 5 1 2 T A F T I j p p Z Q p r i 9 l b A x V Z Q Z m 0 3 B h u C t v r x O W t d l 7 6 Z c q V d K 1 U Y W R x 7 O 4 B w u w Y N b q M I D 1 K A J D B C e 4 RX e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D 4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q px o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 v o Z l 0 l q U L L l o m E q i I n J / H k S c o X M i K k l l C l u b y V s T B V l x k Z U s i F 4 q y + v k / Z V 1 b u u 1 h 5 q l X o z j 6 M I Z 3 A O l + D B D d T h H h r Q A g Y C n u E V 3 px H 5 8 V 5 d z 6 W r Q U n n z m F P 3 A + f w B t c 5 B L < / l a t e x i t > <l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l k Q S z v O 4 + g Z t U 9 3 c L 4 k E 0 = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M e P G Y Q F 6 Q L G F 2 0 k n G z M 4 u M 7 N C W P I F X j w o 4 t V P 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e Q 2 N r e 2 d / K 7 h b 3 9 g 8 O j 4 v F J S 0 e J Y t h kk Y h U J 6 A a B Z f Y N N w I 7 M Q K a R g I b A e T + 7 n f f k K l e S Q b Z h q j H 9 K R 5 E P O q L F S / a p f L L l l d w G y T r y M l C B D r V / 8 6 g 0 i l o Q o D R N U 6 6 7 n x s Z P q T K c C Z w V e o n G m L I J H W H X U k l D 1 H 6 6 O H R G L q w y I M N I 2 Z K G L N T f E y k N t Z 6 G g e 0 M q R n r V W 8 u / u d 1 E z O 8 8 1 M u 4 8 S g Z M t F w 0 Q Q E 5 H 5 1 2 T A F T I j p p Z Q p r i 9 l b A x V Z Q Z m 0 3 B h u C t v r x O W t d l 7 6 Z c q V d K 1 U Y W Rx 7 O 4 B w u w Y N b q M I D 1 K A J D B C e 4 R X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + Π = < l a t e x i t s h a 1 _ b a s e 6 4 = " w P X R F x s G Y 5 n z U K X U C d a 5 S A v r q 0 g = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U C 9 C Q A 8 e E z A P S J Y w O + l N x s z O L j O z Q g j 5 A i 8 e F P H q J 3 n z b 5 w k e 9 D E g o a i q p T 2 d e k z x U y I 8 a W U K a 4 v Z W w I V W U G Z t N w Y b g L b + 8 S p o X Z e + y X K l X S t W 7 L I 4 8 n M A p n I M H V 1 C F e 6 h B A x g g P M M r v D m P z o v z 7 n w s W n N O N n M M f + B 8 / g C Q V Y z M < / l a t e x i t > = ̃ ≡ + < l a t e x i t s h a 1 _ b a s e 6 4 = " F J P 8 T 2 f A P f 5 / y F J 2 U i 9 X 1 r l r u c E = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U C 9 C w I v H B P K C Z A m z k 9 5 k z O z s M j M r h J A v 8 O J B E a 9 + k j f / x k m y B 0 0 s a C i q u u n u C h L B tX H d b y e 3 s b m 1 v Z P f L e z t H x w e F Y 9 P W j p O F c M 1 0 r o q e 9 f l S r 1 S q j a y O P J w B u d w C R 7 c Q B U e o A Z N Y I D w D K / w 5 j w 6 L 8 6 7 8 7 F s z T n Z z C n 8 g f P 5 A 5 U l j N w = < / l a t e x i t > = < l a t e x i t s h a 1 _ b a s e 6 4 = " v 2 W q 4 4 n 2 L S l W t P 1 4F 6 u f L b u Z v o s = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M 6 M F j A u Y B y R J m J 7 3 J m N n Z Z W Z W C C F f 4 M W D I l 7 9 J G / + j Z N k D 5 p Y 0 F B U d d P d F S S C a + O 6 3 0 5 u b X 1 j c y u / X d j Z 3 d s / K B 4 e N X W c K o Y N F o t Y t Q O q U X C J D c O N w H a i k E a B w F Y w u p 3 5 r S d U m s f y w Y w T 9 C M 6 k D z k j B o r 1 S 9 6 x Z J b d u c g q 8 T L S A k y 1 H r F r 2 4 / Z m m E 0 j B B t e 5 4 b m L 8 C V W G M 4 H T Q j f V m F A 2 o g P s W C p p h N q f z A + d k j O r 9 E k Y K 1 v S k L n6 e 2 J C I 6 3 H U W A 7 I 2 q G e t m b i f 9 5 n d S E N / 6 E y y Q 1 K N l i U Z g K Y m I y + 5 r 0 u U J m x N g S y h S 3 t x I 2 p I o y Y 7 M p 2 B C 8 5 Z d X S f O y 7 e n Y 9 F a 8 7 J Z o 7 h D 5 z P H 3 U N j L o = < / l a t e x i t > + = ( + ) < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " F J P 8 T 2 fA P f 5 / y F J 2 U i 9 X 1 r l r u c E = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U C 9 C w I v H B P K C Z A m z k 9 5 k z O z s M j M r h J A v 8 O J B E a 9 + k j f / x k m y B 0 0 s a C i q u u n u C h L B tX H d b y e 3 s b m 1 v Z P f L e z t H x w e F Y 9 P W j p O F c M 1 0 r o q e 9 f l S r 1 S q j a y O P J w B u d w C R 7 c Q B U e o A Z N Y I D w D K / w 5 j w 6 L 8 6 7 8 7 F s z T n Z z C n 8 g f P 5 A 5 U l j N w = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " w PX R F x s G Y 5 n z U K X U C d a 5 S A v r q 0 g = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U C 9 C Q A 8 e E z A P S J Y w O + l N x s z O L j O z Q g j 5 Ai 8 e F P H q J 3 n z b 5 w k e 9 D E g o a i q p x H 5 8 V 5 d z 6 W r Q U n n z m F P 3 A + f w B t c 5 B L < / l a t e x i t > ˜ = ( + )< l a t e x i t s h a 1 _ b a s e 6 4 = " w P X R F x s G Y 5 n z U K X U C d a 5 S A v r q 0 g = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U C 9 C Q A 8 e E z A P S J Y w O + l N x s z O L j O z Q g j 5 A i 8 e F P H q J 3 n z b 5 w k e 9 D E g o a i q p v u r i A R X B v X / X Z y a + s b m 1 v 5 7 c L O 7 t 7 + Q f H w q K n j V D F s s F j E q h 1 Q j Y J L b B h u B L Y T h T Q K B L a C 0 e 3 M b z 2 h 0 j y W D 2 a c o B / R g e Q h Z 9 R Y q X 7 T K 5 b c s j s H W S V e R k q Q o d Y r f n X 7 M U s j l I Y J q n X H c x P j T 6 g y n A m c F r q p x o S y E R 1 g x 1 J J I 9T + Z H 7 o l J x Z p U / C W N m S h s z V 3 x M T G m k 9 j g L b G V E z 1 M v e T P z P 6 6 Q m v P Y n X C a p Q c k W i 8 J U E B O T 2 d e k z x U y I 8 a W U K a 4 v Z W w I V W U G Z t Nw Y b g L b + 8 S p o X Z e + y X K l X S t W 7 L I 4 8 n M A p n I M H V 1 C F e 6 h B A x g g P M M r v D m P z o v z 7 n w s W n N O N n M M f + B 8 / g C Q V Y z M < / l a t e x i t > = < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l k Q S z v O 4 + g Z t U 9 3 c L 4 k E 0 = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M e P G Y Q F 6 Q L G F 2 0 k n G z M 4 u M 7 N C W P I F X j w o 4 t V P 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e Q 2 N r e 2 d / K 7 h b 3 9 g 8 O j 4 v F J S 0 e J Y t h k k Y h U J 6 A a B Z f Y N N w I 7 M Q K a R g I b A e T + 7 n f f k K l e S Q b Z h q j H 9 K R 5 E P O q L F S / a p f L L l l d w G y T r y M l C B D r V / 8 6 g 0 i l o Q o D R N U 6 6 7 n x s Z P q T K c C Z w V e o n G m L I J H W H X U k l D 1 H 6 6 O H R G L q w y I M N I 2 Z K G L N T f E y k N t Z 6 G g e 0 M q R n r V W 8 u / u d 1 E z O 8 8 1 M u 4 8 S g Z M t F w 0 Q Q E 5 H 5 1 2 T A F T I j p p Z Q p r i 9 l b A x V Z Q Z m 0 3 B h u C t v r x O W t d l 7 6 Z c q V d K 1 U Y W R x 7 O 4 B w u w Y N b q M I D 1 K A J D B C e 4 R X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t >+ ̃ = − Σ xc ( − ̃ ) < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D 4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q p x o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 v o Z l 0 l q U L L l o m E q i I n J / H k S c o X M i K k l l C l u b y V s T B V l x k Z U s i F 4 q y + v k / Z V 1 b u u 1 h 5 q l X o z j 6 M I Z 3 A O l + D B D d T h H h r Q A g Y C n u E V 3 p x H 5 8 V 5 d z 6 W r Q U n n z m F P 3 A + f w B t c 5 B L < / l a t e x i t > <l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q p x o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 v o Z l 0 l q U L L l o m E q i I n J / H k S c o X M i K k l l C l u b y V s T B V l x k Z U s i F 4 q y + v k / Z V 1 b u u 1 h 5 q l X o z j 6 M I Z 3 A O l + D B D d T h H h r Q A g Y C n u E V 3 p x H 5 8 V 5 d z 6 W r Q U n n z m F P 3 A + f w B t c 5 B L < / l a t e x i t > ˜< l a t e x i t s h a 1 _ b a s e 6 4 = " w P X R F x s G Y 5 n z U K X U C d a 5 S A v r q 0 g = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o P g K e x K U C 9 C Q A 8 e E z A P S J Y w O + l N x s z O L j O z Q g j 5 A i 8 e F P H q J 3 n z b 5 w k e 9 D E g o a i q p v u r i A R X B v X / X Z y a + s b m 1 v 5 7 c L O 7 t 7 + Q f H w q K n j V D F s s F j E q h 1 Q j Y J L b B h u B L Y T h T Q K B L a C 0 e 3 M b z 2 h 0 j y W D 2 a c o B / R g e Q h Z 9 R Y q X 7 T K 5 b c s j s H W S V e R k q Q o d Y r f n X 7 M U s j l I Y J q n X H c x P j T 6 g y n A m c F r q p x o S y E R 1 g x 1 J J I 9 T + Z H 7 o l J x Z p U / C W N m S h s z V 3 x M T G m k 9 j g L b G V E z 1 M v e T P z P 6 6 Q m v P Y n X C a p Q c k W i 8 J U E B O T 2 d e k z x U y I 8 a W U K a 4 v Z W w I V W U G Z t N w Y b g L b + 8 S p o X Z e + y X K l X S t W 7 L I 4 8 n M A p n I M H V 1 C F e 6 h B A x g g P M M r v D m P z o v z 7 n w s W n N O N n M M f + B 8 / g C Q V Y z M < / l a t e x i t > = < l a t e x i t s h a 1 _ b a s e 6 4 = " c p v L 1 y l k Q S z v O 4 + g Z t U 9 3 c L 4 k E 0 = " > A A A B 6 H i c b V D L S g N B E O y N r x h f U Y 9 e B o M g C G F X g n o M e P G Y Q F 6 Q L G F 2 0 k n G z M 4 u M 7 N C W P I F X j w o 4 t V P 8 u b f O E n 2 o I k F D U V V N 9 1 d Q S y 4 N q 7 7 7 e Q 2 N r e 2 d / K 7 h b 3 9 g 8 O j 4 v F J S 0 e J Y t h k k Y h U J 6 A a B Z f Y N N w I 7 M Q K a R g I b A e T + 7 n f f k K l e S Q b Z h q j H 9 K R 5 E P O q L F S / a p f L L l l d w G y T r y M l C B D r V / 8 6 g 0 i l o Q o D R N U 6 6 7 n x s Z P q TK c C Z w V e o n G m L I J H W H X U k l D 1 H 6 6 O H R G L q w y I M N I 2 Z K G L N T f E y k N t Z 6 G g e 0 M q R n r V W 8 u / u d 1 E z O 8 8 1 M u 4 8 S g Z M t F w 0 Q Q E 5 H 5 1 2 T A F T I j p p Z Q p r i 9 l b A x V Z Q Z m 0 3 B h u C t v r x O W t d l 7 6 Z c q V d K 1 U Y W R x 7 O 4 B w u w Y N b q M I D 1 K A J D B C e 4 RX e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A H n d j M o = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D 4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q p x o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 v o Z l 0 l q U L L l o m E q i I n J / H k S c o X M i K k l l C l u b y V s T B V l x k Z U s i F 4 q y + v k / Z V 1 b u u 1 h 5 q l X o z j 6 M I Z 3 A O l + D B D d T h H h r Q A g Y C n u E V 3 p x H 5 8 V 5 d z 6 W r Q U n n z m F P 3 A + f w B t c 5 B L < / l a t e x i t > ˜< l a t e x i t s h a 1 _ b a s e 6 4 = " 7 Y W Y 1 l Z L T z X r U j 1 u Z A a D s h h X 3 I w = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e C F 4 8 V + g V t K J v N t F 2 6 2 c T d j V B C / 4 Q X D 4 p 4 9 e 9 4 8 9 + 4 b X P Q 1 g c D j / d m m J k X J I J r 4 7 r f T m F j c 2 t 7 p 7 h b 2 t s / O D w q H 5 + 0 d Z w q h i 0 W i 1 h 1 A 6 p R c I k t w 4 3 A b q K Q R o H A T j C 5 m / u d J 1 S a x 7 J p p g n 6 E R 1 J P u S M G i t 1 + 4 a L E L P Z o F x x q + 4 C Z J 1 4 O a l A j s a g / N U P Y 5 Z G K A 0 T V O u e 5 y b G z 6 g y n A m c l f q p x o S y C R 1 h z 1 J J I 9 R + t r h 3 R i 6 s E p J h r G x J Q x b q 7 4 m M R l p P o 8 B 2 R t S M 9 a o 3 F / / z e q k Z 3 v o Z l 0 l q U L L l o m E q i I n J / H k S c o X M i K k l l C l u b y V s T B V l x k Z U s i F 4 q y + v k / Z V 1 b u u 1 h 5 q l X o z j 6 M I Z 3 A O l + D B D d T h H h r Q A g Y C n u E V 3 p x H 5 8 V 5 d z 6 W r Q U n n z m F P 3 A + f w B t c 5 B L < / l a t e x i t > TABLEI .Mathematical expression (left column) and diagrammatic representation (right column) of the Hedin-Baym equations for systems of interacting electrons and phonons.

2 . 1 ( 1 )
If we then multiply Eq. (114) by Ψ * from the left and by Ψ 2 ( 2 ) from the right and we integrate over 1 and 2 we < l a t e x i t s h a 1 _ b a s e 6 4 = " S c f v h k 6 s lC g x d b T i 2 a K 3 1 y + L p 9 I = " > A A A B 8 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l K U Y 8 F P X i s Y D 8 g D W W z 3 b R L N 7 t h d y O U k J / h x Y M i X v 0 1 3 v w 3 b t s c t P X B w O O 9 G W b m h Q l n 2 r j u t 1 P a 2 N z a 3 i n v V v b 2 D w 6 P q s c n X S 1 T R W i H S C5 V P 8 S a c i Z o x z D D a T 9 R F M c h p 7 1 w e j v 3 e 0 9 U a S b F o 5 k l N I j x W L C I E W y s 5 A 8 i h U n m 5 V k j H 1 Z r b t 1 d A K 0 T r y A 1 K N A e V r 8 G I 0 n S m A p D O N b a 9 9 z E B B l W h h F O 8 8 o g 1 T T B d U b D 4 1 a 8 6 6 I o w w n c A r n 4 M E 1 N O E e W t A G A h K e 4 R X e n N R 5 c d 6 d j 0 V r y S l m j u E P n M 8 f p 8 + R d Q = = < / l a t e x i t > + 1 4 w + K e P U P e f P f u G l z 0 N Y H A 4 / 3 Z p i Z F y a c a e N 5 3 0 5 p b X 1 j c 6 u 8 X d n Z 3 d s / q B 4 e t X W c K k J b J O a x 6 o Z Y U 8 4 k b R l m O O 0 m i m I R c t o J J 7 e 5 3 3 m i S r N Y P p p p Q g O B R 5 J F j G C T S x e u 6 w 6 q N c / 1 5 k C r x C 9 I D Q o 0 B 9 W v / j A m q a D S E I 6 1 7 v l e Y o I M K 8 M I p 7 N K P 9 U 0 w d U b D 4 1 a 8 6 6 I o w w n c A r n 4 M E 1 N O E e W t A G A h K e 4 R X e n N R 5 c d 6 d j 0 V r y S l m j u E P n M 8 f p 8 + R d Q = = < / l a t e x i t > + 1 4
e l P x P 6 + b m u g 6 y J h M U k M l m S + K U o 5 M j K Y R o D 5 T l B g + t g Q T x e y t i A y x w s T Y o I o 2 B G / x 5 W X S u q h 4 l 5 X q X b V c q + d x F O A Y T u A M P L i C G t S h A U 0 g o O A Z X u H N e X J e n H f n Y 9 6 6 4 u Q z R / A H z u c P m n K R 9 g = = < / l a t e x i t > H = < l a t e x i t s h a 1 _ b a s e 6 4 = " E Y 6 G n J j N t a n / D S X e A t o h 68 g D k m Q = " > A A A B 8 n i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 l K U Y 8 F L z 1 W s B + Q h r L Z b t q l m 9 2 w u x F K y M / w 4 k E Rr / 4 a b / 4 b t 2 0 O 2 v p g 4 P H e D D P z w o Q z b V z 3 2 y l t b e / s 7 p X 3 K w e H R 8 c n 1 d O z n p a p I r R L J J d q E G J N O R O 0 a 5 j h d J A o i u O Q 0 3 4 4 u 1 / 4 / S e q N J P i 0 c w T G s R 4 I l j E C D Z W 8 o e R w i T z 8 q y R j 6 o 1 t + 4 u g T a J V 5 A a F O i M q l / D s S R p T I U h H G v t e 2 5 i g g w r w w i n e W W Y a p p g M s M T 6 l s q c E x 1 k C 1 P z t G V V c Y o k s q W M G i p / p 7 I c K z 1 P A 5 t Z 4 z N V K 9 7 C / E / z 0 9 N d B d k T C S p o Y K s F k U p R 0 a i x f 9 o z B Q l h s 8 t w U Q x e y s i U 2 x T M D a l i g 3 B W 3 9 5 k / Q a d e + m 3 n x o 1 l r t I o 4 y X M A l X I M H t 9 C C N n S g C w Q k P M M r v D n G e X H e n Y 9 V a 8 k p Z s 7 h D 5 z P H z 3 8 k U I = < / l a t e x i t > self-energy is obtained from the functional derivative of the Ehrenfest functional < l a t e x i t s h a 1 _ b a s e 6 4 = " S c f v h k 6 s l C g x d b T i 2 a K 3 1 y + L p 9 I = " > A A A B 8 n i c b t e x i t s h a 1 _ b a s e 6 4 = " p I t D E B m / M x n f T 7 E L j N 1 s e g C 6 f d 8 = " > A A A B 8 3 i c b ℎ, ( ) ≡ R ( ) − A ( ) = > ( ) − < ( ) = R ( ) Γ ℎ, ( ) + 2 A ( ).

′ 2 , 2 .
and Γ dyn ( ) are real functions.Bearing in mind that [ ( ) + Π clam+stat ] ′ = see Section XII A, we see that the block (1, 1) of the retarded/advanced phononic GF is diagonal in the chosen basis.To lowest order in Λ dyn and Γ dyn Eq. (154) yields the quasi-phonon frequencies Ω ± as the solution of Ω To first order in − Ω ± we can then write ), see Appendix C for the derivation of the symmetry properties of the phononic GF and self-energy.In crystals with time-reversal symmetry we also have Λ dyn ( ) = Λ dyn − ( ), and therefore Λ dyn ( ) = Λ dyn (− ).In this case Ω + = −Ω − and no correlation-induced splitting occurs.