A Functional Approach to the Electronic and Bosonic Dynamics of Many–Body Systems Perturbed with an Arbitrary Strong Electron–Boson Interaction

A functional approach to the electronic and bosonic dynamics of many–body systems perturbed with an arbitrary strong electron–boson interaction

Andrea Marini1, 2, 3 and Yaroslav Pavlyukh4, 5 1Istituto di Struttura della Materia of the National Research Council, via Salaria Km 29.3, I-00016 Monterotondo Stazione, Italy 2European Theoretical Spectroscopy Facilities (ETSF 3Division of Ultrafast Processes in Materials (FLASHit) 4Department of Physics and Research Center OPTIMAS, Technische Universität Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany 5Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle, Germany (Dated: September 16, 2021) We present a formal derivation of the many–body perturbation theory for a system of electrons and bosons subject to a nonlinear electron–boson coupling. The interaction is treated at an arbitrary high order of bosons scattered. The considered Hamiltonian includes the well–known linear coupling as a special limit. This is the case, for example, of the Holstein and Fröhlich Hamiltonians. Indeed, whereas linear coupling have been extensively studied, the scattering processes of electrons with multiple bosonic quasiparticles are largely unexplored. We focus here on a self-consistent theory in terms of dressed propagators and generalize the Hedin’s equations using the Schwinger technique of functional derivatives. The method leads to an exact derivation of the electronic and bosonic self-energies, expressed in terms of a new family of vertex functions, high order correlators and bosonic and electronic mean–field potentials. In the electronic case we prove that the mean–field potential is the nth–order extension of the well–known Debye–Waller potential. We also introduce a bosonic mean–field potential entirely dictated by nonlinear electron–boson effects. The present scheme, treating electrons and bosons on an equal footing, demonstrates the full symmetry of the problem. The vertex functions are shown to have purely electronic and bosonic character as well as a mixed electron–boson one. These four vertex functions are shown to satisfy a generalized Bethe–Salpeter equation. Multi bosons response functions are also studied and explicit expressions for the two and the three bosons case are given.

I. INTRODUCTION cillations normal to this plane cannot be linear. This fact was noticed by Mariani and von Oppen [7] who demonstrated that flexural phonons quadratically Electron–boson (e–b) Hamiltonians are ubiquitous in par- couple to the electron density. ticle, condensed matter physics and optics: the fundamental On the other hands, if the mirror symmetry is broken by the electron–electron interaction is mediated by photons, which presence of a substrate or by the gating, the coupling becomes are bosonic particles; lattice vibrations (phonons) play fun- linear again [8]. damental role in superconductivity [1]; and collective excita- c. Holstein and Fröhlich models. The interplay between tions in many–electron systems (plasmons) as well as bound the effects induced by different orders of the e–b interaction electron–hole states (excitons) have a bosonic nature. Many can have important consequences in the Holstein model [9]. examples of such a duality can also be found in strongly cor- This uses a simplified form of the Fröhlich Hamiltonian, where related systems [2]. The interaction between electrons and carriers couple to a branch of dispersionless optical phonons bosons is typically treated linearly in electronic density and through a momentum–independent coupling. In this case bosonic displacement [3]. The proportionality constant may even small positive nonlinear interaction reduces the effec- have different expressions depending on the microscopic de- tive coupling between the electrons and the lattice, suppress- tails of the system. ing charge–density–wave correlations, and hardening the ef- However, there are cases where nonlinear coupling is com- fective phonon frequency [10, 11]. These finding prompted parable in strength or even dominate the first–order electron– further theoretical investigations of the Holstein model with boson interaction. even more complicated double–well electron–phonon interac- a. Electron–phonon coupling in quantum dots. Very of- tion [12, 13] using a generalization of the momentum aver- arXiv:1807.06949v2 [cond-mat.str-el] 2 Aug 2018 ten the quadratic and linear effects are inseparable, and the age approximation [14], and of general form of interaction former can arise in, e. g., perturbative elimination of the off– using the determinant quantum Monte Carlo approach [15]. diagonal electron–phonon coupling in quantum dots. For Closely connected to these studies are recent experiments em- instance, quadratic coupling of carriers in quantum dots to phasizing the role of nonlinear lattice dynamics as a mean for acoustic phonons modifies the polarization decay and leads to control [16], and as a basis for enhanced superconductivity in exponential dephasing [4]. Linear coupling alone generates MgB2 [17] and some cuprates [18]. They point toward large acoustic satellites in the spectrum, but causes no Lorentzian ionic displacement which is a prerequisite for the nonlinear broadening [5,6]. electron-phonon coupling. b. Flexural phonons. The balance between the first and d. Finite temperature effects. Another prominent exam- the second–order effects can be influenced by the symmetry. ple is the renormalization of electronic structures due to zero or If a system possesses a mirror plane, the coupling to the os- finite temperature phonons. As demonstrated by Heine, Allen 2 and Cardona (HAC) [19, 20] the linear and quadratic couplings the existence of the DW potential, as, for example in Ref. 28. (in atomic displacement) are of the same order in the electron– It is therefore desirable to formulate self-consistent (sc) MBPT ion interaction potential. Moreover they need to be considered for electron–boson system with nonlinear coupling, i. e., in on an equal footing in order for the system to respect the sys- terms of the dressed propagators, in a functional derivative ap- tem translational invariance. The effect of the second–order proach. correction is quite large in carbon materials and can lead to a h. Out–of–equilibrium scenarios Our further motiva- substantial band gap modification [21–23]. tion for this work is experimental feasibility to generate coher- e. Anharmonic effects. Some recent works have also ent phonons [38, 39] and plasmons [40–42]. For such scenar- demonstrated that, potentially, even simple systems like dia- ios the notion of transient spectral properties is of special inter- mond [24, 25] or palladium [26] show remarkable nonlinear est [43–45]. A powerful method to deal with time-dependent effects. However, at the moment, these anharmonic effects can processes is the non-equilibrium Green’s function (NEGF) ap- be treated only by using an adiabatic approach based on finite proach [46]. The method relies on solving the Kadanoff-Baym displacements of the atoms. This approach ignores dynamic equations (KBE) of motion for the Green’s functions (GFs) on effects that, however, have been shown to be relevant in the the Keldysh time contour [47–51]. To the best of our knowl- linear coupling case [22] and, therefore, cannot be neglected, edge, for systems with nonlinear coupling such theory is not a priori in the case of nonlinear coupling. available. Manuscript organization f. Existing theoretical approaches. Nonlinear electron– . Our manuscript is organized as boson models have been treated theoretically by essentially follows: In Sec.II we introduce the Hamiltonian and its prop- stretching methods developed for pure electronic case or lin- erties. Given the Hamiltonian, in Sec. III A, we derive the cor- ear coupling scenario: quantum Monte Carlo [11], the av- responding equation of motion for the bosonic and electronic erage momentum approximation [14], and the cumulant ex- operators. The equation of motion are analyzed in terms of pansion [4]. Since only electronic spectrum was of inter- functional derivatives in Sec. III B. The Green’s functions are est, they rely on diagrammatic methods, without system- introduced in Sec.IV. atically exploring the renormalization of phononic proper- We first discuss the electronic case whose self-energy is de- exactly ties due to electrons. However, as has been shown in the rived to all orders in the electron–boson interaction, in linear case using perturbative expansions of both electron Sec.V. We derive the form of a generalized Debye–Waller po- and phonon propagators, electrons typically overscreen bare tential in Sec.VA which, in turns, define the remaining non- phonon frequencies leading to the conclusion that renormal- local and time–dependent mass operator, Sec.VB. ized phonon frequencies must be fitted to experiments [27]. The bosonic subsystem is, then, split in single–boson and Thus, Marini et al. [28] have recently extended many-body multi–boson case in analogy with the electronic case. In ex- perturbation theory (MBPT) for electron–phonon interaction Sec.VI we introduce the bosonic self-energy that we split actly including quadratic terms and using Density Functional The- in a mass operator, Sec.VIC, and a mean–field poten- ory [29] as a starting point. This is a remarkable achieve- tial, Sec.VIA. The exact bosonic mass operator is rewritten ment since even ab initio determination of momentum- in terms of four generalized vertex functions whose coupled dependent electron-phonon linear coupling function is a non- equation of motion is derived in Sec.VID. trivial task [30]. The Born–Oppenheimer approximation is The presented exact formulation is illustrated by the commonly used as a starting point. However, the seminal derivations of the lowest order approximations for the elec- works of Abedi et al. [31] and Requist et al. [32] on the ex- tronic (Sec. VII A) and bosonic (Sec. VII B) self-energies. act factorization of the fermionic and bosonic wave–function The last part of the work is devoted to the electronic show that alternative paths beyond the Born–Oppenheimer ap- and bosonic response functions (Sec. VIII). We derive a proximation are possible. Bethe–Salpeter like equation for the electronic response in g. Diagrammatic perturbation theory Nonexistence of Sec. VIII A. In Sec.VIII B we discuss the bosonic case by the Wick theorem for bosons [33], which is a consequence of showing how to reduce the general bosonic dynamics to diago- the fact that averages of the normal product of bosonic opera- nal number conserving response functions. Then, the cases of tors are non-zero, makes it difficult to develop a diagrammatic two and three bosons are studied, respectively, in Sec. VIII B 2 perturbation theory [34]. To circumvent this difficulty, sys- and Sec. VIII B 3. tems above the Bose-condensation temperature are implicitly Finally, in AppendixA we motivate our treatment of assumed [35]. Method of functional derivatives is a comple- electron–electron correlation, in AppendixB we formally con- mentary method [36]. In contrast to diagrammatic construc- nect the formalism to the electron–phonon problem. In Ap- tions based on the series expansions of the evolution oper- pendixE we finally list some key mathematical quantities and ator on a contour, it yields functional relations between the approximations used throughout the whole manuscript. Logi- dressed propagators. They do not rely on the Wick theorem. cal flow of the whole work is depicted on Fig.1. In the seminal works of L. Hedin [37] and R. van Leeuwen[27], the Schwinger technique of functional derivatives is used to II. NOTATION AND HAMILTONIAN derive the linear electron–boson coupling and no Debye– Waller (DW) potential is found. This is in stringent disagree- ment with the HAC theory where this potential naturally ap- We start from the generic form of the total Hamiltonian of pears. On the other hand any diagrammatic approach predicts the system that we assume to be composed by fermions and 3

† Hamiltonian standard way in terms of the creation (b̂ ) and annihilation (b̂ ) (Sec. II) operators: ̂ ̂ ̂ ̂ 퐻 = 퐻e + 퐻b + 퐻e-b 1 ̂† ̂ Sec. III Sec. III Q = b + b , (3a) 2 Auxiliary ﬁelds (Sec. III. A) i † P = √ b̂ − b̂ . (3b) d d2 (퐱 ) ̂ ( ) = ̂ + 푛( ) ̂푛 + Ω2 ̂ ( ) 푖 ̂휓 , 푧 퐻휉,휂 푧 퐻 휉휈 푧 푄휈 2 휈 푄휈 푧 2 d푧 휈 d푧 ∑ [ ] + 푑퐱 휂(퐱, 푧) ̂휌(퐱) √ ∫ The electron–boson interaction is taken to have the general

EOM for 퐺 (Sec. V) EOM for 퐷 (Sec. VI) form:

† n n Matrix BSE (Sec. IV. D) He−b = dx ̂ (x) V (x) ̂ (x) Q , (4) Ê Electron self-energy: i-j i-j Bosonic self-energy: n, Γ = Γ0 Σe = Φ + 푀 Σb = 푈 + Π É + 퐾i-e퐺퐺Γe-j Mean-ﬁeld: 휂 + Φ Mean-ﬁeld: Ξ + 푈 + 퐾i-b퐷퐷Γb-j with Sec. VII. A Sec. VII. B n n 푀0 훿Σe 훿Π Π0 Q = Q , (5a) 퐾e-e = 퐾b-e = i 훿퐺 Response (Sec. VIII) 훿퐺 i=1 e-b 훿푀 훿Σb Ç 퐾 = 훿퐺 퐾b-b = n 훿퐷 휒 = − 훿퐷 1 훿휂 V n (x) = ) V (x) . 훿퐷푚푖,푛푖 i e–b (5b) 퐷푚,푛 = 퐹 퐷푚푖,푛푖 , … , n! 푝푖 H i=1 I [ 훿휉 ] Ç eq

Here, Ve–b (x) is a generic potential that dictates the electron– FIG. 1. Logical structure of the work. boson interaction. The connection to the electron–phonon problem is given in the AppendixB. Eq.(5b) makes it clear n (x) symmetric tensor bosons with a nonlinear interaction that V is a with respect to indices . The differentiation is performed with respect to the bosonic coordi- H = H + H + H . e b e–b (1) nates evaluated at the equilibrium point. The physical form of the potential depends on the specific problem. Therefore the The unperturbed part of H is He + Hb and can be rewrit- equilibrium coordinates are specific to the kind of physics the ten in terms of corresponding energies (i is the energy of the n electronic state i, Ω is the energy of the bosonic mode ) and bosons are describing. In the case of phonons =1 ) is i i eq eigenstates obeying fermionic, bosonic statistics, respectively: evaluated at the equilibrium atomic configuration,∏ as defined in AppendixB. = † He i ̂ci ̂ci, (2a) Averaging the total Hamiltonian, Eq. (1), with respect i to electronic coordinates leads to the effective anharmonic É 1 2 2 bosonic Hamiltonian. Solving such a model leads to, among H = Ω P + Q . (2b) b 2 other effects, the prediction of the temperature dependence É of the averaged displacement. While interesting and well- In general, the partitioning of a physical Hamiltonian in the discussed problem on its own, we will not consider this effect form of Eq. (1) is an highly nontrivial problem [31, 32]. In here assuming that for each given temperature an Hamiltonian the present context, we are interested in the nonlinear e–b cou- of the type defined by Eq. (1) can be derived such that pling and, to keep the formulation simple, we assume that such a partition does exist and that the electronic correlation can Q = 0. (6) be approximatively described with a mean–field potential that ( ) renormalizes the free electrons and bosons. This is a com- In contrast, as will be shown using our diagrammatic approach, mon practice, for example, in the DFT approach to electrons other correlators of the position operator will be modified by and phonons. The DFT mean–field potential is defined in Ap- electron-boson interaction in nontrivial way. pendixA. Eq. (5) highlights an important and crucial aspect of the no- In Eq.(2b) we have introduced the operators for the bosonic tation. The symbol represents a generic vector of bosonic coordinates, Q, and momenta, P. The fermions are de- n † indices of dimension , which is indicated as a superscript and scribed by the corresponding creation ( ̂ci ) and annihilation ( ̂ci) should not be confused with power. Therefore we consider operators. These are used to expand the electronic field opera- the most general case where the nth–order e–b interaction is a tor ̂ (x) = i i (x) ̂ci, with i (x) eigenfunctions of the elec- nonlocal function of n bosonic coordinates. tronic Hamiltonian in the first quantization (denoted as ℎ (x)). For convenience we also introduce the electronic operator ∑ e i and Ω are the independent electrons and bosons ener- † gies. They are assumed to incorporate the mean–field poten- ̂ n ≡ dx ̂ (x) V n (x) ̂ (x) , (7) ̂ ̂ Ê tials embodied in He + Hb. Q and P are expressed in the 4 such that He−b can be written as In Eq. (11c), the combinatorial prefactor m follows from the fact that ̂ m also is a symmetric tensor of rank m. This equation n n He−b = ̂ Q. (8) is formally demonstrated in AppendixC. n, É In Eq. (11c) we have introduced a general definition for a multi–dimensional operator whose index is a composition of Having introduced the general electron-boson Hamiltonian (1) m two subgroups of indexes. In the case of ̂ , the vector of in- and specified its ingredients, our goal now is to obtain a self- ⊕ consistent set of equations that relate well-defined objects such dices has m−1 components, and ⊕ = 1, … m−1, as electron and boson propagators. To this end, we generalize is correctly m dimensional. By combining the last two of the Schwinger’s method of functional derivatives [52], which Eqs. (11) we obtain a second–order differential equation for allows to express more complicated correlators that appear in the displacement operator [53] with a source term: their equations of motion (the Martin-Schwinger hierarchy) in terms of functional derivatives. 2 d + Ω2 = −Ω m m−1 2 Q z1 m ̂ ⊕ z1 Q z1 . Ldz1 M m, III. THE EQUATION OF MOTION FOR THE É ELECTRONIC AND BOSONIC OPERATORS (12)

A. Time Dependence More compicated operators appearing on the right hand side of Eqs. (11,12) can be expressed using the method of functional For our purpose we deﬁne operators in the Heisenberg pic- derivatives. ture (indicated here by the H subscript) with time–arguments running on the Keldysh contour (z ∈ ):

H (z) ≡  z0, z   z, z0 , (9) B. Functional derivatives where z0 is arbitrary initial time and ̂ z, z0 is the time- In order to introduce the functional derivatives approach evolution operator from the initial time z0 to z. In this pic- we couple the Hamiltonian to time-dependent auxiliary fields ture, the operators are explicitly time–dependent, whereas n (z) and (x, z) wave-functions not. This allows to make a connection with the many-body perturbation theory, which relies on the time- evolution on the contour and on the Wick theorem. In what ̂ ( ) = ̂ + n ( ) n ( ) + x (x ) (x ) follows, the picture in which operators are given is not explic- H, z H z Q z d , z ̂ , z , n, Ê itly indicated when it can be inferred from the corresponding É arguments. (13) The electronic, bosonic operators satisfy standard anticom- + − n mutation (denoted with ), commutation (denoted with ) where a superscript in (z) indicates that is an n- rules, respectively: dimensional vector of indices. We introduced the electron den- † † sity operator ̂(1) = ̂ (1) ̂ (1). ̂ ̂ x1 , ̂ x2 + = x1 − x2 , (10a) Consider now the time-evolution in the presence of these Q , P = i . (10b) external fields. The corresponding time-evolution operator is − denoted as , z0, z . Now in the definition of the average x operator We now introduce a short-hand notation i, zi ≡ i so (1) x that ̂ ≡ ̂ 1, z1 . The Heisenberg equations of mo- ̂ tion (EOM) for ̂ , Q and P follow by applying Eqs.(10) to Tr  exp − i ∫ d ̄zH,( ̄z) , (z) evaluate commutators with the full Hamiltonian H: ̂ (z) = , , , $ % Tr  exp − i ∫ d ̄zH ( ̄z)  , d n n $ % i ̂ (1) = ℎe (1) + V x1 Q z1 ̂ (1) , (11a) (14) dz1 L n, M É d the and functions occur twice signaling that both: Q z1 = ΩP z1 , (11b) dz1 the operator Ô in the Heisenberg picture ̂ , (z) = d m m−1 , z0, z ̂, z, z0 and the density matrix are defined P z1 = −Ω Q z1 − m ̂ z1 Q z1 . dz ⊕ with respect to the perturbed Hamiltonian. Starting from this 1 m, É form various functional derivatives can be computed. We (11c) write … for … , where it does not lead to ambiguities.

⟨ ⟩ ⟨ ⟩ 5

̂ (i) Let us consider the case of a generic, contour–ordered product of operators: i  zi . Constituent operators depend, in general, on diﬀerent times z and are distinguished by the subscript (i). By the formal diﬀerentiation, one can prove that i ∏

(i) ( ) ( ) i  ̂ z =  ̂ i z Qn ( ̄z) −  ̂ i z Qn ( ̄z) , (15) n ( ̄z) , i i i i ó n i i ( $ Ç %)ó=0, =0 ( $ Ç %) ( $ Ç %)( ) ó ó ó where  denotes the contour ordering operator.ó The second term in Eq. (15) stems from the variation of denominator, i. e., it assures correct normalization. In general, this identity can contain side by side electronic and bosonic operators and also operators with equal time arguments. For the latter, the standard deﬁnition of  needs to be amended with a rule that equal-time operators do not change their relative order upon contour-ordering. For mixed operators, only the permutations of the electronic ones induce a sign–change [46]. A similar expression holds for the derivative with respect to :

(i) ( ) ( ) i  ̂ z =  ̂ i z (1) −  ̂ i z (1) . (16) (1) , i i i i n i i ( $ %)ó=0, =0 ( $ %) ( $ %)( ) Ç ó Ç Ç ó ó ó

Here and in the following we always assume that the limit of This equation can be further generalized to zero auxiliary fields is taken after variations. In practice, however, this means that during derivations all Green’s functions m,n m−k,n D z1, z2 = i D z1, z2 are formally dependent on the auxiliary fields. This will be ev- , k , z1 ident from the form of the electronic and bosonic Dyson equations with mean–fields that include the auxiliary fields. + k m−k,n Q z1 D, z1, z2 ( ) + m−k k,n Q z1 D, z1, z2 , (21) IV. GREEN’S FUNCTION AND DIAGRAMMATIC ( ) NOTATION for k < m and = ⊕ . Eq. (21) is proved in AppendixD. The last two terms represent a contraction of symmetric tensors of ranks m − k and k yielding a symmetric tensor of rank We use the standard definitions of the electronic Green’s m (with respect to the first argument). We will make an ex- function (GF) on the Keldysh contour: tensive use of these differential form of Dm,n as well as of the representation in terms of Feynman diagrams. We introduce (1 2) = − { (1) † (2)} G , i  ̂ ̂ , (17) ad hoc graphical objects to easily represent the multi-fold aspects of the nonlinear e–b interaction; in Fig.2 all ingredients where … is the trace evaluated with the exact density matrix. of the diagrammatic representation are showed. The bosonic propagators on the Keldysh contour extend the In general, the selection of k bosonic operators out of m, m definition⟨ ⟩ of the electronic case that appear on the r.h.s. of Eq. (21), can be performed in k ways. These corresponds to all the possible choices of k ele- m,n m n ments out of m. However Eq. (21) is exact for any choice of the D, z1, z2 = −i  ΔQ z1 ΔQ z2 , (18) elements. Therefore no combinatorial prefactor is needed ( $ %) whenever Eq. (21) is used. where Δ̂ ≡ ̂ − ̂ is the fluctuation operator. In the case By using Eq. (6) we can write m = n = 1 the standard bosonic propagator is recovered ⟨ ⟩ m = m−1,1 + 1,1 Q⊕ z1 iD, z1, z1 . (22) D, z1, z2 = D z1, z2 . (19) , ( ) + + We use here z1 = z1 + 0 . It is important to note, here, that in Thanks to Eq. (14), we can rewrite Dm,n as n = 0 = 0 m the limit , we have that Q⊕ z1 is constant because of the time–translation invariance.( However) during m,n = m n iD, z1, z2  Q z1 Q z2 the derivation the time-dependence is induced by the auxiliary ( ) fields. m Q z1 m n The EOM for bosonic displacement operators (12) leads us − Q z1 Q z2 = i . (20) m−1,1 ( n ) to consider a specific case of D , which can be reduced z2 ( )( ) to simpler propagators by the application of Eq. (21) with k = 6

x (a) vertices: = 1 = 푧1 = (x1, 푧1) 1 1 1

푛 ν1, ν2, … , ν푛 푛 ν 푛 (b) Vν (x1) = = 1 1 푛 ν 푛 μ ν (a) ΦDW(1) = (푚, 푛) 1 푚,푛 (c.1) Dμ,ν (푧1, 푧2) = 1 2 μ ν 3 α (푛) 푛,푛 (c.2) Dμ,ν(푧1, 푧2) = 1 2 2 μ ν

(c.3) Dμ,ν(푧1, 푧2) = 1 2 3 ν 3 (b) ΦDW(1) ≈ 푚 μ 1 ̂푚 (d) 푖 ⟨Qμ (푧1)⟩ = 1 FIG. 3. (a) Diagrammatic form of the nth–order DW potential. (b) Perturbative expansions in term of bare bosonic propagators (they are (e) G(1, 2) = denoted as dashed lines) lead to complicated diagrams. The inclusion 1 2 of nonlinear e-b interaction leads to non-vanishing odd-order terms that are zero in the linear interaction case. FIG. 2. Definition of the diagrammatic elements used in this work. ○ × (a) and represent a generic time and position point respectively. Using Eq. (15), the correlator on r.h.s. of Eq. (24) can be ex- These two symbols can be combined to indicate a time and position pressed as the functional derivative vertex ⊕1 equivalent to 1 = x1, z1 . (b) Finally a box around a spa- tial point represents the scattering integral V n (x) with two fermionic − (1) n †(2) and n bosonic dangling lines. (c) Bosonic propagators can be repre- i  ̂ Q z1 ̂ sented in three different forms depending on their order. (d) Expecta- ( ) tion value of the bosonic coordinates expressed in terms of a bosonic n = i + Q z1 G (1, 2) . (25) propagator. (e) Electronic Green’s function. n L z1 M ( ) Our goal is to rewrite Eq. (24) in the form of a Dyson equa- m − 2: tion, which involves a dressed mean–field potential Φ and correlated mass operator M: m−1,1 D z1, z2 = i D z1, z2 , m−2 , z1 ) i − ℎe (1) − (1) − Φ (1) G(1, 2) = (1, 2) m−2 )z1 + Q z1 D, z1, z2 , (23) 4 5 ( ) + d3 M (1, 3) G (3, 2) . (26) Ê where we used the fact that Q is zero in the limit of vanishing auxiliary fields and = ⊕ . The potential Φ follows from the second term on the r.h.s. of ⟨ ⟩ Eq. (25) V. ELECTRON DYNAMICS n n Φ (1) = V x1 Q z1 . (27) n, É ( ) The EOM for G is obtained with the help of EOMs for the d The mass operator is implicitly written as constituent operators and using the relation z1 − z2 = dz1 n − d3 M (1, 3) G (3, 2) = i V x1 G (1, 2) . z1 z2 . Thus, we have Ê n n, z1 É ) (28) i − ℎe (1) − (1) G (1, 2) = (1, 2) 4 )z1 5 The potential Φ and the mass operator M can be conveniently n n † combined in the electronic self–energy operator Σe: − i V x1  ̂ (1) Q z1 ̂ (2) . (24) n, e É ( $ %) Σ (1, 2) = Φ (1) (1, 2) + M (1, 2) . (29) 7

푚 μ 4 (푛, 푚)

e-b,푚 푛 ν Γ 푛 ν (a) Σ푒(1, 2) = + 1 3 2 1 2 2 4 6

푚 μ 푚 μ 푚 μ 푒 e-b,푚 δ푡Σ (1,2) e-b,푚 (b) Γ = + δ푡G(4,5) Γ 3 1 3

1 1 5 7

FIG. 4. Diagrammatic form of the self–energy operator (a) and of the vertex function (b) for arbitrary orders of the electron–boson interaction and arbitrary number of bosons involved in the scattering. In order to close this set of equations, expressions for the bosonic propagator Dm,n e-b,n (Secs.VIC,VIII) and the vertex function Γ (Sec.VID) are additionally needed. The lowest order approximation for the electron self-energy is described in Sec. VII.

A. The nth–order Debye–Waller potential self-energy originating from the first–order coupling (due to translational invariance). Φ The present approach extends its definition to arbitrary or- In order to rewrite in terms of the bosonic Green’s func- ders and, also, highlights its physical origin. The Schwinger’s tion, we apply Eq. (22) to Eq. (27). It follows that we can in- variational derivative technique has the merit of showing that troduce a nth–order bosonic mean field, Φn (1), defined as: DW the mean–field potential is due to the dressing of the potential induced by the nth–order fictitious interaction n. Physically n n n−1,1 + this corresponds to the dressing of the electronic potential in- Φ (1) = i V x1 D z1, z , (30) DW ,n 1 duced by strongly anharmonic effects. É This also clarifies why the DW potential is not present in any previous treatment [27, 54] of the electron–phonon inter- = Φn with ⊕ n. DW is showed in diagrammatic form in action performed using the Schwinger’s variational derivative Fig.3(a) in the general case. technique. The reason is that in these works the e–b interaction Eq. (30) provides a generalization of the Debye– is treated at the first order only. Waller (DW) potential to arbitrary orders. The expression In conventional theories involving linear electron–boson in- n of this potential is well–known in the electron–phonon case teractions the Q z1 averages are, in general, connected = 2 only when n , and it has been derived only by using to the boson mean( displacement) (n = 1) and the population a diagrammatic approach. In the present case, it naturally (n = 2). As a consequence, it is zero for any odd value of n. appears as the mean–field electronic potential induced by the The presence of higher–order e–b interactions deeply modifies nonlinear electron–boson interaction: n this simple scenario. Q z1 is a nth–order bosonic tad- 2 2 + pole whose dynamics( includes ) nontrivial contributions, like Φ (1) = i V x D z , z . DW 1,2 1 1,2 1 1 (31) the one showed in Fig.3(b). These tadpoles are, in general, 1,2 É nonzero.

The DW potential has a long history in the electron–phonon B. The mass operator context. Early developments are nicely summarized in the HAC approach. They present a very simple perturbation theory derivation that also emphasizes a close connection with

The mass operator requires additional manipulations. We integrate by parts 8

n −1 n −1 M (1, 2) = i V x1 d3 G (1, 3) G (3, 2) = −i V x1 d3 G (1, 3) G (3, 2) . (32) Ê n Ê n n, L z1 M n, z1 É É This equation is exact. Now the problem is how to evaluate this variational derivative. By noticing that

) −1 i − ℎe (1) − (1) − Φ (1) (1, 2) = G (1, 2) + M (1, 2) , (33) 4 )z1 5 we have that

−1 Φ (2) M (3, 2) Φ (2) tM (3, 2) − G (3, 2) = (2, 3) + = (2, 3) − d4567 n n n n Ê G (4, 5) z1 z1 z1 z1 t G−1 (6, 7) × G (4, 6) G (7, 5) . (34) n z1

In Eq. (34) we have introduced the t symbol to make clear that lowing integro-differential equation: we are using a total derivative. In this way the derivation of the electronic self–energy and vertex function closely follows e-b,n e-b,n Γ 1, 2; z3 = Γ 1, 2; z3 + the well–established procedure introduced in the case of the 0 linear e–b coupling [27]. In the next section we will further M (1, 2) e-b,nó + 4567 t (4 6) Γ ó 6 7; (7 5) discuss this subtle but important aspect. d G , ó , z3 G , , (38) Ê tG (4, 5) ó We can now define a vertex function that extends to the e–b case the known electronic vertex function. In order to do so with we start by expanding the first term appearing on the r.h.s. of e-b,n Φ (1) Eq. (34) using Eq. (20): Γ 1, 2; z3 = (1 − 2) 0 Qn z ó 3 m ó Q z2 ó = ((1 − 2) ) − n x Φ (2) m ó z1 z3 V 1 . (39) = V x2 n ( n ) z1 m, z1 e É In Eq. (38) appears M (defined in Eq. (29)) instead of Σ as m m,n does not depend on and the lowest order derivative comes = V x2 D, z2, z1 . (35) m, through Φ. This, in practice, means that in the independent É particle approximation (IPA), (Σe = 0), the mixed e–b vertex It is natural to define the electron–boson vertex function, is zero, as it should be. e-b,m By analogy with electronic case, it can be regarded as Γ (1, 2; z) [55] as the Bethe-Salpeter equation for the vertex function. It was discussed in the linear electron-phonon coupling by R. van −1 e-b,n − G (3, 2) ≡ Γ 3, 2; z1 Leeuwen [27]. Eq. (38) also defines the electron–electron ker- n z1 nel Σe (1 2) m e-e t , −1 Q z4 K (1, 5; 2, 4) ≡ , (40) G (3, 2) G (4, 5) = − dz4 t ( n ) m, Ê m z1 É Q z4 that will also appear in Sec. VIII A in the case of the equa- ( ) tion of motion for the electronic response function. Note that e-b,m m,n = dz4 Γ 3, 2; z4 D, z4, z1 . (36) in this section we have already introduced a specific notation m, Ê É for the vertex and for the kernel. Indeed, in both cases we have that the vertex/kernel is defined as the functional deriva- Here, we have also introduced an alternative form of the e–b tive of electronic/bosonic observable (the inverse GF for the vertex function: vertex and the self–energy for the kernel) with respect to an −1 electronic/bosonic potential (for the vertex) or GF (for the ker- ,n (1 2) e-b G , nel). In the present case Ke-e is purely electronic, while in Γe-b Γ 1, 2; z3 ≡ − . (37) n the field, , is bosonic. In the following sections we will in- Q z3 troduce other vertexes and associated kernels and demonstrate ( ) e-b that they are connected via matrix generalization of the Bethe- From Eq. (35) and Eq. (37) it follows that Γ satisfies the fol- Salpeter equation. 9

Σe(1,2) Σe(1,2) The full mass operator can be ﬁnally written as the variation is implicitly included in the t . This D(4,5) tG(4,5) is the main diﬀerence from the pure electronic case, where the n screened interaction explicitly depends on the electron Green’s M (1, 2) = i d3 dz4 V x1 G(1, 3) n, m, Ê Ê function. Thus, although Eq. (41) is exact, it is not practical. É É A better approach is to consider from the beginning the elec- e-b,m m,n × Γ 3, 2; z4 D, z4, z1 . (41) tronic self–energy to be a functional of both propagators, i.e. Σe = Σe [G,D], which requires the introduction of other ver- By comparing the expression for the electron self-energy tex functions. This procedure will be implemented below in with the expression in a pure electronic case one observes combination with the bosonic self–energy. m mn n that m, n, V (3)D z3, z1 V (1) plays the role of the screened∑ Coulomb∑ interaction. Eq. (41) is not the most convenient representation of the VI. SINGLE-BOSON DYNAMICS electron self-energy because there is no simple way of comput- ing the kernel Ke-e (1, 5; 2, 4) even though the diagrammatic e form of Σ is known. As can be seen from the exact formula, Starting from the equation of motion (12) for Q we derive Eq. (41), and its diagrammatic representation in Fig.4(a), the the equation of motion for the bosonic propagator D, in a self-energy contains the bosonic propagator D, and, therefore, similar way to the electronic case:

2 1 ) 2 − + Ω D z , z = z − z Ω 2 , 1 2 1 2 L)z1 M n n−1 n n−1 − i n dx1 V x1  Δ ̂ (1) Q̂ z1 Q̂ z2 + z1  ΔQ̂ z1 Q̂ z2 , (42) Ê ⊕ ⊕ ,n L M É «¯¬( ) («¯¬ ) (n) (n) JV J

n 1 = 1 ̂ = 0 The last term is driven by the auxiliary fields . According to order (n > ) terms. In the n case, Q , and we can n use the chain rule to write the rules specified above, the limit of zero is to be taken at (1) ̂ (1) ⟨ ⟩ the end of derivations. = (1) ̂ = (n) JV  ̂ Q z2 i 1 In Eq. (42) we have schematically represented with JV and z2 (n) ⟨ −1⟩ J , respectively, the term induced by the scattering potential G (3, 4) = − d34 G (1, 3) G (4, 1) . (45) and by the auxliary field. The goal of this section is to rewrite Ê 1 exactly z2 We can now use the definition of the electronic vertex, (1) 1 (n) (n) Π − i n dx V n x J + n z J Eq. (36), and rewrite JV in terms of the mass operator : Ê 1 ⊕ 1 V ⊕ 1 ,n 4 5 É 1 1 Π z , z = d34 dx V x , 1 2 Ê Ê 1 1 = dz3 Π, z1, z3 Ê e-b,1 (1 3) (4 1) Γ 3 4; É G , G , , z2 , (46) + U, z1 + Ξ, z1 z1 − z3 that is diagrammatically represented in Fig. (6a). This con- ×D , z3, z2 . (43) tribution to the bosonic mass operator does not require further In Eq. (43) we have introduced the generalized bosonic mass manipulations and is explicit function of the single–boson cor- Π(1) operator, Π and the mean–field potentials, U and Ξ. Ξ is driven relator D. represents the generalization to the case of by the fictitious external field and vanishes when → 0. Π, U non–linear e-b coupling of the first–order e-b mass operator and Ξ sum in the total bosonic self–energy Σb that, consistently well known and widely used in the literature [56, 57] to calcu- with Eq. (29), is defined as late, for example, phonon linewidths [58]. We now move to the n > 1 case. We observe that, thanks to b Σ, z1, z2 = Π, z1, z2 + U, z1 z1 − z2 . (44) Eq. (6), In order to find the explicit expression for Π , U and Ξ, we start n−1 n−1,1  ΔQ̂ z1 Q̂ z2 = iD z1, z2 , (47) by observing that Eq. (42) includes linear (n = 1) and higher- , ( ) 10

= = −2 (n>1) and, by using Eq. (23) with m n and k n we can express The JV correlator can be evaluated by using Eq. (16): (n>1) J as (n>1) n−2 J = Q z1 + i V n−2 H z1 I (n>1) n n−1,1 ( ) J = n z D z , z ⊕ 1 , 1 2 × ̂(1) + i D z1, z2 . (49) É (1) , 0 1 n n−2 = n z1 Q z1 + i ⟨ ⟩ ⊕⊕ n−2 In Eq. (49) the derivative is made acting before the one. , H z1 I In this way the limit of zero external ﬁeld can be safely taken É ( ) and the last term of Eq. (23) vanishes. It is, indeed, important × D , z1, z2 . (48) to remind that Q z1 = 0 only when = 0. If we now collect( Eq.) (48) and Eq. (49) and plug them in with = ⊕ . Eq. (42) we can recast the EOM for D in the form

2 1 ) 2 −2 − + Ω D z , z = z − z + n ̂ n z + n z Qn z D z , z Ω 2 , 1 2 1 2 ⊕⊕ 1 ⊕⊕ 1 1 , 1 2 )z1 n>1, , É «¯¬ ( ) ⟨ ⟩ U+Ξ (1) + dz3 Π, z1, z3 D , z3, z2 Ê É (1) 2 n D , z1, z2 n−2 D , z1, z2 D , z1, z2 + n dx1 V x1 i + i Q z1 − . (50) ⊕⊕ n−2 n−2 1 Ê L z1 (1) (1) z1 M n> , , «¯¬( ) É «¯¬ «¯¬ Π(3) Π(2) Π(4)

Eq. (50) represents a key result of this work. We have already with schematically identified the different terms that compose the (n) n → 0 = n n−2 EOM for D. The J term reduces, when only to the U, z1 n ̂ ⊕⊕ z1 Q z1 , (52a) ( ) ⩾2 n nÉ, ( ) U potential, while the JV term reduces to the sum of three mass operators. In the following we study them in detail in ⟨ n ⟩ n−2 Ξ, z1 = n⊕⊕ z1 Q z1 . (52b) order to recast Eq. (50) in the form of a Dyson equation for D. n⩾2, É ( ) n n We remind the reader that ⊕⊕ and ̂ ⊕⊕ are symmetric tensors of rank n, and is an n − 2 dimensional vector. Eq. (52a) is represented diagrammatically in Fig.5 in the limit of vanishing auxiliary fields. A. Mean-field potentials

B. The pure bosonic vertex function Γb-b The first contribution to the EOM for D is through the mean–field potentials, U and Ξ. These potentials are A key ingredient of Eq. (50) is the first order derivative due to the first term on the r.h.s. of Eq. (48) and to the D(z1,z2) n . This term shows some remarkable properties that n−2 (z1) Q z1 D , z1, z2 term in Eq. (50). The sum of these we study here in detail. We start from the term (two terms can) be rewriten as the action of two local potentials on the bosonic propagator: D , z1, z2 = dz3dz4 D z1, z3 n−2 Ê , z1 , É b-b,n−2 U z1 + Ξ z1 D z1, z2 , × Γ ; , , , (51) , ; z3, z4 z1 D , z4, z2 . (53)

É 11

( − 1) n Eq. (56) gives, in practice, only n n terms as all n−2 per- m−2 (푛 − 2) mutations of … inside the Q z1 gives I2 In−1 I …I @ 2 n−1 A b-b,n the same contribution. The final form of Γ 0 is, therefore: 푛 μ ⊕ κ ⊕ ν ó ó 1 Uμ,ν = b-b,n b-b,n Γ z1, z2; z3 ≡ Γ z1 z1 − z2 z1 − z3 , ; 0 , ; 0 n! FIG. 5. The nth–order bosonic mean–field potential is one of the 1ó ó = ó n−2 ó constituents of the total bosonic self–energy. ó Q …ó z1 ( − 1)! I2 In−1 n I=1 É ( ) × , , z1 − z2 z1 − z3 . (57) Eq. (53) introduces a further vertex with an entire bosonic I1 In character: −1 D z1, z2 b-b,n , Γ z1, z2; z3 ≡ − . (54) , ; n z3 Γb-b,n Note the contracted single–time form of 0 introduced in Eq. (57). It will be used in the zeroth–order approximations The lowest–order contribution to this vertex function is from for Σb, cf. Eq. (44). ó the variational derivative of the driving field entering the C. Nonlinear self–energies ó mean–field potential, Eq. (52a):

Γb-b,n ; = , ; z1, z2 z3 0 The ﬁrst term we analyze is Π(2). With the help of Eq. (53), m ó m−2 it follows that ⊕⊕ ó z1 Q z1 m ó z1 − z2 . (55) n ( ) m⩾2, z3 É → 0 m In the limit only the derivative of ⊕⊕ z1 gives a Π(2a) = n nonzero contribution. As written previously the function is , z1, z2 i n ̂ ⊕⊕ z1 n⩾2, , totally symmetric. In practice this means that, if we call I the É n–dimensional vector containing a generic permutation of the ⟨ ,n−2 ⟩ n n b-b = × dz3 D , z1, z3 Γ ; z3, z2; z1 . (58) indexes, we have that ,…, . It follows that Ê , I1 In É m ⊕⊕ z1 = − n z1 z3 nm z3 n! By expressing the electron density in terms of the equal times + × n ⋯ Green’s function as (1) = −iG 1, 1 , we compute the vari- 1, 1, , , . (56) ! I2 In−1 I1 In ̂(1) n I=1 ation n−2 . It yields É (z1) ⟨ ⟩

(2b) n e-b,l l,n−2 + Π z1 = n dx1 V x1 d34 G(1, 3) dz5 Γ 3, 4; z5 D z5, z1 G 4, 1 . (59) , Ê ⊕⊕ Ê Ê , n⩾2, ,,l É This mass operator is local and can be seen as a correlated correction to U. There is no analogous contribution to the mean–field potential in pure electronic systems, and to the best of our knowledge, it was not discussed in the context of e-b interactions. n−2 D Next we consider the Q variation ( ) (3) n n−2 b-e Π z1, z2 = i n dx1 V x1 Q z1 dz3 D z1, z3 Γ z3, z2; 1 , (60) , Ê ⊕⊕ Ê , , n⩾2, , É ( ) É 12 where we used the chain rule of differentiation and introduced a new vertex function with two bosonic and one fermionic coordinates: −1 −1 D z1, z2 D z1, z2 b-e , , ̂ (4) b-e Γ z1, z2; 3 ≡ − = − d4 ≡ d4 Γ z1, z2; 4 (4, 3) . (61) , (3) Ê ̂(4) (3) Ê , ⟨ ⟩ Notice, that similarly to the other mixed vertex, Eq. (36), we pulled out the common part of the functional derivative from the ⟨ ⟩ definition. The common part is given by the electron density response function

+ ̂ (1) G 1, 1 (1, 2) = = −i . (62) (2) (2) ⟨ ⟩ Γb-e ; 3 Other terms as well as contributions to the vertex function , z1, z2 from the bosonic self–energy will be considered in the next section. 2 Our next contribution results from the application of double diﬀerential operators n−2 and consists of three terms (z1)(1)

Π(4a) z , z = − n dx V n x dz dz dz D z , z , 1 2 Ê 1 ⊕⊕ 1 Ê 3 4 5 , 1 4 n⩾2, , ,, É É × Γb-b,n−2 ; Γb-e ; 1 , ; z4, z5 z1 D , z5, z3 , z3, z2 , (63a)

Π(4b) z , z = − n dx V n x dz dz dz D z , z , 1 2 Ê 1 ⊕⊕ 1 Ê 3 4 5 , 1 3 n⩾2, , ,, É É × Γb-e ; 1 Γb-b,n−2 ; , z3, z4 D, z4, z5 ,; z5, z2 z1 , (63b) b-e Γ z3, z2; 1 (4c) (4d) n , Π z1, z2 + Π z1, z2 = − n dx1 V x1 dz3 D z1, z3 . (63c) , , Ê ⊕⊕ Ê , n−2 n⩾2, , z1 É É Note that Eq. (63c) produces two terms, Π(4c) and Π(4d) as it will be demonstrated in the next section.

D. Vertex functions

In the preceeding sections we derived the equation of motion of the bosonic propagator D,, Eq. (50). Its important ingredients are the mean–ﬁeld potentials U, and Ξ,, Eq. (52a) and Eq. (52) and the bosonic mass operator Π consisting of eight terms Π(1), Π(2a), Π(2b), Π(3), Π(4a), Π(4b), Π(4c) and Π(4d). They, in turn, explicitly depend on three vertex functions: Γe-b, Γb-e, and Γb-b. Γe-e appears implictly through the response function , in Eq. (61). The vertex functions contain one, two or three external bosonic indices. In order to close the functional equations, we still need to express these vertex functions in terms of already deﬁned correlators. In order to do so, let us rewrite the vertex function as components of a Jacobian matrix:

−1 −1 G (1,2) G (1,2) e-e e-b,n n Γ (1, 2; 3) Γ 1, 2; z3 (3) (z3) (1, 2; 3) ≡ − −1 −1 = , (64) D (z1,z2) D (z1,z2) b-e b-b,n ⎡ , , ⎤ ⎡ Γ z1, z2; 3 Γ ; z1, z2; z3 ⎤ n , , ⎢ (3) (z3) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ and ⎣ ⎦ ⎣ ⎦

Σe(1,2) M(1,2) e-e e-b K (1, 5; 2, 4) K 1, z5; 2, z4 G(4,5) D (z4,z5) K (1, 5; 2, 4) ≡ , = . Σb (65) Π, (z1,z2) , (z1,z2) b-e 5; 4 b-b ; ⎡ ⎤ ⎡ K z1, z2, K z1, z5 z2, z4 ⎤ ⎢ G(4,5) D, (z4,z5) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ K ⎥ Here, is built of the vertex functions,⎣ and is the matrix of⎦ kernels.⎣ ⎦ 13

λ β β 5 3 3 e-b 3 Γ b-e b-b 3 4 Γ Γ (푙, 푛 − 2) 푛 μ ⊕ κ ⊕ α 푛 μ ⊕ κ ⊕ α ν 1 μ ν ν e-b Γ 1 2 1 2 1 2 푛 μ ⊕ κ ⊕ ν

4 1 푛 − 2 (a) Π(1) (b) Π(2a) (c) Π(2b) (d) Π(3)

β ϕ ψ ψ 3 ψ 4 λ β β 5 5 8 8 푛 μ ⊕ κ ⊕ α 3 6 푛 μ ⊕ κ ⊕ α 3 6 b-e b-b 1 1 b-b 푛 μ ⊕ κ ⊕ α Γ b-e 푛 μ ⊕ κ ⊕ α Γ δΓb-e δΓb-e Γ ν Γ ν Γb-b Γe-b δD δD 1 2 1 2 ϕ β ϕ 7 ν 7 ν 4 3 χ 2 5 2 5

(e) Π(4a) (f) Π(4b) (g) Π(4c) (h) Π(4d)

FIG. 6. A total of eight diagrams constituting the exact bosonic mass operator Π, (z1, z2).

2 2 2 4 6 2 4 6

i-j i-j i-e e-j i-b b-j Γ = Γ0 + K Γ + K Γ 3 3 3 3

1 1 1 5 7 1 5 7

FIG. 7. Diagrammatic form of the generalized Bethe–Salpeter equation. Black dots denote generic electron or boson indexes, i, j = (e, b).

푚 μ 푛 κ e-e e-b,푛 (푚, 푛) Γ0 (1, 2; 3) = Γκ (1, 2; 푧3) = |0 1, 2, 3 1, 2 3 (푚 − 2) (푚 − 2)

b-b Γμ,ν,κ(푧1, 푧2; 푧3) = b-e 푚 μ ⊕ λ ⊕ ν Γμ,ν (푧1, 푧2; 3)|0 = χ0(1, 3) μ ⊕ κ ⊕ ν 1, 2 3 1, 2, 3

FIG. 8. Diagrammatic representation of the lowest-order vertex functions, Eq. (67). .

The deﬁnitions introduced with Eq. (64) and Eq. (65) make clear that the electronic and bosonic degrees of freedom are totally symmetric and treated on equal footing. Indeed the rows and columns of the two matrices can be labelled with the kind of 14

e-e e-b input/output legs of the vertex/kernel . b-e b-b 4 5 For a given diagrammatic expression of the electronic and bosonic self-energies, the corresponding partial variations can be easily computed. Finally, we introduce the free term given by the derivatives of the mean-ﬁeld electronic and bosonic potentials:

(1) Φ(1) e-e e-b (1 − 2) n (1 − 2) Γ (1, 2; 3) Γ 1, 2; z (3) (z3) (1 2; 3) 0 0 3 0 , ≡ ≡ U (z1) Ξ (z1) , (66) Γb-e ; 3 Γb-b ; ⎡ , − , − ⎤ ⎡ 0 z1, z2 0 z1, z2 z3 ⎤ (3) z1 z2 n z1 z2 ⎢ (z3) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ with ⎣ ⎦ ⎣ ⎦ e-e Γ0 (1, 2; 3) = (1 − 2)(1 − 3), (67a) e-b e-b,n m m,n Γ 1, 2; z3 = Γ 1, 2; z3 = V x1 D z1, z3 (1 − 2), 0 0 , (67b) m, ó É b-e b-e ó m−2 m Γ z1, z2; 3 = Γ z1, z2; 3 = m óQ̂ z1 dx1 V x1 (1, 3) (z1 − z2), (67c) 0 , 0 Ê ⊕⊕ m, ( ) ó É ó n! b-b b-b,n ó 1 n−2 Γ0 z1, z2; z3 = Γ,; z1, z2; z3 = Q … z1 , , z1 − z2 z1 − z3 . (67d) 0 (n − 1)! I2 In−1 I1 In I=1 ( ) ó É ó These four quantities are related by a systemó of linear equations:

i-j i-j i-e e-j Γ (1, 2; 3) = Γ0 (1, 2, 3) + K (1, 5; 2, 4) G (4, 6) G (7, 5) Γ (6, 7; 3) + i-b 1 5; Γb-j ; 3 K , z2, z4 D , z4, z6 D, z7, z5 , z6, z7 , (68) where the summation and the integration over the repeated arguments is assumed, and the generic indexes are i, j = (e, b). This is the sought generalized Bethe-Salpeter equation (GBSE) for the vertex functions. b-e Now we are in the position to evaluate Eq. (63c), which, in fact, contains the variation n . Since is a solution of the complicated equation, its explicit form is not known. Therefore we use again the chain rule: Γb-e ; 1 Γb-e ; 1 Γb-e ; 1 , z3, z2 , z3, z2 D, z5, z6 , z3, z2 G (5, 6) = dz5dz6 + d56 n Ê n Ê (5 6) n z4 D, z5, z6 z4 G , z4 É, Γb-e ; 1 , z3, z2 = dz dz dz dz D z , z Γb-b,n z , z ; z D z , z Ê 5 6 Ê 7 8 , 5 7 ,; 7 8 4 , 8 6 , D, z5, z6 , É É b-e Γ z3, z2; 1 , e-b,n + d5678 G (5, 7) Γ 7, 8; z4 G (8, 6) . (69) Ê G(5, 6)

With this ingredient, the theory of interacting fermions and of electron-phonon interactions (see Sec. V.A of Giustino [1]). bosons is formally complete: the self-energies are expressed In the case of linear electron-phonon interactions the vertex is in terms of propagators and vertex functions. Note that we do renormalized solely due to the electron-electron interactions not have yet determining equations for higher-order bosonic (e.g. Fig. 2 of Leeuwen [27]). In the nonlinear case considered propagators and for the electron density response functions. here, the four vertex functions inevitably arise from a single n For the former, one would have to study the equation of motion electron-boson vertex, V (x) . At a marked diﬀerence with n for Q which, is rather compicated. Therefore, in Sec. VIII we these simpler theories, there are now four ways to renormalize use again the method of functional derivatives to recast and the bare vertex. In the next Sec. VII we consider what form the m,n D, in terms of the simplest propagators G and D. electron and the boson propagators take when the lowest-order The vertex funcions are related by the generalized Bethe- approximations (Eqs. 67) are adopted for the vertex functions. Salpeter equation which retains a surprisingly simple structure pertinent to the pure electronic case. The relation between bare and dressed vertex functions is a nontrivial point in the theory 15

self–consistent loop of Fig.9 by simply looking at the Qn G Φ + M as contractions of bosonic response function. Therefore, n⟨ ⟩ for the zeroth order vertexes will use the IBA, Q ≈ n−1,1 D z, z+ . (1…n−1),n 0 ⟨ ⟩ ó ó D푚,푛| Γi-j ó (푚,푛)>1 |(푖, 푗)={e, b} A. Electrons:ó the generalized Fan approximation

(푚, 푛)

D U + Π 푚 μ 푛 ν

M (1, 2) = FIG. 9. Schematic representation of the self-consistent cycle involv- 0 1 2 ing the diﬀerent components of the generalized Hedin’s equations. The dashed lines correspond to the generalized GW approximation where the vertex functions Γ are taken to their lowest order approxi- FIG. 10. The lowest–order fermionic self–energy. m,n m,n mation and D ≈ D 0 b-e . By using the zeroth–order Γ vertex function, Eq. (39) in ð the mass operator expression, Eq. (41), allows to introduce a generalization of the Fan approximation [1, 20]. Indeed we VII. LOWEST–ORDER APPROXIMATIONS FOR THE get M (1, 2) ≈ M0 (1, 2) with BOSONIC AND ELECTRONIC SELF-ENERGIES n m M0 (1, 2) = i V x1 V x2 n, m The solution of the Dyson equations for fermions and É É bosons are considerably more involved than in the case of lin- m,n × G (1, 2) D, z2, z1 . (70) ear electron–boson coupling. The equations have two level of 0 internal consistency that we schematically represent in Fig.9. ó ó Eq. (70) represents the generalization of the usual Fanó approx- ó Let us take the electronic case as an example. The Dyson imation which is known only in the linear coupling case (cor- = = 1 equation is itself nonlinear. For a given approximation for M responding to m n ). Its diagrammatic form is shown in Fig. 10. the Dyson equation must be solved and the new G plugged in n,m M for a new solution. This process must be continued up to In Eq. (70) D 0 is the zeroth–order approximation for the when self-consistency is reached. Besides this internal con- bosonic propagator which can be recast as a functional of non- e-b interacting bosonic propagators, as described in Sec.VIII B 1 Γ ð sistency the mass operator depends on the vertex function for some speciﬁc cases. and on the multi–boson propagators Dn,m. The usual approach to cut this self-consistent loop is based on approximating the vertexes to their lowest order and to take the independent bo- B. Bosons: a generalized polarization self-energy son approximation (IBA) for Dn,m. A similar procedure can be applied in the bosonic case. As sketched in Fig.9, the lowest order approximation for It is interesting to note that, at variance with the purely the bosonic self-energy is obtained by using the zeroth–order electronic case, the zeroth order bosonic vertex functions generalized vertex functions, Eq. (67), and the IBA (Dn,m ≈ n,m n n,m are still dependent on D through the Q terms appear- D 0) and IPA ( ≈ 0) for for bosons and electrons, re- ing in Eqs. (67c, 67d). This dependence is resolved in the spectively. ⟨ ⟩ ð These approximation must be used in Eq. (46), Eq. (52), Eq. (58), Eq. (59), Eq. (60) and Eq. (63). Eq.(63b) and Eq.(63c) need not be considered because it contains variations of other vertex functions. In total we obtain six terms:

(1) 1 1 Π z1, z2 = dx1 dx2 V x1 G (1, 2) G (2, 1) V x2 , (71a) , 0 Ê ó ó (2a) ó n b-b,n−2 Π z1 = n ̂ z1 D z1, z1 Γ z1 , (71b) , 0 ⊕⊕ , ,; 0 n⩾2, , ó É É ó ó ⟨ ⟩ ó ó ó 16

푙 λ (푛 − 4)

3 푛 μ ⊕ κ ⊕ α (푙, 푛 − 2) 1 μ ν 푛 μ ⊕ κ ⊕ ν

1 2 1 (1) (2a) (2b) (a) Π0 (b) Π0 (c) Π0

(푛 − 2) 푛 μ ⊕ κ ⊕ ν (푚 − 2) 3 (푛 − 2) 푚 β ⊕ ζ ⊕ ν 푛 μ ⊕ κ ⊕ α 푚 β ⊕ λ ⊕ ϕ 푚 β ⊕ λ ⊕ ν 푛 μ ⊕ κ ⊕ α 1 2 1 (푚 − 4) 1 2 (푛 − 4)

(푚 − 2) (3) (4a) (4b) (d) Π0 (e) Π0 (f) Π0

Π z1, z2 FIG. 11. Lowest–order approximation for the bosonic mass operators, , 0. ó ó ó (2b) n l l,n−2 + Π z1 = n dx1 V x1 d3 G (1, 3) V x3 D z3, z1 G 3, 1 , (71c) , 0 Ê ⊕⊕ Ê , 0 n, l, ó É É ó ó ó ó ó (3) n n−2 m m−2 Π z1, z2 = i (nm) dx1 V x1 Q dx2 D z1, z2 V x2 Q 0 (2, 1) , (71d) , 0 Ê ⊕⊕ Ê , ⊕⊕ n,m, ( ) ( ) ó , ,É ó ó

(4a) n b-b,n−2 Π z1, z2 = − n dx1 V x1 dz3D z1, z1 Γ z1 D z1, z3 , 0 Ê ⊕⊕ Ê , , ; 0 , n,, ,, ó É É ó ó m ó m−2 ó × m dx2 V x2 Qó D z1, z2 0 (2, 1) , (71e) Ê ⊕⊕ , m, , É ( )

(4b) n b-b,n−2 Π z1 = − n dx1 V x1 Γ z1 , 0 Ê ⊕⊕ ,; 0 n,, ,, , ó É ó ó m ó m−2 ó × m dx3dz3 V ó x3 Q D z1, z3 D z3, z1 0 (3, 1) . (71f) Ê ⊕⊕ , , m, É ( ) These equations are depicted diagrammatically in Fig. 11.

VIII. RESPONSE FUNCTIONS with n > 1 or m > 1. These response functions are more

The electronic and bosonic self–energies are written, also, m,n in terms of the response functions, (1, 2) and D, z1, z2 17 involved to calculate compared to the single–body case. In- A. Electronic response deed, in the purely electronic case, the single electronic GF satisfies the Dyson equation, while the two–bodies GF solves The electronic response, Eq. (62), can be rewritten in terms a more complicated Bethe–Salpeter equation [59]. This is the of the purely electronic vertex, Γe-e by means of the usual chain contracted form of the equation of motion for the electronic rule and connecting to the trace of G: vertex. G−1 (3, 4) However, when the electronic and bosonic degrees of free- (1, 2) = i d34 G (1, 3) G 4, 1+ dom are considered on equal footing as in Sec.VI D, the four Ê (2) vertex functions are mutually connected via a matrix integro- + = i d34 G (1, 3) G 4, 1 Γe-e (3, 4; 2) . (72) differential, Eq. (68) — the generalized Bethe-Salpeter equa- Ê tion. From Eq. (68) we do know that the equation of motion for Γe-e In the following sections our goal is investigate the form corresponds to the e-e channel of GBSE. In practice this means which take the electronic and the bosonic response functions that, at variance with the purely electronic case, it is not pos- as a consequence of the GBSE. In addition, thanks to the power sible to write the equation of motion for the response function of the Schwinger technique of functional derivatives, we will solely in terms of . Indeed, will depend, in general, on rewrite the equation of motion for the response function in Dn,m and, also, on the two mixed generalized response func- terms of single fermion and single boson self-energies. tions obtained by contracting Γb-e and Γe-b with bosonic and We have two aspects that complicate enormously the goal of fermionic operators. this section: (i) the electronic and bosonic response functions An alternative path, that we follow here, is to find an explicit are mutually dependent, (ii) the D may contain an arbitrary form of Γe-e and use Eq. (72) to obtain . From Eq. (68) we pair of incoming and outgoing bosonic lines, (n, m). know that

Γe-e (3, 4; 2) = Γe-e (3, 4, 2) + d5678 Ke-e (3, 6; 4, 5) G (5, 7) G (8, 6) Γe-e (7, 8; 2) 0 Ê

e-b b-e + d56 dz7z8 K 3, z6; 4, z5 D z5, z7 D z8, z6 Γ z7, z8; 2 . (73) Ê Ê , , , , É The ﬁrst two terms in Eq. (73) represent a generalization of the usual Bethe–Salpeter equation, widely used in the context of optical absorption [59], to the case of an arbitrary number of bosons that mediate the electron–hole interaction. The second term, instead, is new and represents a boson–mediated electron–hole propagation. The electron–hole pair annihilates producing a number of bosons, which are subsequently scattered giving rise to a particle-hole pair.

e-e e-e In order to visualize this important modifications we con- specific case where we use Γ (6, 7; 3) ≈ Γ0 (6, 7; 3) and sider the case where M is approximated with the generalized b-e b-e Γ z6, z7; 3 ≈ Γ z6, z7; 3 in the r.h.s. of Eq. (73). Fan form, Eq. (70), to evaluate Ke-e and Ke-b: 0 If we plug Eqs.(74), (75) in Eq. (73) and the resulting Γe-e in ó e-e e-e Eq. (72), a closed form expressionó for follows. K (3, 5; 4, 6) ≈ K0 (3, 4) (3, 5) (4, 6) ó In Fig. 12 we consider two interesting cases of Eq. (73): (a) n m m,n Ke-e = i V x3 V x4 D, z4, z3 , (74) the contribution from the first integral and evaluated with nm 0 n = m = 3, (b) the contribution from the second integral when É É ó e-b b-e ó n = m = 1 in K and n = 2 in Γ . ó and ó e-b 3 ; 4 B. Bosonic response K, , z5 , z6 e-b ≈ K z3; z4 z3 − z6 z4 − z5 , 0 We start from Eq. (21), applied to Dn+Δn,n. Thanks to this equation it is possible, for a given n, to reduce the evalua- = iG (3, 4) ó dx x V n x V m x Êó 3 4 ⊕ 3 ⊕ 4 tion of Dn+Δn,n to the one of Dn,n, DΔn,n and the functional nm ó É É derivative of Dn,n. If we assume Δn ⩽ n (the derivation can m−1,n−1 D z4, z3 . (75) be easily extended to the case Δn > n) Eq. (21) lowers the , 0 order of n + Δn. If we further apply the same procedure to ó Δn,n n,Δn Δn+(n−Δn),Δn We can now use Feynman diagrams to make theó different D = D = D the initial problem of evalu- contributions to more transparent. Let us consideró the ating Dn+m,n can be cast in an expression which includes only 18

2,2 2,2 3 ν 2,2 D D D F D,D , 1 , 1 , 3 . From this simple example it follows that it is enough to study diagonal bosonic response func- 4 tions and their functional derivatives in order to calculate any non–diagonal response functions. In the following we discuss the IBA and give as an example 2,2 3,3 1 2 the case of D and D . (a) 3 μ

4 1. The independent bosons approximation 1 ϕ The limit of independent bosons is instructive to understand 3 the actual number of diagrams that can be expected at any level 2 ϕ ⊕ η of the perturbative expansion. In order to evalute this number in the IBA we observe that: 1 7 2 (b) 1 ψ m,n = − Δ …Δ D, z1, z2 i  Q1 z1 Q z1 0 m 4 ó ( $ ó ΔQ z …ΔQ z . ó 1 2 n 2 (79) ó 0 FIG. 12. First order contributios to the electronic response function. %) e-e = 2 (a) is the contribution from K0 when n , while (b) is the con- with ⋯ 0 the thermal average corresponding to the free– Ke-b n = 1 M m,n tribution from 0 when . Both terms are calculated with bosons Hamiltonian. D reduces to the sum of all possible approximated with the generalized Fan approximation, Eq. (70). Al- , 0 contractions⟨ ⟩ of two bosonic operators. From simple combina- ready at this simple order of perturbation theory (a) shows the si- ó ó multaneous electron–hole interaction mediated by three bosons. (b), torics arguments we know tható the number of possible ordered instead, is totally new and shows how the electron–hole dynamics pairs of two operators out of a product of n ≥ 2 is given by the is temporarely transformed in a two bosons dynamics already in the number of the so-called chord diagrams [60] linear coupling. (n − 1)!! even n, Nn = (80) m,m 0 odd n. diagonal response function, of the form D with m an arbi- < trary integer m ⩽ n. 5,2 Let us take as an example the D z1, z2 case. From By doing simple diagrammatic expansion we see, indeed, that 2,2 Eq. (21) it follows that D z1, z2 produces a total of N4 = 3 terms. One , 0 of them is disconnected and corresponds to the complete 5,2 2,2 ó D z1, z2 = i D z1, z2 ó Δ Δ , 3 , contractions ofó the two terms Q1 z1 Q2 z1 and z1 0 ( ) 3 2 2 ΔQ z2 ΔQ z2 . + , 1 2 0 Q z1 D, z1, z2 ( 3,3 ) In theD z1, z2 case, instead, all contractions are con- ( ) 2 3,2 , 0 + Q z1 D z1, z2 , (76) , nected because there is always at least one contraction with ( ) ó = different time-arguments.ó This means that we have in total with ⊕ . We can now apply again Eq. (21) on ó 3,3 3 2 N6 = 15 terms. The explicit form of D z1, z2 will be , , 0 D, z1, z2 . It follows that given in the Sec.VIII B 3. ó n,m ó 3 2 2 2 We can therefore generally state that D z1, z2 is com- , = , , ó0 D, z1, z2 i 1 D , z1, z2 z1 posed of N + − N N connected diagrams. n m n m ó This simple combinatorics discussion allows usó to derive + 2 1,2 ó Q z1 D , z1, z2 , (77) some general rule on the strength of the nth order of the per- ( ) turbative expansion. As it is clear from the derivation done in = with ⊕ . A last application of Eq. (21) finally gives the precedent sections at any order of the perturbative expan- m,n n m 1,2 2,1 sion, a D, appears multiplied by V V . These potentials D z1, z2 = D z2, z1 = i D , z2, z1 . , , 1 1 include a 1∕ (n!m!) prefactor. 2 z2 Overall, we can deduce that the (n, m) order in the bosonic (78) propagator will be weighted with a Nn+m∕ (n!m!) prefactor. We have finally reduced D5,2 to an explicit functional of only When n increases this term decays fast enough to make the diagonal response functions and their derivatives: D5,2 = overall expansion controllable. 19

2 σ By applying the chain rule we get:

2 + 3 Q z1 D z1, z b-b,2 2,2 1,2 1 δD (푧 , 푧 ) Γ D z1, z2 = = i . (83) μν 1 2 , 2 2 (a) = μ ν ⟨ z2 ⟩ z2 δξ2(푧 ) σ 3 1 2 We can now follow the procedure for the electronic case and connect D2,2 to the Γb-b vertex:

D1,2 z1, z1 i = i dz dz D z , z 2 Ê 3 4 1 1 1 3 z2 É Γb-b,2 ; 1, 2; z3, z4 z2 D 2,2 z4, z1 . (84) Eq. (84) is represented diagrammatically in Fig. 13(a). 1 2 b-b,2 Eq. (68) provides the equation of motion for Γ 1, 2 that, in a similar way to Eq. (73), is written in terms of the pure bosonic (b − b) and the mixed boson–electron (b − e) vertex (b) functions. This equation involves the kernels Kb-b and Kb-e. We can now follow the same path of the purely electronic case and use the lowest-order bosonic self–energy, Eq. (71), to derive the corresponding expression for the b–b and b–e kernels and, consequently, of Γb-b,2. Two representative diagrams contributing to D2,2 are shown in Fig. 13(b) and Fig. 13(c). The IBA for D2,2 can be easily evaluated by using the zeroth Γb-b,2 (c) 1 2 order expression for . From Eq. (57) we know that when n = 2 we have only n!∕(n − 1)! = 2 terms,

b-b,2 Γ z1, z2; z3 = 1, 2; 0 FIG. 13. (a) Digrammatic representation of the ﬁrst order functional z −z z −ó z + , derivative of D, Eq. (83). (b) and (c) represent two terms contributing 1 2 1 ó 3 1,1 2,2 1,2 2,1 (85) 2,2 ó to D which show the connection between the single–boson self– energy and the bosonic response function. Indeed the diagram (b) which gives Π(3) b-b 0 comes from the scattering term in K due to D . Similarly dia- (1) Π 0 gram (c) is induced by the contribution of toðKb-e. Both terms 2,2 = + G D, z1, z2 i D1,1 z1, z2 D2,2 z2, z1 are treated, in (b) and (c), at the ﬁrst order in the generalized Bethe- 0 ð ó Salpeter equation, Eq. (68). ó ó D1,2 z1, z2 D2,1 z2, z1 . (86) ó 2. The two–bosons case Eq. (86) coincides with the expression that can be derived by using the diagrammatic approach.

The case of D2,2 can be easily worked out following an approach similar to what has been used in Sec.VIII A. From 3. The three–bosons case Eq. (35) we know that

3,3 2 In the three–bosons case the calculation of D may appear Q z1 2,2 to be prohibitively complicated. Still, the present scheme al- D z1, z2 = . (81) , 2 lows, via the functional derivative approach to derive it in an z2 ⟨ ⟩ elegant and compact way. We start by applying Eq. (21) to D3,3: 2 At the same time we can rewrite the Q z1 in terms of the single–body GF, using Eq. (20): 3,3 2 1,3 D, z1, z2 = i + Q z1 Dt, z1, z2 , ⟨ ⟩ 2 z ⎡ 1 ( )⎤ Q2 z = iD z , z+ . 1 12 1 1 (82) ⎢ ⎥ (87) ⎢ ⎥ ⎣ ⎦ ⟨ ⟩ 20

to the IBA expression for D3,3 which is, indeed, composed of

2 a total of 15 terms. δ D(푧1, 푧2) (a) = b-b,2 b-b,2 δξ2(푧 )δξ2(푧 ) Γ Γ 1 2 1 2 IX. CONCLUSIONS

In this work we applied Schwinger’s variational derivative 2 technique to calculate the coupled electronic and bosonic dy- b-b,2 b-b,2 + Γ Γ namics induced by an electron–boson Hamiltonian with cou- 1 pling linearly proportional to the electronic density ̂n(x) and to all orders in the bosonic displacement Q. The complex and coupled electronic and bosonic dynamics is formulated in the form of a system of functional rela- 2 2 1 2 single boson (b) ⟨Q (푧1)⟩⟨Q (푧2)⟩D(푧1, 푧2) = tions between the dressed electronic G (1, 2), the D, z1, z2 propagators and the generalized electronic and 2 Σe (1 2) Σb bosonic self–energies, , and z1, z2 . These are expressed as closed functionals of the electron Γb-b,2 δD(푧1, 푧2) (c) ⟨Q2(푧 )⟩ = density-density response , the multi–boson response func- 1 δξ2(푧 ) n,m e-e b-e e-b 2 1 tions D , and four different vertex functions: Γ , Γ , Γ and Γb-b. These vertex functions are shown to have either a b-e e-b (2) mixed electron–boson character (Γ and Γ ), or a purely electronic (Γe-e) and bosonic (Γb-b) character. The exact equa- (d) D2,2(푧 , 푧 )D(푧 , 푧 ) = 1 2 2 1 tions of motion for all these quantities are formally derived. 1 2 Sound and controlled approximations are also proposed in order to make the calculations feasible. FIG. 14. Diagrammatic representation of the terms in Eq. (89) con- The present formulation allows us to tackle the very am- tributing to D3,3. bitious problem of deriving using the Schwinger’s technique coupled equations of motion for the electronic and bosonic response functions and provide several interesting conclusions with ≡ ⊕ t. By using Eq. (21) again we get that and new concepts. We extend to the nonlinear e–b interaction known concepts 1,3 3,1 like the Debye–Waller potential and the Fan approximation. Dt, z1, z2 = D,t z2, z1 = We further extend the Bethe–Salpeter equation to a 2 × 2 non– linear system of integro-differential equations for the four ver- + 2 tex functions. Thanks to this equation we show that there is no i 2 Q z2 Ds,t z2, z1 , (88) L z2 M simple way to decouple the electronic and bosonic dynamics. ( ) We demonstrate, by using simple diagrammatic examples, that where ≡ ⊕ s. Eq. (87) and Eq. (88) show that D3,3 is electrons and bosons can equally well mediate the electron– composed of five terms hole and boson–boson interaction. The present scheme, indeed, demonstrates a full and deep symmetry between the elec- 2 tronic and bosonic degrees of freedom. 3,3 = 2,2 − The final result is an important generalization of the well– D, z1, z2 D, z1, z2 2 2 L z1 z2 known Hedin’s equations with a wealth of potential applica- tions in different areas of condensed matter physics, optics and + 2 2 + 2 Q z1 Q z2 i Q z2 2 chemistry. z1 ( )( ) ( ) + 2 ACKNOWLEDGEMENTS i Q z1 2 Ds,t z2, z1 . (89) z2 M ( ) AM acknowledges the funding received from the Euro- The construction of diagrammatic form of Eq. (89) can be done pean Union project MaX Materials design at the eXascale by using a simple diagrammatic form of the Eq. (84), as shown H2020-EINFRA-2015-1, Grant agreement n. 676598 and in Fig. 13(a). This shows that any of the functional derivatives Nanoscience Foundries and Fine Analysis - Europe H2020- appearing in Eq. (89) can be rewritten in terms of a second INFRAIA-2014-2015, Grant agreement n. 654360. Y.P. ac- order b–b vertex function. In this way it is possible to rewrite knowledges funding of his position by the German Research D3,3 in terms of known quantities, as shown in Fig. 14. All Foundation (DFG) Collaborative Research Centre SFB/TRR Γb-b,2 ≈ Γb-b,2 diagrams represented in Fig. 14 reduce, when 0 173 “Spin+X”. ó ó ó 21

Appendix A: The mean–field treatment of the electron–electron In introducing Eq. (B1) it is important to stress that Hn (R) interaction includes both the kinetic and nuclear–nuclear interaction while We–n (R) represents the electron–nuclei interaction, whose ex- In order to describe how we treat the correlation induced pansion in the atomic displacements leads, as well known, to by the electron–electron interaction let us start from the full the diagrammatic expansion. Moreover, in the spirit of Ap- Hamiltonian in the first quantization and make explicit the dis- pendixA we have assumed, in Eq. (B1), to use DFT to de- tinction between dressed and undressed operators: scribe the effect of the electron–electron correlation via the well–known exchange–correlation potential. = 0 + 0 + 0 + R H He Hb He–b He–e, (A1) We split, now, the generic atomic position operator, I , in its reference plus displacement with the 0 superscript indicating bare operators. Indeed the dressing of the different components of the Hamiltonian (when R I ≡ RI 1̂ + ΔR I . (B2) possible) is a product of the dynamics and cannot be, a priori, inserted from the beginning. The Cartesian components of ΔR I play the role of the bosonic In Eq. (A1) we introduced coordinate operators, Q. We can, indeed, write that 1 H = v x − x , (A2) ̂ −1∕2 e–e 2 i j ΔRI = NMI Ω ( I) Q, (B3) i≠j É É ð with v the bare Coloumb potential. It is well documented in with N the number of atoms in the system, MI the mass of the literature that one of the effects of He–e is to screen itself atom I, is the phonon mode polarization vector. We assume and all other interactions, including the e–b one. This has been here, for simplicity, a finite system that can be generalized to extensively demonstrated, for example, in Ref. 28. an periodic solid using periodic boundary conditions. The path we take here is, therefore, to embody He–e in a Our initial system is, therefore, characterized by a set of 0 dressed, electronic and bosonic single–particle states with en- mean–field correction to He and, consequently, dressing of 0 0 ergies { } and frequencies {Ω }. We have in total 3N H and H : i b e–b bosonic coordinates. (R) 0 We have now all ingredients to expand the We–n in terms H ⇒ H + V + H + H , (A3) e mf b e–b of ̂ (x) and Q. Indeed we can, formally, write that 0 ( ) with He = H + Vmf. Eq. (A3) is the connection with Eq. (1). (R) = n (R) = e We–n We-n V n The specific form of mf depends on the physical problem. An É example is to use DFT, where Vmf = VHxc is the Hartree plus † (n) n = dx ̂ (x) V (x) ̂ (x) Q, (B4) the Kohn–Sham exchange–correlation potential [61]. In this n Ê 0 0 É É case also the dressing of He–b and He-e is well–known and widely documented. In the case of the electron–phonon prob- with lem, for example, the self–consistent dressing of the electron– nuclei interaction is described by the Density–Functional per- n turbation theory (DFPT) [62, 63]. (n) (x) = (x − R) V )i Vscf . (B5) H =1 I Çi eq Appendix B: Connection with the electron–phonon problem In Eq. (B5) Vscf is the dressed DFPT electron–nuclei potential and the derivative is taken at the equilibrium position R = R. A specific physical application of the present theoretical scheme is represented by the coupled electron–phonon system. This is a very wide field with a wealth of application in several Appendix C: Proof of Eq. branches of physics. (11c) The Hamiltonian of the coupled electron–phonon system is obtained by starting from the total Hamiltonian of the system, The equation of motion for P can be derived by using some that we divide in its independent bare electronic He, nuclear care. Indeed Eq.(10b) implies that Hn (R), electron–nucleus (e–n) We–n (R) parts m m P z1 , Q z1 = P z1 , Q z1 H (R) = He + Hn (R) + We–n (R) , (B1) − i L i=1 M− Ç where R is a generic notation representing positions of the nu- m m = (− ) clei. The notation used in this paper is the same adopted in i , j Q i z1 . (C1) j=1 i≠j,i=1 Ref. 28. É Ç 22

If we now plug Eq. (C1) into the P z1 , H z1 commu- We start by expanding the three terms resulting from the func- − tional derivative of the three components of D: tator we get d (−i) P z1 , H z1 = P z1 = −ΩQ z1 − dz1 m m m−k n − , z1 Q z1 . (C2)  Q z1 Q z2 = j i k m, j=1 i≠j,i=1 z1 É É Ç ( ) m−k n k Now we reorder the components of vector ( is a fully sym-  Q z1 Q z2 Q z1 metric tensor) so that ( ) − m−k n k  Q z1 Q z2 Q z1 . (D2) z1 = … … z1 = … z1 . , j 1, , j−1,, j+1, , m 1, , m−1, ( )( ) (C3)

We now rename by introducing the m−1 dimensional vector The second and third term are due to the derivative of the two ≡ , … , 1 m−1 . Thanks to Eq. (C3) we have that single displacement operator averages: m = m−1 Q i z1 Q z1 , (C4) i≠j,i=1 Ç and we ﬁnally get m−k m−k k Q z1 = Q z1 Q z1 k z1 d −1 ( ) ( ) P z = −Ω Q z − m ̂ m z Qm z . 1 1 ⊕ 1 1 m−k k dz1 m, − Q z1 Q z1 . (D3) É ( )( ) (C5) and

Appendix D: Proof of Eq. (21)

We start by observing that n n k Q z2 =  Q z2 Q z1 k z1 i m−k,n ( ) ( ) D z1, z2 = k , − n k z1 Q z2 Q z1 . (D4) m−k n ( )( )  Q z1 Q z2 k z1 ( ) m−k n − Q z1 Q z2 . (D1) If now we put together all components of Eq. (D1) we get ( )( )

i m−k,n m n m n D z1, z2 =  Q z1 Q z2 − Q z1 Q z2 k , ⊗ ⊗ z1 ( ) ( )( ) − m−k n k + m−k k n  Q z1 Q z2 Q z1 Q z1 Q z1 Q z2 ( )( ) ( )( )( ) − n k m−k + n k m−k  Q z2 Q z1 Q z1 Q z2 Q z1 Q z1 . (D5) ( )( ) ( )( )( ) Eq. (D5) ﬁnally gives

i m−k,n m,n k m−k,n m−k k,n D z1, z2 = D z1, z2 − Q z1 D z1, z2 − Q z1 D z1, z2 . (D6) k , ⊗ , , , z1 ( ) ( )

In Eq. (D5) we have used the fact that Eq.(D6) proves Eq. (21). m−k n k = m n  Q z1 Q z2 Q z1  Q ⊗ z1 Q z2 . ( ) ( (D7)) 23

Appendix E: Summary of deﬁnitions f. Bosonic mass operator

a. Bosonic coordinates and the interaction vertex (I) Π z1, z2 = Π z1, z2 n , , I=(1,2a,3,4a,4b,4c,4d) n = É Q Qi . (2b) =1 + Π z1 z1 − z2 . Çi , n 1 n (x) = (x) V )i Ve–b . n! H =1 I g. Vertex functions Çi eq n † n ̂ = dx ̂ (x) V (x) ̂ (x) . Ê G−1 (1, 2) Γe-e (1, 2; 3) = . b. Auxiliary ﬁelds (3) −1 e-b,k G (1, 2) Γ 1, 2; z3 = . Ĥ (z) = Ĥ + n (z) Qn + dx (x, z) ̂ (x) . k (3) , Ê n, −1 É e-b,k G (1, 2) Γ 1 2; = c. Correlators and electronic response , z3 . Qm z 3 G(1, 2) ≡ −i  { ̂ (1) ̂ † (2)} . (−1 ) D, z1, z2 Γb-e z , z ; 3 = . n,m n m , 1 2 (3) D, z1, z2 ≡ −i  ΔQ z1 ΔQ z2 . −1 ( $ %) D z1, z2 ̂ (1) Γb-b,k ; = , (1, 2) ≡ . ,; z1, z2 z3 k . (2) (3) ⟨ ⟩ h. Kernels d. Mean–ﬁeld potentials

e m m e-e Σ (1, 2) Φ(1) = V x1 Q z1 . K (1, 5; 2, 4) = . m, G (4, 5) É ( ) e-b M (1, 2) U z = n ̂ n z Qn−2 z . K 1, z5; z2, 4 = . , 1 ⊕⊕ 1 1 D z , z n, , 4 5 É ( ) Π z , z ⟨ ⟩ b-e , 1 2 e. Electronic mass operator K z1, 5; 2, z4 = . G (4, 5) b Σ z1, z2 M (1, 2) = i d3 dz V n x G(1, 3) b-b , Ê Ê 4 1 K z1, z5; z2, z4 = . n, m, D, z4, z5 É É e-b,m m,n ×Γ 3, 2; z4 D, z4, z1 .

[1] F. Giustino, Rev. Mod. Phys. 89, 015003 (2017). [11] S. Li, E. A. Nowadnick, and S. Johnston, Phys. Rev. B 92, [2] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosoniza- 064301 (2015). tion and strongly correlated systems (Cambridge University [12] C. P. J. Adolphs and M. Berciu, Phys. Rev. B 89, 035122 (2014). Press, Cambridge, U.K.; New York, NY, 1998). [13] C. P. J. Adolphs and M. Berciu, Phys. Rev. B 90, 085149 (2014). [3] G. Mahan, Many-particle physics, 3rd ed. (Kluwer Aca- [14] M. Berciu, Phys. Rev. Lett. 97, 036402 (2006). demic/Plenum Publishers, New York, 2000). [15] S. Li and S. Johnston, Eurphys. Lett. 109, 27007 (2015). [4] E. A. Muljarov and R. Zimmermann, Phys. Rev. Lett. 93, [16] M. Först, C. Manzoni, S. Kaiser, Y. Tomioka, Y. Tokura, 237401 (2004). R. Merlin, and A. Cavalleri, Nature Phys. 7, 854 (2011). [5] B. I. Lundqvist, Phys. Kondens. Mater. 9, 236 (1969). [17] A. Y. Liu, I. I. Mazin, and J. Kortus, Phys. Rev. Lett. 87, 087005 [6] D. C. Langreth, Phys. Rev. B 1, 471 (1970). (2001). [7] E. Mariani and F. von Oppen, Phys. Rev. Lett. 100, 076801 [18] R. Mankowsky, A. Subedi, M. Först, S. O. Mariager, M. Chol- (2008). let, H. T. Lemke, J. S. Robinson, J. M. Glownia, M. P. Minitti, [8] T. Gunst, K. Kaasbjerg, and M. Brandbyge, Phys. Rev. Lett. A. Frano, M. Fechner, N. A. Spaldin, T. Loew, B. Keimer, 118, 046601 (2017). A. Georges, and A. Cavalleri, Nature 516, 71 (2014). [9] T. Holstein, Ann. Phys. 8, 325 (1959). [19] P. B. Allen and V. Heine, J. Phys. C 9, 2305 (1976). [10] C. P. J. Adolphs and M. Berciu, Eurphys. Lett. 102, 47003 [20] M. Cardona, Solid State Communications 133, 3 (2005). (2013). [21] F. Giustino, S. G. Louie, and M. L. Cohen, Phys. Rev. Lett. 105, 24

265501 (2010). [43] D. N. Basov, R. D. Averitt, D. van der Marel, M. Dressel, and [22] E. Cannuccia and A. Marini, Phys. Rev. Lett. 107, 255501 K. Haule, Rev. Mod. Phys. 83, 471 (2011). (2011). [44] A. S. Moskalenko, Y. Pavlyukh, and J. Berakdar, Phys. Rev. A [23] S. Poncé, Y. Gillet, J. Laflamme Janssen, A. Marini, M. Ver- 86, 013202 (2012). straete, and X. Gonze, J. Chem. Phys. 143, 102813 (2015). [45] E. Perfetto, D. Sangalli, A. Marini, and G. Stefanucci, J. Phys. [24] G. Antonius, S. Poncé, E. Lantagne-Hurtubise, G. Auclair, Chem. Lett. 9, 1353 (2018). X. Gonze, and M. Côté, Phys. Rev. B 92, 085137 (2015). [46] G. Stefanucci and R. van Leeuwen, Nonequilibrium Many-Body [25] B. Monserrat, N. D. Drummond, and R. J. Needs, Phys. Rev. B Theory of Quantum Systems: A Modern Introduction (Cam- 87, 144302 (2013). bridge University Press, Cambridge, 2013). [26] I. Errea, M. Calandra, and F. Mauri, Phys. Rev. Lett. 111, [47] N. E. Dahlen and R. van Leeuwen, Phys. Rev. Lett. 98, 153004 177002 (2013). (2007). [27] R. van Leeuwen, Phys. Rev. B 69, 115110 (2004). [48] P. Myöhänen, A. Stan, G. Stefanucci, and R. van Leeuwen, [28] A. Marini, S. Poncé, and X. Gonze, Phys. Rev. B 91, 224310 Phys. Rev. B 80, 115107 (2009). (2015). [49] M. P. von Friesen, C. Verdozzi, and C.-O. Almbladh, Phys. Rev. [29] R.M.Dreizler and E.K.U.Gross, Density Functional Theory Lett. 103, 176404 (2009). (Springer-Verlag, 1990). [50] M. Schüler, J. Berakdar, and Y. Pavlyukh, Phys. Rev. B 93, [30] C. Verdi and F. Giustino, Phys. Rev. Lett. 115, 176401 (2015). 054303 (2016). [31] A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. [51] N. Schlünzen, S. Hermanns, M. Bonitz, and C. Verdozzi, Phys. 105, 123002 (2010). Rev. B 93, 035107 (2016). [32] R. Requist and E. K. U. Gross, Phys. Rev. Lett. 117, 193001 [52] P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959). (2016). [53] In the case of out–of–equilibrium dynamics the one boson [33] A. A. Abrikosov, L. P. Gor’kov, and I. Y. Dzyaloshinskii, Meth- displacement operator fulfills an harmonic oscillator equation ods of quantum field theory in statistical physics, rev. english ed. driven by the time-dependent electron density [50]. ed. (Dover Publications, New York, 1975). [54] L. Hedin and S. Lundqvist, in Solid State Physics, Vol. 23, edited [34] Y. Pavlyukh, Sci. Rep. 7, 504 (2017). by D. T. a. H. E. Frederick Seitz (Academic Press, New York, [35] R. van Leeuwen and G. Stefanucci, Phys. Rev. B 85, 115119 1970) pp. 1–181. (2012). [55] There is discrepancy in the literature about the sign in its def- [36] G. Strinati, Riv. Nuovo Cimento 11, 1 (1988). inition, for instance Strinati [36] defines it with a plus sign, [37] L. Hedin and S. Lundqvist (Academic Press, 1970) pp. 1 – 181. Eq. (4.4). Here we follow a convention in the book of Stefanucci [38] E. Papalazarou, J. Faure, J. Mauchain, M. Marsi, A. Taleb- and van Leeuwen [46], Eq. (12.34). Ibrahimi, I. Reshetnyak, A. van Roekeghem, I. Timrov, N. Vast, [56] G. Mahan, Many-Particle Physics (Plenum, 1990). B. Arnaud, and L. Perfetti, Phys. Rev. Lett. 108, 256808 (2012). [57] J. D. W. Alexander L.Fetter, Quatum Theory of Many-Particle [39] L. X. Yang, G. Rohde, T. Rohwer, A. Stange, K. Hanff, C. Sohrt, Systems (McGraw-Hill, 1971). L. Rettig, R. Cortés, F. Chen, D. L. Feng, T. Wolf, B. Kam- [58] L. Paulatto, I. Errea, M. Calandra, and F. Mauri, Phys. Rev. B ble, I. Eremin, T. Popmintchev, M. M. Murnane, H. C. Kapteyn, 91, 054304 (2015). L. Kipp, J. Fink, M. Bauer, U. Bovensiepen, and K. Rossnagel, [59] G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 Phys. Rev. Lett. 112, 207001 (2014). (2002). [40] R. Huber, C. Kübler, S. Tübel, A. Leitenstorfer, Q. T. Vu, [60] Y. Pavlyukh and W. Hübner, J. Math. Phys. 48, 052109 (2007). H. Haug, F. Köhler, and M.-C. Amann, Phys. Rev. Lett. 94, [61] R. M. Dreizler and E. K. U. Gross, Density Functional Theory 027401 (2005). an Approach to the Quantum Many-Body Problem (Springer [41] C. Lemell, S. Neppl, G. Wachter, K. Tőkési, R. Ernstorfer, Berlin Heidelberg, Berlin, Heidelberg, 1990). P. Feulner, R. Kienberger, and J. Burgdörfer, Phys. Rev. B 91, [62] X. Gonze, Phys. Rev. A 52, 1096 (1995). 241101 (2015). [63] S. Baroni, S. de Gironcoli, A. D. Corso, and P. Giannozzi, Rev. [42] M. Schüler, Y. Pavlyukh, P. Bolognesi, L. Avaldi, and J. Berak- Mod. Phys. 73, 515 (2001). dar, Sci. Rep. 6, 24396 (2016).