Sum Rules and Vertex Corrections for Electron-Phonon Interactions

Sum rules and vertex corrections for electron-phonon interactions

O. Rösch,∗ G. Sangiovanni, and O. Gunnarsson Max-Planck-Institut für Festkörperforschung, Heisenbergstr. 1, D-70506 Stuttgart, Germany

We derive sum rules for the phonon self-energy and the electron-phonon contribution to the electron self-energy of the Holstein-Hubbard model in the limit of large Coulomb interaction U. Their relevance for finite U is investigated using exact diagonalization and dynamical mean-field theory. Based on these sum rules, we study the importance of vertex corrections to the electron- phonon interaction in a diagrammatic approach. We show that they are crucial for a sum rule for the electron self-energy in the undoped system while a sum rule related to the phonon self-energy of doped systems is satisfied even if vertex corrections are neglected. We provide explicit results for the vertex function of a two-site model.

PACS numbers: 63.20.Kr, 71.10.Fd, 74.72.-h

I. INTRODUCTION the other hand, a sum rule for the phonon self-energy is fulfilled also without vertex corrections. This suggests that it can be important to include vertex corrections for Recently, there has been much interest in the pos- studying properties of cuprates and other strongly corre- sibility that electron-phonon interactions may play an lated materials. important role for properties of cuprates, e.g., for The Hubbard model with electron-phonon interaction 1–3 superconductivity. In particular, the interest has fo- is introduced in Sec. II. In Sec. III, sum rules for the cused on the idea that the Coulomb interaction U might electron and phonon self-energies are derived focusing on enhance effects of electron-phonon interactions, e.g., due the limit U and in Sec. IV we numerically check 4 to interactions with spin fluctuations. Effects of the their accuracy→ for ∞ large but finite U. These sum rules electron-phonon coupling are described by the electron- then form the basis for the discussion of the effects of ep phonon part Σ of the electron self-energy Σ and the vertex corrections in Sec. V. The results are illustrated phonon self-energy, Π. We have previously derived sum in Sec. VI for a two-site model. rules for these quantities for the t-J model, and used the sum rules to demonstrate that the electron-phonon in- ep teraction influences Σ and Π in quite different ways for II. HUBBARD MODEL IN THE LIMIT OF strongly correlated systems.5 Here, we extend this work LARGE U and derive sum rules for the related half-filled Holstein- Hubbard model in the limit of a large U. We obtain sum Strongly correlated electrons are often described by the ep rules for Σ integrating either over all frequencies or Hubbard model only over frequencies in the photoemission energy range. The latter sum rule shows a very strong dependence on H = ε n t (c† c +H.c.)+U n n , (1) d iσ − iσ jσ i↑ i↓ U, increasing by a factor of four in going from U = 0 i,σ i X hi,jXi,σ X to U = . From numerical calculations, using both exact diagonalization∞ and dynamical mean-field theory, we where εd is the level energy, t(> 0) is the hopping integral show that the U = result is also relevant for inter- between nearest-neighbor sites i, j , U is the Coulomb ∞ h i † mediate values of U 3D, where D is half the band repulsion between two electrons on the same site, ciσ cre- ≈ † width. ates an electron on site i with spin σ, and niσ = ciσciσ. In a diagrammatic many-body language, the electron- In addition, we introduce an electron-phonon interaction phonon interaction could be enhanced by U via cor- 1 † iq·Ri 6–9 H = gq(n 1)(b + b )e , (2) rections to vertex functions or dressing of Green’s ep √ i − q −q 10 et al.7 8 N i,q functions. Huang and Koch and Zeyher studied X how U changes an effective vertex function in the static where N is the number of sites, n = n + n , and b† limit (ω = 0) and found a suppression, although it was i i↑ i↓ q creates a phonon with the wave vector q and energy ω concluded in Ref. 7 that the suppression is reduced for q as described by the free phonon Hamiltonian a large U and a small wave vector q. Often, one is not only interested in these special cases but in properties H = ω b† b . (3) q ph q q q that depend on integrals over ω and containing ver- q tex functions. Here, we study to what extent the sum X rules above are fulfilled when vertex corrections are ne- We assume a q-dependent on-site coupling with the ep glected. We find that the sum rule for Σ , integrating strength gq. The coupling to hopping integrals is ne- over the photoemission energy range, is underestimated glected, which, e.g., has been found to be a good approx- by a factor of four if vertex corrections are neglected. On imation for the planar oxygen (half-)breathing mode in 2

11 the high-Tc cuprates. The special case of a Holstein In the limit of large U, states with double occupancy coupling is obtained by setting gq = g and ωq = ωph for can be projected out. If certain terms are assumed to be all q. negligible,12 this leads to the t-J model13 as an effective To describe photoemission (PES) and inverse photoe- low-energy model for Eq. (1). Since double occupancy is mission (IPES) within the sudden approximation, we excluded in the t-J model, its electron Green’s function consider the one-electron removal (-) and addition (+) has no contribution from inverse photoemission for the − spectra undoped system, i.e., Gt-J (k,z) = Gt-J (k,z). If we assume that the photoemission spectra of the Hubbard and e−βEm A−(k,ω)= n c m 2δ(ω+ ), (4) the t-J model are identical (apart from a trivial energy Z |h | kσ| i| En−Em mn shift U/2), we obtain X ≈ e−βEm G−(k,z)= G (k,z + U/2). (12) A+(k,ω)= n c† m 2δ(ω+ ), (5) t-J Z |h | kσ| i| Em−En mn X The inverse photoemission in the half-filled Hubbard model can be related to the photoemission in the un- where the energy ω is measured relative to the chemi- doped t-J model because of particle-hole symmetry, cal potential µ, m and n are eigenstates of the grand | i | i + canonical Hamiltonian = H µ N with eigenenergies G (k,z)= Gt-J (kAF k,U/2 z) (13) H − h i − − m/n, Z = m exp( β m) is the corresponding parti- Etion sum, and β = 1−/TE. We assume that there is no for a two-dimensional square lattice where kAF = P (π/a,π/a). Terms of order t U are neglected in the explicit dependence on the electron spin σ. From the ≪ total spectral density energy shifts U/2 in Eqs. (12) and (13). The electron± self-energy in the t-J model is defined − + A(k,ω)= A (k,ω)+ A (k,ω), (6) via Gt-J (k,z)=0.5/[ω Σt-J (k,z)] and it follows from Eqs. (10), (12), and (13)− that we obtain the one-electron Green’s function − ∞ Σ (k,z) = Σt-J (k,z + U/2), (14) A(k,ω) + G(k,z)= dω (7) Σ (k,z) = Σt-J (kAF k,U/2 z). (15) −∞ z ω − − Z − In the following we derive sum rules for the electron which depends on the electronic wave vector k and the self-energy of the Hubbard model, and then use the ap- complex energy z. It is related to the electron self-energy proximate relations above to make contact to the t-J Σ(k,z) via the Dyson equation model. Starting from a sum rule for the charge-charge re- 1 sponse function in the t-J model, we also obtain a corre- G(k,z)= , (8) sponding (approximate) sum rule for the Hubbard model z εk Σ(k,z) − − by using these relations. ik(R −R ) where εk = ε t (e i j + H.c.)/N is the bare d − hi,ji electronic dispersion. We can split up G(k,z) into two P III. SUM RULES parts,

G(k,z)= G+(k,z)+ G−(k,z), (9) In this section, we discuss sum rules for the electron and phonon self-energies. They will allow us to ad- where the (I)PES Green’s functions G±(k,z) are deﬁned dress the importance of vertex corrections to the electron- by replacing the spectral density in Eq. (7) by A±(k,ω). phonon interaction in a diagrammatic treatment of the We also deﬁne corresponding self-energies by Hubbard model.

a± G±(k,z)= k , (10) z ε± Σ±(k,z) A. Electron self-energy − k − ± ± where ak is the integrated weight and εk absorbs energy- First, we consider a sum rule which gives the total independent contributions to the self-energy. We con- weight of the spectral density of the electron self-energy sider the half-filled system and choose εd = U/2 to Σ(k,z) integrated over all frequencies. − have explicit particle-hole symmetry. Then, as detailed ∞ ± ± 1 + 2 in the appendix, ak 1/2, εk U/2+εk for U . k → →± → ∞ dω Im Σσ( ,ω i0 )= U ni−σ (1 ni−σ ) In this limit, one finds from inserting Eqs. (8) and (10) π −∞ − h i − h i Z into Eq. (9) to leading order in 1/U gq † † +2U (b + b )ρq gq 0 b + b n ± √ q −q −σ = i i i−σ Im Σ(k,z U/2)=2 Im Σ (k,z U/2), (11) " q N h i− h ih i# ≈± ≈± X 1 2 † 2 2 † 2 relating the spectral densities of the different self-energies + gq b + b g b + b , (16) N q −q q=0 i i for energies z U/2. q | | h| | i− h i ≈± X 3

−iqRi √ 2 2 2 where we defined ρqσ = i niσe / N and bi = 2g −3g 2g Hubbard b eiqRi /√N. σ is the electron spin for which the q q P self-energy is calculated but in our case the results do g2/2 g2/2 no VC notP depend on it. The sum rule in Eq. (16) which is derived in App. A 1 using spectral moments is valid for any 2 U and interestingly, it is independent of the electronic g t−J wave vector k. For a Holstein coupling, it simplifies to 2 2 g /2 g /2 non−int. 1 ∞ dω Im Σ (k,ω i0+)= U 2 n (1 n ) ω π σ − h i−σi − h i−σi Z−∞ PES −U/4 0 U/4 IPES † 2 † 2 +2Ug (bi + bi )ni−σ + g (bi + bi ) . (17) h i h i FIG. 1: Weights obtained by integrating Im Σep(k,ω−i0+)/π In the derivation of Eqs. (16) and (17), we have assumed over the indicated frequency intervals for the half-filled Hub- translation invariance so the expectation values on the bard and undoped t-J models. Also shown are the result for right hand side of the equations do not depend on the the Hubbard model without vertex corrections (no VC) and site i at which they are evaluated. the lowest-order result for the Hubbard model with U = 0 In the following, we focus on the half-filled system (non-int.). For the t-J model, the photoemission spectrum has been shifted by -U/2 and for the U = 0 Hubbard model where the mean occupation per site and spin n = i−σ the photoemission and inverse photoemission spectra have 1/2 and consider the limit U . Because ofh the com-i → ∞ been shifted by -U/2 and U/2, respectively. The results for plete suppression of double occupancies and the specific U = 0 refer to the k-averaged self-energy. form of the electron-phonon coupling in Eq. (2), there are no phonons excited in the ground state and expectation values involving phonon operators in Eqs. (16) and full self-energy Σ(k,z) in the Hubbard model, † (17) greatly simplify, e.g., U (bi + bi )ni−σ 0 and † 2 h i → 1 ep + 2 (bi + bi ) 1. For the electron-phonon contribution lim dω Im Σ (k,ω i0 )=2g , (20) h ep i → U→∞ π − Σ (k,z) to the electron self-energy, i.e. the difference Z(I)PES between the self-energies for systems with and without where we integrate over the (I)PES energy range around electron-phonon coupling, one then obtains the following ω U/2. An explicit choice in the limit of large U sum rule: would≈ ± be, e.g., to integrate from to U/4 (from U/4 ∞ −∞ − 1 ep + 1 2 2 to ) in order to fully include the energy range of the lim dω Im Σ (k,ω i0 )= gq g . ∞ U→∞ π N PES (IPES) spectrum. −∞ − q | | ≡ Z X Together with Eq. (18), it follows from Eq. (20) that (18) the corresponding partial sum rule for integrating over In the special case of a Holstein coupling, g = g. the central energy range ω 0 is given by Besides this sum rule over all frequencies, we now want ≈ to derive in addition partial sum rules where the integra- 1 lim dω Im Σep(k,ω i0+)= 3g2. (21) tion runs only over a certain energy range. We start U→∞ π ω≈0 − − from total sum rules for the electron-phonon contribu- Z tions to the (I)PES self-energies Σ±(k,z) that can be The negative value can be understood as follows. For + derived analogously as described in App. A 2. g¯ = 0, Im Σ(k,ω i0 )/π has a pole in this energy range with a large positive− weight ( U 2/4). When the 1 ∞ electron-phonon coupling is switched≈ on, the strength of lim dω Im Σ±,ep(k,ω i0+)= g2. (19) U→∞ π −∞ − this pole is slightly reduced which shows up as a pole with Z a negative weight in the spectral function of Σep(k,z). Because of the relations in Eqs. (14) and (15), this result In summary, we have derived different sum rules for the corresponds to a sum rule for the t-J model which was electron-phonon contribution to the electron self-energy obtained in Ref. 5. As Eqs. (14) and (15) involve certain in the half-filled Hubbard model in the limit of large U. assumptions needed to relate the undoped t-J model and A total sum rule (Eq. (18)) is complemented by three the half-filled Hubbard model in the limit of large U (see partial sum rules that correspond to integrating over the discussion in Sec. II), a derivation of Eq. (19) from the PES, IPES and central energy ranges (Eqs. (20) and result in Ref. 5 would be only approximate. Our present (21)). These sum rules and the corresponding one in the derivation, however, is entirely within the framework of t-J model are summarized schematically in Fig. 1. For the Hubbard model and exact. comparison, we also show the result for non-interacting The spectral functions of Σ±(k,z) are non-zero only in electrons (U = 0) to lowest order in g2. Integrating the ± ep the energy range where the (I)PES spectra Im G (k,ω k-averaged Σnon-int over the range of the photoemission i0+)/π are located, i.e., around ω U/2. With− spectrum for a half-filled model gives the contribution Eq. (11), we can therefore derive from≈ Eq. ± (19) also par- g¯2/2, which is a factor of 4 smaller than what is ob- tial sum rules for the electron-phonon contribution to the tained in the large-U model (cf. Eq. (20)). Results for 4 the large-U Hubbard model obtained by neglecting ver- IV. NUMERICAL RESULTS AT FINITE U tex corrections (no VC) are also shown in Fig. 1 and will be discussed in Sec. V. The sum rules in Sec. III A for the electron-phonon contribution to the electron self-energy have been derived in the limit U . In practice, however, we are inter- B. Phonon self-energy ested in strongly→ correlated ∞ systems with large but finite U. In order to check the usefulness of the sum rules for The phonon self-energy Π(q,z) can be expressed in such cases we have performed numerical calculations us- terms of the exact charge-charge response functionχ ˜(qz) ing exact diagonalization (ED) and dynamical mean-field as theory (DMFT). (g2 /N)˜χ(q,z) The ED calculations are done on a two-dimensional Π(q,z)= q , (22) tilted 10-site square cluster with periodic boundary con- 2 q q 1 + (gq/N)˜χ( ,z)D0( ,z) ditions. We consider a weak electron-phonon coupling such that it suffices to include only states with at most where D (q,z) is the free phonon Green’s function. To 0 one phonon excited, thereby limiting the size of the lowest order in g , the denominator of Eq. (22) can be ne- q phonon Hilbert space. These exact results are only influ- glected andχ ˜(q,z) can be replaced by the exact charge- enced by finite size effects. charge response function χ(q,ω) for a system without electron-phonon interaction. At T = 0, In addition, we also consider the thermodynamic limit in the dynamical mean-field approximation,15 which also

2 1 1 allows us to consider larger electron-phonon couplings. χ(q,z)= ν ρq 0 , (23) |h | | i| z ω − z + ω Most DMFT calculations only consider the paramag- ν ν ν 16 X − netic (P) phase. This approach neglects antiferromag- netic (AF) correlations, which play an important role for where ν is an eigenstate of H with eigenenergy ων rel- | i weakly doped systems with a large U. In particular, ative to the ground state energy and ρq = ρqσ is the σ AF correlations are essential for the interplay between Fourier transform of ni. For U t and a half-filled system, the ground state has exactly≫ one| | electronP per site to electron-phonon and Coulomb interactions. Thus it has been found that AF-DMFT calculations predict only a lowest order in t/U. Applying ρq to the ground state 0 , | i moderate suppression of the electron-phonon part of the it then follows that ρq 0 is zero to this order if we con- 17 sider q = 0. The sum| rulei for Im χ(q,ω + i0+) /(πN) electronic self-energy due to the Coulomb repulsion, 16 integrated6 over all frequencies is| then also zero to| lowest while P-DMFT calculations give a strong suppression. order in t/U: We therefore use the AF-DMFT method. We consider a Bethe lattice with a semi-elliptical den- 1 ∞ sity of states (half bandwidth D). The self-consistent An- dω Im χ(q = 0,ω + i0+) = (t/U). (24) πN −∞ | 6 | O derson impurity model, which appears in this approach, Z is solved using exact diagonalization and a continued In the t-J model, an exact sum rule for the spectral fraction expansion18 We employ up to 14 discrete bath function of χ(q,z) has been derived:14 levels and allow for up to 30 excited phonons. Specif- ∞ ically, we study the undoped Hubbard-Holstein model 1 + dω Im χt-J (q,ω + i0 ) =2δ(1 δ), (25) with D = 4t = 1 and ωph = 0.025D = 0.1t. In both πN 2 | | − q=06 Z−∞ approaches, we first calculate the (I)PES spectral den- X sities A± which directly lead to the Green’s function G where δ is the doping. We now assume that the pho- using Eqs. (6) and (7). The inversion of the Dyson equa- toemission spectra of the large-U Hubbard and the t-J tion (Eq. (8)) then gives the self-energy Σ whose spectral model are identical. They extend over an energy range density we integrate over different energy ranges. Alter- which is equal or smaller than the width of the lower Hub- natively, the total sum rule can be obtained from calcu- bard band, ∆ = (t) U. Then, Eq. (25) also leads lating the ground state expectation values appearing in O ≪ to a sum rule for χ(q,z) if the integration is limited to Eq. (17). ω 2∆ which excludes transitions from the lower to the We first consider a weak electron-phonon coupling cor- | |≤ upper Hubbard bands not captured by the t-J model: responding to a dimensionless coupling constant λ = 2 2 g /(ωphD) = g /(ωph4t)=0.0025. Using results from 1 2∆ dω Im χ(q,ω + i0+) =2δ(1 δ). (26) both ED and DMFT calculations, Fig. 2 shows how much πN 2 | | − the total spectral weight of the electron-phonon contri- q=06 Z−2∆ X bution to the local (k-averaged) electron self-energy at fi- As discussed in Ref. 5, these results indicate a strong sup- nite U deviates from the sum rule for U (Eq. (18)). pression of the phonon self-energy in weakly doped sys- Relative to the latter, the deviation is less→ ∞ than 10% for tems with strong correlations because one would obtain U as small as 3.5D, a value often considered appropriate unity if non-interacting electrons (U = 0) were assumed for the cuprates. This difference decreases like 1/U 2 as instead. can be seen from Fig. 2, in agreement with expectations 5

← U/D -2.6% ∞ 15 8 5 3.5 total: (∫ dω Im Σep (ω-i0+)/π - g2)/g2 0% −∞ loc. (I)PES: (∫ dω Im Σep (ω-i0+)/π - 2g2)/(2g2)

2 ED (I)PES loc. DMFT ∫ ω Σep ω + π 2 2 )/g -3% central: ( ω≈0 d Im loc. ( -i0 )/ + 3g )/(-3g ) 2 -2% PT - g π )/ + -3.4% -i0 -4% ω ( ep loc. Σ -6% -3.8% Im ω d ∞ −∞ ∫ -8% -4.2%

( 0 0.02 0.04 0.06 0.08 0 0.2 0.4 0.6 0.8 1 1/(U/D)2 → λ

FIG. 2: Relative deviation of the total sum rule for the lo- FIG. 4: Relative deviations of total and partial sum rules from cal (k-averaged) Σep from its large-U limit (Eq. (18)) as a their large-U limits as a function of λ using DMFT for U = function of U using ED and DMFT with λ = 0.0025. 5D. For the total sum rule, also results from perturbation theory (PT) for a simpliﬁed model (see text) are shown.

3g2 central are accessible because of the smaller phonon Hilbert space compared to ED. In Fig. 4, we plot the λ depen- 2g2 dence of the relative deviations of total and partially inte- (I)PES grated spectral weights of the electron-phonon contribution to the local electron self-energy from their respective sum rules for U (Eqs. (18), (20), and (21)). For 2 → ∞ g both the (I)PES and the central sum rules, the relative total deviations are comparable in size to that of the total sum g2/2 rule (this similarity is also found when the U dependence is considered) and are less than 10%. In all cases, the 0 0 1 2 3 4 5 6 deviations decrease linearly with λ. Again, for the to- U/D tal sum rule, the results can be quite well described by a result from perturbation theory for a simplified model FIG. 3: U dependence of total and partial sum rules for Σep which we introduce in the following. using DMFT with λ = 0.0025. The dotted lines indicate the We consider the self-consistent Anderson impurity expected small-U behavior. model to be solved in AF-DMFT of the Hubbard- Holstein model and replace the bath by a single (but spin-dependent) level. With the impurity level at U/2, from a simple perturbational approach which we discuss the self-consistent bath level is located at U/−2 de- in more detail at the end of this section. We note that pending on the spin orientation; the hopping≈ amplitude ± results from ED and DMFT agree rather well. This con- between impurity and bath is fixed to V D/2. We treat sistency indicates that the finite size effects of the former both this hybridization and the electron-phonon≈ interac- and the approximations of the latter method are proba- tion as perturbations of the atomic limit and find for bly not strongly influencing the results discussed here. large U The full U dependence of both total and partial sum 2 rules is illustrated in Fig. 3 using results from DMFT cal- † 2V g −4 (bi + bi )ni−σ = 2 + (U ) (27) culations. Already for U larger than the non-interacting h i −U (U + ωph) O bandwidth 2D, the sum rules clearly approach their respective large-U limits (Eqs. (18), (20), and (21)). As and discussed at the end of Sec. IIIA, we expect for U 0 2 → 2 2 that the total sum rule again approaches g with the † 2 4V g /ωph −4 (bi + bi ) =1+ + (U ). weight equally distributed over the PES and IPES en- h i U(U + ωph)(U +2ωph) O ergy ranges and no central contribution to the spectrum (28) of Σep at ω = 0. These trends are manifest in the calcu- Using these results in Eq. (17), we expect the total spec- lated results. tral weight of the electron-phonon contribution to the Next, we study the dependence on the strength of the electron self-energy to be proportional to U −2 as was ob- electron-phonon coupling for fixed U = 5D. We restrict served in Fig. 2. Although we have replaced the bath by ourselves to DMFT calculations where larger couplings a single level, the expressions in Eqs. (27) and (28) give 6 a) q q

g Γ g Γ g Γ k q k+q q k+q −q

FIG. 5: Diagrammatic representation of the vertex function b) Γ(k, q), where k (q) stands for the incoming electron (phonon) k+q momentum and frequency. The full and dashed lines represent electron and phonon Green’s functions, respectively. Γ a rather accurate description of the numerical results as can be seen in Fig. (4). In conclusion, we ﬁnd that for typical values of U the relative deviations from the sum rules derived for U k → ∞ are smaller than 10%, and therefore these sum rules can 2 FIG. 6: a) Lowest-order contribution (in |gq| ) to the electron- be used semi-quantitatively also for ﬁnite U. phonon part of the electron self-energy Σep and b) the charge- charge response function χ. The full and dashed lines represent electron and phonon Green’s functions, respectively, and V. EFFECTS OF VERTEX CORRECTIONS the circles the vertex functions (gq)Γ.

We now use the sum rules introduced in Sec. III to study the effects of vertex corrections in a diagrammatic tron per spin. As a result, we find calculation of the electron and phonon self-energies. We 1 ep + 1 2 define the vertex function Γ(k, q) as the sum of all irre- lim dω Im Σno VC(k,ω i0 )= g¯ , (30) U→∞ π − 2 ducible vertex diagrams connecting two electron Green’s Z(I)PES functions with a phonon propagator taking out one cou- where we have also used that at half-filling, the phonon pling constant gq explicitly (see Fig. 5). We use the 4- Green’s function D(q) is not dressed in the large-U limit. vectors k and q as shorthand notation for the momenta Comparing this result for the diagram in Fig. 6a with- and frequencies involved. out vertex corrections (no VC) with the corresponding exact sum rule, Eq. (20), shows that this approximation underestimates the sum rule by a factor of four. This A. Electron self-energy result is schematically indicated in Fig. 1. When vertex corrections are neglected, the diagram in Fig. 6 has no 2 An important lowest-order (in gq ) contribution to contributions in the energy range ω 0. 19 ≈ the electron-phonon part of the electron| | self-energy is We have elsewhere used the self-consistent Born ap- shown in Fig. 6a, although there are also other, more proximation to study electron-phonon interaction in the complicated lowest-order diagrams.19 The diagram in undoped t-J model which is closely related to the half- Fig. 6a is filled Hubbard model in the large-U limit. In this approach, fairly good agreement with exact results is ob- Σep(k,ω)= (29) tained although vertex corrections are neglected. Al- ′ ready the lowest-order (in the electron-phonon coupling) i 2 dω gq G(k + q)D(q)Γ(k, q)Γ(k + q, q), diagram for the electron-phonon contribution to the elec- N | | 2π − q Z tron self-energy fulfills an exact sum rule for the total X spectral weight. The reason for this result contrasting where q stands for a wave vector q and a frequency ω′. the strong violation of the sum rule in the large-U Hub- G and D are fully dressed electron and phonon Green’s bard model when vertex corrections are neglected can be functions. traced to the use of a Green’s function for spinless holons We now neglect the vertex corrections, i.e, we put in the self-consistent Born approximation. Its spectral Γ = 1. After using Eq. (7) to express the electron Green’s function integrates to unity over the photoemission en- function in terms of its spectral function, the ω′ integral ergy range whereas the spin-dependent electron Green’s can be performed. For the half-filled Hubbard model in function in the large-U Hubbard model has the spectral the large-U limit, the spectral function integrated over weight one half in both the photoemission and the inverse the lower or the upper Hubbard band gives half an elec- photoemission energy range. 7

B. Phonon self-energy TABLE I: Sum rules equivalent to Eq. (32) but for individual values of q, using ED of the t-J model on an 18-site cluster For q = 0, the charge-charge response function, 6 with two holes and J/t = 0.3. The “No VC” results were χ(q,ω), for the Hubbard model can be obtained from obtained from wP(k) and wIP(k), using the second line of the diagram in Fig. 6b: Eq. (32), and the exact ones result from the direct calculation of χ. “Ratio” shows the ratio of these results. dω′ χ(q,ω)= 2i G(k+q)G(k)Γ(k+q, q). (31) q/ π (1, 1) (2, 0) (2, 2) (3, 1) (3, 3) − 2π − 3 k X Z No VC 0.1848 0.1927 0.2103 0.2025 0.2285 We consider the large-U limit for a hole-doped Hubbard Exact 0.2100 0.1961 0.2191 0.2085 0.2212 model. The k-averaged photoemission spectrum for a Ratio 0.8804 0.9825 0.9597 0.9714 1.0330 given spin integrates to (1 δ)/2. As earlier, we use the assumption that this spectrum− agrees with the spectrum of the t-J model and that it only extends over an energy For the half-ﬁlled Hubbard model, there is no contri- range ∆ U. In inverse photoemission, the probability bution to Im χ(q,ω + i0+) for ω U. We therefore ≪ | | ≪ of adding an electron to an unoccupied site is δ. We focus on contributions for ω U. If Imχno VC(q,ω + therefore assume that U is so large that the integral of i0+) /(πN) is integrated over| | ≈all frequencies,| we obtain the inverse photoemission spectrum for a given spin up to unity.| The exact result in Eq. (24), however, is zero to ∆ is given by δ. We neglect vertex corrections and replace lowest order in t/U. This dramatic disagreement shows Γ(p+q, q) by unity. Introducing spectral functions, the the importance of vertex corrections in this case. ω′ integration− is performed. We consider the integral of the spectrum of the charge-charge response function over ω 2∆, thereby excluding transitions between the two VI. EXAMPLE: TWO-SITE MODEL Hubbard| |≤ bands. Then,

1 2∆ To study the vertex corrections more explicitly, we con- q + 2 dω Im χno VC( ,ω + i0 ) sider a two-site Hubbard model. The electron-phonon πN q −2∆ | | X Z interaction in Eq. (2) can then be split in q = 0 and 2 q = π terms. The q = 0 term has just the rather triv- = [w (k)w (k + q)+ w (k + q)w (k)] N 2 P IP P IP ial but important effect of convoluting the spectra with q k X X phonon satellites, while the q = π term introduces dy- =2δ(1 δ), (32) − namics, scattering electrons between bonding and antibonding orbitals. We therefore only keep the the more where wIP(k) is the integrated weight of the photoemis- interesting q = π term here. k k sion spectrum for the wave vector and wIP( ) is the cor- Following Huang et al.,7 we can calculate the vertex responding quantity for inverse photoemission, exclud- function explicitly for the two-site model in the limit of ing the upper Hubbard band. For a large system, the U/t very large. We consider an incoming electron in the q 0 = term gives a negligible contribution in Eq. (32). It bonding orbital (+) with the frequency ω scattered by then agrees with the sum rule in Eq. (26), derived from the antibonding (q = π) phonon with the frequency ω′ the corresponding sum rule in the t-J model, although into the antibonding orbital (-) with the frequency ω +ω′ vertex corrections have been neglected. It is important, and obtain however, to use dressed Green’s functions in calculating χ(q,ω). Otherwise, 2δ in Eq. (32) would have been re- Γ(ω, +; ω′)=Γ(ω + ω′, ; ω′) (33) placed by (1 + δ), and there would have been a strong − − ω(ω + ω′)+ ω′t + U 2/4 disagreement with Eq. (26) in the low-doping regime. = , The sum rule in Eq. (32) refers to an average over q. (ω + t)(ω + ω′ t) − We next study individual values of q for a t-J model on a two-dimensional tilted 18-site square cluster with two where various terms of higher order in t/U have been holes, periodic boundary conditions, and J/t = 0.3. We neglected. have calculated wP(k) and wIP(k), using exact diagonal- Using this result for the vertex function, we can cal- ization. From the second line of Eq. (32), we can obtain culate the diagram in Fig. 6a according to Eq. (29). In sum rules for each q similar to Eq. (32). The result is the limit of large U, we find poles with weight 2g2 both shown in the line “No VC” of Table I. The results are at ω U/2 and at ω U/2. Therefore, the sum rules ≈− ≈ compared with the sum rule for χt-J (q,z) calculated in for integrating over either the PES or the IPES energy the t-J model (“Exact” in Table I). As can be seen from range, Eq. (20), are exactly fulfilled. Without vertex cor- the ratios of these results, the sum rule for individual rections, these sum rules are underestimated by a factor values of q that can be deduced from Eq. (32) is also of four, cf. Eq. (30). This can be understood by noting rather well fulfilled (typically, with a deviation of 5-10%) that for electronic energies in the range of the lower or although vertex corrections are neglected. upper Hubbard band, ω U/2, it follows from Eq. (33) | |≈ 8 that substantially influenced by vertex corrections. For the U half-filled system, we have to integrate Im Π over all fre- Γ( ω , +; ω′) 2 (34) | |≈ 2 ≈ quencies to obtain a nontrivial sum rule. This sum rule is only satisfied if vertex corrections are included. when the phonon frequency ω′ is assumed to be small The sum rule for Im Σep, integrating over frequencies compared to U. Therefore, including vertex corrections corresponding to photoemission only, is violated by a fac- effectively increases the weight of poles around ω U/2 tor of four if vertex corrections are neglected. These re- by a factor Γ2 = 4. In addition, the self-energy calculated| |≈ sults have been illustrated by an explicit calculation of using vertex corrections also has poles at ω = t ω and ph the vertex function in a two-site model. We have studied ω = t, the latter being a double pole. Except− for a dif- integrated quantities where all values of q and ω enter, ferent− sign, to leading order in U they give the same con- both in terms of their relative ratio and| their| absolute tribution, g2(U/2)2/(2t ω )2, to the integral over the ph magnitude. Therefore, our findings cannot be directly spectral function∓ of the self-energy.− The sum of the two compared with previous ones which focused on the static contributions, however, is not zero, but one finds, taking limit7,8 or on small q and ω.20 Our results show that into account also terms which involve lower powers of U, the inclusion of vertex| corrections| can be essential to cor- that it equals 3g2 as expected from Eq. (21). There- rectly describe effects of electron-phonon interaction in fore, also the sum− rule over all frequencies in Eq. (18) is strongly correlated systems. fulfilled. The sum rule for the charge-charge response function in Eq. (26) applies to finite dopings. For the two- Acknowledgments site model, this implies the uninteresting filling one, for which there is no Coulomb interaction. It is interesting, however, to study the sum rule over all frequencies We acknowledge useful discussions with Erik Koch, for the half-filled two-site model. As expected, we find Claudio Castellani, and Massimo Capone. that the neglect of vertex corrections, incorrectly, gives contributions at ω U with weight 1/2, respectively. Only when the vertex≈ ± function from Eq. (33) is used in APPENDIX A: DERIVATION OF SUM RULES Eq. (31), χ(q = π,ω) vanishes to lowest order in t/U. USING SPECTRAL MOMENTS This is the correct result from Eq. (24). 1. Total sum rule

VII. SUMMARY To derive the sum rule for the total integrated weight of the spectral density of the electron self-energy, Eq. (16), We have derived exact sum rules for the electron- we expand (z ω)−1 in Eq. (7) in powers of 1/z. One phonon contribution Σep to the electron self-energy of the obtains − half-filled Holstein-Hubbard model in the limit of large ∞ (m) U. In particular, we consider integrations both over all M G(k,z)= k , (A1) frequencies and over frequencies in the photoemission en- zm+1 m=0 ergy range. Comparing results for U = and U = 0, X ∞ we find identical sum rules when integrating over all fre- where the moments of the spectral density are defined as quencies but a difference by a factor of four when consid- ering frequencies corresponding to photoemission only. ∞ (m) m k Using different numerical methods, we find that these Mk = dω ω A( ,ω). (A2) −∞ sum rules are relevant also for systems with intermediate Z values of U 3D that are typically of interest. These On the other hand, using the Heisenberg equations of ≈ sum rules should be useful for testing approximate cal- motion for the time-dependent operators in the defini- culational schemes. tion of the spectral density, these moments can also be We have also used sum rules for studying the impor- obtained from21 tance of vertex corrections in a diagrammatic approach to (m) m † properties in the Hubbard-Holstein model. For a weakly Mk = ckσ,ckσ , (A3) doped Hubbard-Holstein model in the large-U limit, the h{L }i phonon self-energy Π is strongly reduced compared to with = [ , H]. [ , ′] and , ′ denote the com- LO O O O {O O } ′ the non-interacting case. This is described by a sum mutator and the anticommutator of two operators , , O O rule which integrates Im Π over a finite frequency range respectively. such that transitions between the Hubbard bands are not For the Hubbard model with electron-phonon interac- involved. This sum rule is satisfied if properly dressed tion which was defined in Eqs. (1)-(3), one obtains from Green’s functions are used to calculate the phonon self- Eq. (A3) energy, even if vertex corrections are neglected. The en- (0) ergy dependence of Im Π(q,ω) could, nevertheless, be Mk =1, (A4) 9

(1) † Mk = εk + U ni−σ + gq=0 bi + bi , (A5) model from Eqs. (1)-(3). It turns out, however, that they h i h i simplify considerably once the limit U is taken. For and the zeroth moments which correspond→ to ∞ the integrated ± spectral weights ak of the Green’s functions in Eq. (10), (2) 2 2 1 2 † 2 M = ε + U n + gq b + b one finds k k i−σ N q −q h i q | | h| | i X † +,(0) + 1 +2εkU ni−σ +2εkgq=0 bi + bi M = a =1 nk for U , (A10) h i h i k k − h σi → 2 → ∞ 1 † +2U gq (b + b )ρq , 1 √ q −q −σ M −,(0) = a− = n for U , (A11) N q h i k k kσ X h i → 2 → ∞ where ε , ρ , and b have already been defined after k qσ i where the large-U limit applies to the half-filled system Eqs. (8) and (16). without spin polarization which is of interest for us here. When the 1/z expansion of the self-energy, In this limit, one obtains the following first and second ∞ moments: C(m) Σ(k,z)= k , (A6) zm m=0 +,(1) M (U + ε + εk)/2 for U , (A12) X k → d → ∞ −,(1) is inserted into Eq. (8), a comparison with Eq. (A1) leads Mk (εd + εk)/2 for U (A13) to and → → ∞ 2 (1) (2) (0) (1) (0) C = M /M M /M . (A7) +,(2) 2 2 k k k k k M ((U + ε + εk) + g )/2 for U , (A14) − k → d → ∞ −,(2) 2 2 (1) Mk ((εd + εk) + g )/2 for U . (A15) As Ck corresponds to the zeroth moment of the spectral → → ∞ density of Σ(k,z), we arrive at the sum rule, Eq. (16), when Eqs. (A4)-(A6) are used in Eq. (A7). where we used that double occupancy of sites and phonon excitations are suppressed in the groundstate of the half- filled system at large U. As in the previous section, these 2. Sum rules for (I)PES self-energies ±,(m) moments can be related to the coefficients Ck in a 1/z expansion of the self-energies Σ±(k,z). At half-filling The derivation of sum rules for the (I)PES self-energies and for εd = U/2, − Σ±(k,z) can be done in full analogy with the previous section. Only the moments of the (I)PES spectra ±,(1) ± 22 ±,(0) Mk ± A (k,ω) are now obtained from C = = ε εk U/2 for U , (A16) k ±,(0) k → ± → ∞ Mk M +,(m) = ( mc )c† , (A8) k h L kσ kσi −,(m) † m which gives the z-independent contributions ε± to the Mk = ckσ ckσ (A9) k h L i self-energies defined in Eq. (10). Using the analog to instead of Eq. (A3). Because Eqs. (A8) and (A9) do not Eq. (A7) and focusing on the electron-phonon contribu- contain an anticommutator, much more complicated re- tion to the self-energies, one then arrives at the result in sults are obtained for these moments when applied to our Eq. (19).

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