Sum rules and vertex corrections for electron-phonon interactions O. R¨osch,∗ G. Sangiovanni, and O. Gunnarsson Max-Planck-Institut f¨ur Festk¨orperforschung, 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 ex- act 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 as- sume 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- tionE 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.