There are several signs of a rather strong electron-phonon interaction in high Tc cuprates.
In neutron scattering experiments, several phonons show an appreciable softening under doping, suggesting a substantial coupling to the doped holes. This is in particular the case for the oxygen half-breathing phonon at about 70 meV, where two of the four oxygen atoms, surrounding a Cu atom in the CuO2 plane, move towards the Cu atom. While a shell model describes other phonons quite well, it fails to describe the softening of the half-breathing phonon. In the doped system, the width of this phonon is about an order of magnitude larger than predicted by calculations in the local density approximation (LDA). This anomalous behavior suggests that many-body effects due to electron-electron interaction are important.
Photoemission experiments show a large kink in the hole dispersion at about 70 meV binding energy or less. This suggests a strong coupling to bosons. Natural candidates are phonons, in particular since there is an isotope effect. Photoemission also shows formation of polarons for undoped cuprates, indicating strong electron-phonon coupling for these systems.

Starting from the three-band model, we have derived a t-J model with phonons. In the t-J model, Cu d-holes and O p-holes form so-called Zhang-Rice singlets. For a rigid lattice, the large singlet energy (about 5 eV) leads to an uninteresting constant in the total energy. Due to the strong distance-dependence of the hopping integrals and Coulomb integrals, however, displacements of the atoms lead to large changes of the singlet energies. The result is a strong electron-phonon interaction. We find that the coupling to the site energies due to the modulation of the hopping integrals dominates. This coupling explains the anomalous softening of the half-breathing phonon (Electron-Phonon Interaction in the t-J Model, Phys. Rev. Lett. 92, 146403 (2004)).

To connect to calculations in the LDA, we have studied the three-band model in the Hartree-Fock (HF) approximation. Due to its mean-field character, the HF approximation has some similarities to the LDA, but it has the advantage of giving an antiferromagnetic insulating solution for the undoped system. While the HF approximation and the t-J model give comparable softenings for intermediate dopings (x~0.1), the doping and q-dependence are quite different. The presence of many low-lying excitations in the t-J model is important for the large width of the half-breathing phonon (Electron-phonon interaction in the three-band model, Phys. Rev. B 70, 224518 (2004)).

Recent experimental work gives strong evidence that undoped cuprates are in the polaronic limit. The electron spectral function has a very weak quasi-particle (not visible in photoemission) and a broad phonon side-band. There is both experimental and theoretical evidence that the quasi-particle dispersion in the t-J model without phonons describes the dispersion of the phonon side-band. To address this surprising result, we have introduced an adiabatic approximation for treating models with electron-phonon interaction. This formalism becomes particularly simple if the electron-phonon interaction can be neglected in the initial state of the photoemission experiment. The photoemission spectrum can then be described as a broadened version of the spectrum without electron-phonon interaction (Dispersion of incoherent spectral features in systems with strong electron-phonon coupling, Eur. Phys. J. B 43, 11 (2005)).

Using a shell model, we have calculated the electron-phonon coupling in undoped La₂CuO₄ and found that it is strong enough to give polarons. We developed an efficient method for calculating angle-resolved photoemission spectra in undoped systems. Using the calculated couplings, we found the width of the phonon side band in good agreement with experiment. The dependence of the width on the binding energy is analyzed. Our results show that the electron-phonon coupling plays an important role for properties of undoped cuprates (Polaronic behavior of undoped high-T_c cuprates from angle-resolved photoemission spectra, Phys. Rev. Lett. 95 227002 (2005)).

There has been much interest in the issue of whether the Coulomb interaction suppresses or enhances the electron-phonon interaction. Charge fluctuations are strongly suppressed by the Coulomb repulsion, and it has therefore been argued that phonons coupling to the net charge of an atom has a weak interaction. In agreement with this, it is found that in the t-J model with an on-site electron-phonon coupling, the phonon self-energy goes to zero with the doping x. This raises questions if the electron-phonon part of the electron self-energy is also strongly suppressed. We have addressed this issue by deriving a sum rule for the electron self-energy and by performing exact diagonalization and DMFT calculations.

We have derived an exact sum rule for the imaginary part of the electron-phonon contribution to the electron self-energy. This sum rule is identical to the lowest order contribution to the corresponding sum rule for noninteracting electrons. Thus the electron-phonon part of the electron self-energy is not suppressed by correlation effects in a similar way as the phonon self-energy. This can also be seen by comparing with a similar sum rule for the phonon self-energy. Estimates of the electron-phonon coupling from phonon properties can therefore underestimate the effects on the electrons (Apparent electron-phonon interaction in strongly correlated systems , Phys. Rev. Lett. 93, 237001 (2004)).

We have also studied the effects of the Coulomb repulsion on the electron-phonon interaction in the Holstein-Hubbard model, using the antiferromagnetic dynamical mean-field approximation. We find that antiferromagnetic correlations strongly enhance electron-phonon coupling effects on the electron Green's function with respect to the paramagnetic correlated system, but the net effect of the Coulomb interaction is a moderate suppression of the electron-phonon interaction. Doping leads to additional suppression. In contrast, the Coulomb interaction strongly suppresses electron-phonon interaction effects on phonons, but the suppression weakens with doping. These results are in agreement with sum rules derived above. (Electron-Phonon Interaction and Antiferromagnetic Correlations, Phys. Rev. Lett. 97 046404 (2006) (pdf-file)).

This work has been described in a review article J. Phys.: Cond. Mat. 20 043201 (2008). Some results concerning the coupling to breathing phonons in cuprates are shown in a poster.

The pseudogap in the high Tc cuprates has attracted much interest. It is of electronic origin (see, e.g., our work here ). Nevertheless we have found that there is a substantial isotope effect on the pseudogap (Phys. Rev. B, Rapid Commun. 84, 100505 (2011).

A list of some relevant publications is given here.

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