Interest in the spectral properties of strongly coupled electron-phonon systems has increased due to the discovery of a kink in the quasiparticle dispersion (2,3,4,1) measured by angle-resolved photoemission (ARPES) experiments in cuprate superconductors. The energy of this kink is characteristic of a specific optical phonon mode. Furthermore, a peculiar isotope-effect, manifesting in a shift of the quasiparticle dispersion at high energy upon oxygen isotope substitution, was recently measured in the high T materials (5), undoubtedly showing that the electron-phonon interaction plays an important role in the spectral properties of cuprates.

The electron-phonon interaction has been investigated in a variety of models (7,8,6). Although the ground state properties of these models received great attention, less was paid to the spectral features. Noticeable exceptions include the Holstein (9,10) and the Fröhlich (11) models which are characterized by a momentum independent or a weakly momentum dependent electron-phonon coupling. However, in many systems, such as cuprates superconductors or organic materials, the electron-phonon coupling is strongly momentum dependent. This can give rise to distinctive properties of the single particle spectral features, as we will discuss in this chapter. In particular we will consider a sinusoidal dependence of the coupling on the phonon momentum, which is realized in many systems including the SSH coupling in polyacetilens (7) and the coupling to the breathing mode in cuprate superconductors (14,12,13).

The treatment of a multi-electron system coupled with phonons is an extremely complex and difficult problem. In general, phonons mediate an effective attraction between electrons and consequently the system becomes susceptible to various kinds of instabilities (16,15), which might have a significant effect on the photoemission spectra. However, in this chapter we investigate the effect of phonons on a single quasiparticle only, thus neglecting electron-electron or electron-hole scattering processes at the Fermi surface. We believe this to be suitable in describing the properties at energy or temperature scales larger than those associated with the ground state instability, e.g. the normal state of superconductors.

In fact, as different investigations have shown (10,17), the polaron models, which consist of a single electron interacting with phonons (18), capture much of the physics seen in the ARPES experiments on materials with significant electron-phonon interaction (2,19). This includes the kink in the quasiparticle dispersion observed at the phonon characteristic frequency. One expects that the single particle description of the influence of electron-phonon coupling would be valid to describe the photoelectron or inverse photoelectron spectral function in insulators and semiconductors in which the bands are either full or empty. However things are more complicated for a strongly correlated insulator in which the conductivity gap is a result of electron correlations and the material is an insulator in spite of having a half filled band. In systems described by the Hubbard or t-J models the single hole spectral function will be influenced by the interaction with spin fluctuations. This results in strongly dressed quasiparticles even without the electron-phonon coupling, as calculations employing self consistent Born approximation (SCBA) (20) and exact diagrammatic quantum Monte Carlo (QMC) (21) shows. The later calculation also shows that the SCBA, which neglects all the magnon crossing diagrams, is a good approximation to the quasiparticle mass renormalization and its dispersion. When the phonons are considered, in the simplest approximation we can neglect all the crossing diagrams involving phonons and magnons and treat the interaction with phonons in a single particle fashion by taking the hole band dispersion as that given by the t-J model calculations. The approximations involving the polaron models under investigation here are done in this spirit.

In this chapter we investigate the spectral properties of two polaron
models: The one-dimensional Holstein (H) model (6) and a
one-dimensional version of the breathing model (B) ^{4.1}, relevant for cuprates
superconductors (22,23). The electron-phonon
coupling in the H Hamiltonian is a constant, thus independent of the
phonon momentum. The B model has a sinusoidal momentum dependent
electron-phonon coupling, being small (large) for scattering with
small (large) momentum phonons. In both cases we consider
dispersionless optical phonons with frequency . Although
for the sake of simplicity our calculations are done in one dimension,
the conclusions and the qualitative properties of the spectra are
independent of dimensionality.

The difference between the properties of the two models emphasizes the importance of momentum dependent couplings. The specific momentum dependence of the B bare electron-phonon coupling results in an effective coupling which is an increasing function of the total polaron momentum, such that the small momentum polaron is in the weak coupling regime while the large momentum one is in the strong coupling regime. This might be germane to the peculiar behavior of the high energy quasiparticle dispersion (5) or the temperature dependency of the photoemission linewidth (24) in cuprate superconductors.

The method we use in our calculations is the self-consistent Born approximation (SCBA)(25). Although this method is an uncontrolled approximation which neglects electron-phonon vertex corrections in the self-energy calculation, the results at small coupling are in good agreement with numerically exact quantum Monte Carlo calculations. Because the contribution of configurations with an infinite number of phonons is considered, this approximation manages to capture the physics at large momentum, such as the gap and the flattening of the quasiparticle dispersion at the phonon energy (26). It can easily provide information about excited states and thus determine the polaron spectral properties, unlike most QMC methods (11,27,28) fit to calculate only the ground state properties.

The chapter is organized as follows. In Sec. 4.2.1 we introduce the H and B Hamiltonians. The dimensionless electron-phonon coupling is defined in Sec. 4.2.2. The SCBA method is discussed in Sec. 4.2.3. The results are presented in Sec. 4.3 and their significance is discussed in Sec. 4.4. A short summary and the conclusions are given in Sec. 4.5.