The Holstein Hamiltonian in real space is given by
where () is the electron (phonon) annihilation operator at site . The first term describes the electron kinetic energy. The second term describes a set of independent oscillators with frequency at every site. The electron-phonon coupling in the Holstein model is local and is described by the last term of Eq. 4.1 where the electron density couples with the lattice displacement with strength .
Our breathing Hamiltonian is a one-dimensional version of the model
which describes the coupling of Zhang-Rice (ZR) singlets (29)
with the Cu-O bond-stretching vibrations in high T
superconductors (22,23). In cuprates a ZR singlet
is a bound state between a hole on the Cu and a hole on the four
neighboring O atoms. It's energy is stabilized by the Cu-O
hybridization term and therefore is influenced by the Cu-O distance.
In our one dimensional model we consider a set of independent,
in-between sites oscillators (the analogue of the O atoms) which
modulate the charge carrier's (the analogue of the ZR singlet) on-site
energy. Therefore we define the B Hamiltonian as
Both the H and the B model can be written in the momentum representation as
where
is the H electron-phonon coupling and
is the B electron-phonon coupling. Notice that in the momentum representation the H coupling is a constant and the B coupling is an increasing function of the phonon momentum for small momenta.
In Eq. 4.1 and Eq. 4.2 the free electron part of the Hamiltonian was introduced as a tight-binding hopping. However in order to study the influence of electron dispersion on the polaron properties in this chapter we also employ calculations which consider different forms of electron dispersion.