4.2.1 Model

The Holstein Hamiltonian in real space is given by

$$H_H=-t\sum_{<ij>}(c^\dagger_{i}c_{j}+h.c.)+\omega_0 \sum_{i} b^\dagger_i b_i +g \sum_{i}n_i x_i~,$$ (4.1)

where $c_{i}$ ($b_{i}$) is the electron (phonon) annihilation operator at site $i$. The first term describes the electron kinetic energy. The second term describes a set of independent oscillators with frequency $\omega _0$ 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 $n_i=c^\dagger_{i}c_{i}$ couples with the lattice displacement $x_i=\frac{1}{\sqrt{2 M \omega_0}}(b^\dagger_i+b_i)$ with strength $g$.

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$_c$ 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

$$H_B = -t\sum_{<ij>}(c^\dagger_{i}c_{j}+h.c.)+ \omega_0 \sum_{i} b^\dagger_{i+\frac{1}{2}} b_{i+\frac{1}{2}}$$ (4.2)

$$+ g \sum_{i}n_i(x_{i-\frac{1}{2}}-x_{i+\frac{1}{2}})~~.$$

Both the H and the B model can be written in the momentum representation as

$$H _{(H,B)}= \sum_{k} \epsilon(k) c^{\dagger}_k c_k + \sum_q \omega_0 b^{\dagger}_q b_q +$$ (4.3)

$$\frac{1}{\sqrt{N}} \sum_{k,q} \gamma_{(h,b)} (q) c^{\dagger}_{k-q} c_k (b^{\dagger}_q+b_{-q})$$

where

$$\gamma_H(q)=\frac{g}{\sqrt{2 M \omega_0}}$$ (4.4)

is the H electron-phonon coupling and

$$\gamma_B(q)=-i \frac{2g}{\sqrt{2 M \omega_0}} \sin{\frac{q}{2}}$$ (4.5)

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.