Electron-Phonon Interaction

 In free system, hamiltonian is 

$$\hat{\mathcal{H}}=-t\sum_{i,j} \left( \hat{c}_{i\sigma}\hat{c}_{j\sigma}+\text{h.c.}\right) + \sum_i\left( \frac{p_i^2}{2m}+\frac{1}{2}m\omega^2x_i^2 \right)$$

First term is electron hopping, second term is phonon.

Hopping is orbital intersection of lattice.

So when phonon exists, displacement $x_i$ change orbital intersection.

This make hopping perturbation.

$$\hat{\mathcal{H}}_I=\sum_{i,j}\delta t |x_i-x_j|\hat{c}^\dagger_{i\sigma}\hat{c}_{j\sigma}$$

By $\hat{x}_i=\hat{b}^\dagger_i+\hat{b}$, $$\left(\hat{b}_i^\dagger+\hat{b}_i-\hat{b}^\dagger_j-\hat{b}_j\right) \hat{c}^\dagger_i\hat{c}_j$$

In momentum space, Fourier transform $$\sum_{i,j}e^{-ikr_i}e^{-iqr_i}e^{-ipr_j}\hat{b}^\dagger_k\hat{c}^\dagger_q\hat{c}_p$$ makes delta function $\delta(p-k-q)$.