# Dipole operator in a lattice model

If I have a lattice Hamiltonian, say for example the Hubbard model $$H = \sum_{j,k, \sigma} t_{j,k} \hat{c}^\dagger_{j \sigma}\hat{c}_{k \sigma} + U\sum_{j} \hat{n}_{j \uparrow}\hat{n}_{j \downarrow},$$ and I want to make it interact with an electromagnetic field, in the dipolar approximation I would write it $$H_I = -\vec{\hat{D}} \cdot \vec{\hat{E}},$$ where the electric field is given by boson operators: $$\vec{\hat{E}}=\sum_{\vec{k},\lambda}w_k^{-1/2} \vec{e}_\lambda(\vec{k}) \left(\hat{b}_\lambda(\vec{k}) - \hat{b}^\dagger_\lambda(\vec{k}) \right)$$ (up to a prefactor).

But how can I write in the lattice model the dipolar operator $$\vec{\hat{D}}$$? That is, I would like to write the dipolar operator, usually written in space representation as $$\sum_i q_i \hat{r}_i,$$ in terms of the lattice operators $$\hat{c}_{j\sigma}$$ \ $$\hat{c}^\dagger_{j\sigma}$$

## 1 Answer

If each lattice site is considered a state, then the usual second quantization procedure means expanding the wave function as $$\psi_\sigma(\mathbf{r})=\sum_{j,\sigma}\phi\left(\mathbf{r}-\mathbf{R}_j\right)\chi_\sigma c_{j,\sigma},$$ where $$\phi\left(\mathbf{r}-\mathbf{R}_j\right)$$ is the orbital or a Wannier function centered on the $$j$$-th site. (Wannier functions are orthogonal, and this form a real basis, but one usually uses non-orthogonal functions, whose oevrlap is related to tight-binding approximation hopping integral.)

Given any operator in space representation, e.g., $$H_I=-e\mathbf{r}\mathbf{E},$$ we can now convert it to the second quantized form.