# Coulomb interaction in 2D crystal

My question is very simple. What is the correct way of modelling a Coulomb interaction on a 2D lattice?

Usually for a system that is infinitely big $$(N\to\infty)$$ and not discrete $$(a_0\to 0)$$, the two particle operator is simply $$V=\sum_{\vec{k}_1,\vec{k}_2,\vec{p}}V(\vec{p})c^{\dagger}_{\vec{k}_1+\vec{p},\uparrow}c^{\dagger}_{\vec{k}_2-\vec{p},\downarrow}c_{\vec{k}_2,\downarrow}c_{\vec{k}_1,\uparrow}$$ where I guess $$V(\vec{p})\propto\frac{1}{\sqrt{\Vert\vec{p}\Vert^2+\kappa^2}}$$ and $$\kappa$$ is the screening parameter. But obviously this can't be true on a lattice where the momenta are merely crystal momenta such that $$\vec{k}$$ and $$\vec{k}+\vec{K}$$ is considered equivalent when $$\vec{K}$$ is a reciprocal lattice vector.

What is the correct discretized form of the Coulomb interaction on a 2D reciprocal lattice?

Not sure why the number of dimensions matters here. When dealing with a lattice we can attribute to each lattice site wave functions of its orbitals, $$\phi_{n,j}(\mathbf{x})=\phi_{n,0}(\mathbf{x}-\mathbf{x}_j),$$ and calculate the matrix elements of the Coulomb interaction between the orbitals - just the way we would do when, e.g., studying Helium molecule. Assuming for simplicity only one orbital per site, the Coulomb interaction this takes form: $$V=\frac{1}{2}\sum_{j_1,j_2,j_4,j_3}\sum_{\sigma_1,\sigma_2} U_{j_1j_2j_3j_4}c_{j_1,\sigma_1}^\dagger c_{j_2,\sigma_2}^\dagger c_{j_3,\sigma_2}c_{j_4,\sigma_1},\\ U_{j_1j_2j_3j_4}=\int d\mathbf{x}d\mathbf{x}' \phi_{j_1}(\mathbf{x})^*\phi_{j_2}(\mathbf{x}')^*v(\mathbf{x}-\mathbf{x}')\phi_{j_4}(\mathbf{x})\phi_{j_3}(\mathbf{x}').$$ Remark: the usual caveat here is different order of indices for the operators and the matrix element.
This is often further simplified, neglecting the Coulomb scattering or even keeping only on-site Coulomb interaction, in which case one obtains a Hubbard model: $$V=\frac{1}{2}\sum_j\sum_\sigma U c_{j,\sigma}^\dagger c_{j,\bar{\sigma}}^\dagger c_{j,\bar{\sigma}}c_{j,\sigma},$$ where the interaction between the equal spins gives a trivial constant energy shift, due to the Pauli principle $$c_{j,\sigma}^\dagger c_{j,\sigma}^\dagger c_{j,\sigma}c_{j,\sigma}=c_{j,\sigma}^\dagger c_{j,\sigma}.$$ The Hubbard model ahs been extensively studied in different numbers of dimensions - e.g., for a square or honeycomb lattice in 2D.