describe the algorithm

docs/source/mf_notes.md

 # Algorithm overview
 
-## Derivation of the mean-field Hamiltonian
+## The mean-field Hamiltonian
 
 ### Interacting problems
 
@@ -82,8 +82,46 @@ V^\text{MF}_{nm} (R) =  \sum_{i} \rho_{ii} (0) v_{in} (0) \delta_{nm} \delta(R)
 
 where now indices $i, n, m$ label the internal degrees of freedom of the unit cell and $R$ is the relative position between the two unit cells in terms of the lattice vectors.
 
-## Numerical solution of the mean-field Hamiltonian
+## Numerical implementation
+
+### Self-consistency loop
 
 In order to calculate the mean-field interaction in {eq}`mf_infinite`, we require the ground-state density matrix $\rho_{mn}(R)$.
 However, the density matrix in {eq}`density` is a functional of the mean-field interaction $\hat{V}^{\text{MF}}$ itself.
-Therefore,
+Therefore, we need to solve for both self-consistently.
+
+We define a single iteration of a self-consistency loop:
+
+$$
+\text{SCF}(\hat{V}^{\text{init, MF}}) \to \hat{V}^{\text{new, MF}},
+$$
+
+such that it performs the following operations given an initial mean-field interaction $\hat{V}^{\text{init, MF}}$:
+
+1. Calculate the total Hamiltonian $\hat{H}(R) = \hat{H^0}(R) + \hat{V}^{\text{init, MF}}(R)$ in real-space.
+2. Fourier transform the total Hamiltonian to the momentum space $\hat{H}(R) \to \hat{H}(k)$.
+3. Calculate the ground-state density matrix $\rho_{mn}(k)$ in momentum space.
+    1. Diagonalize the Hamiltonian $\hat{H}(k)$ to obtain the eigenvalues and eigenvectors.
+    2. Calculate the fermi level $\mu$ given the desired filling of the unit cell.
+    3. Calculate the density matrix $\rho_{mn}(k)$ using the eigenvectors and the fermi level $\mu$ (currently we do not consider thermal effects so $\beta \to \infty$).
+4. Inverse Fourier transform the density matrix to real-space $\rho_{mn}(k) \to \rho_{mn}(R)$.
+5. Calculate the new mean-field interaction $\hat{V}^{\text{new, MF}}(R)$ via {eq}`mf_infinite`.
+
+### Self-consistency criterion
+
+To define the self-consistency condition, we first introduce an invertible function $f$ that uniquely maps $\hat{V}^{\text{MF}}$ to a real-valued vector which minimally parameterizes it:
+
+$$
+f : \hat{V}^{\text{MF}} \to f(\hat{V}^{\text{MF}}) \in \mathbb{R}^N.
+$$
+
+Currently, $f$ parameterizes the mean-field interaction by taking only the upper triangular elements of the matrix $V^\text{MF}_{nm}(R)$ (the lower triangular part is redundant due to the Hermiticity of the Hamiltonian) and splitting it into a real and imaginary parts to form a real-valued vector.
+
+With this function, we define the self-consistency criterion as a fixed-point problem:
+
+$$
+f(\text{SCF}(\hat{V}^{\text{MF}})) = f(\hat{V}^{\text{MF}}).
+$$
+
+To solve this fixed-point problem, we utilize a root-finding function `scipy.optimize.anderson` which uses the Anderson mixing method to find the fixed-point solution.
+However, our implementation also allows to use other custom fixed-point solvers by either providing it to `solver` or by re-defining the `solver` function.