solve conflict

docs/source/index.md

@@ -28,6 +28,8 @@ hubbard_1d.md
 :maxdepth: 1
 :caption: Documentation
 
+mf_notes.md
+algorithm.md
 documentation/pymf.md
 ```
 

docs/source/mf_notes.md

-# Algorithm overview
+# Theory overview
 
-## The mean-field Hamiltonian
-
-### Interacting problems
+## Interacting problems
 
 In physics, one often encounters problems where a system of multiple particles interact with each other.
 In this package, we consider a general electronic system with density-density interparticle interaction:
 
 :::{math}
 :label: hamiltonian
-\hat{H} = \hat{H^0} + \hat{V} = \sum_{ij} h_{ij} c^\dagger_{i} c_{j} + \frac{1}{2} \sum_{ij} v_{ij} c_i^\dagger c_j^\dagger c_j c_i
+\hat{H} = \hat{H_0} + \hat{V} = \sum_{ij} h_{ij} c^\dagger_{i} c_{j} + \frac{1}{2} \sum_{ij} v_{ij} c_i^\dagger c_j^\dagger c_j c_i
 :::
 
 where $c_i^\dagger$ and $c_i$ are creation and annihilation operators respectively for fermion in state $i$.
-The first term $\hat{H^0}$ is the non-interacting Hamiltonian which by itself is straightforward to solve in a single-particle basis by direct diagonalizations made easy through packages such as `kwant`.
+The first term $\hat{H_0}$ is the non-interacting Hamiltonian which by itself is straightforward to solve in a single-particle basis by direct diagonalizations made easy through packages such as [kwant](https://kwant-project.org/).
 The second term $\hat{V}$ is density-density interaction term between two particles, for example Coulomb interaction.
 In order to solve the interacting problem exactly, one needs to diagonalize the full Hamiltonian $\hat{H}$ in the many-particle basis which grows exponentially with the number of particles.
 Such a task is often infeasible for large systems and one needs to resort to approximations.
 
-### Mean-field approximaton
+## Mean-field approximaton
 
 The first-order perturbative approximation to the interacting Hamiltonian is the Hartree-Fock approximation also known as the mean-field approximation.
 The mean-field approximates the quartic term $\hat{V}$ in {eq}`hamiltonian` as a sum of bilinear terms weighted by the expectation values the remaining operators:
 :::{math}
 :label: mf_approx
-\hat{V} \approx \hat{V}^{\text{MF}} \equiv \sum_{ij} v_{ij} \left[
+\hat{V} \approx \hat{V}_{\text{MF}} \equiv \sum_{ij} v_{ij} \left[
 \braket{c_i^\dagger c_i} c_j^\dagger c_j - \braket{c_i^\dagger c_j} c_j^\dagger c_i \right]
 :::
 we neglect the superconducting pairing and constant offset terms.
@@ -32,11 +30,11 @@ The expectation value terms  $\langle c_i^\dagger c_j \rangle$ are due to the gr
 The ground-state density matrix reads:
 :::{math}
 :label: density
-\rho_{ij} \equiv \braket{c_i^\dagger c_j } = \text{Tr}\left(e^{-\beta \left(\hat{H^0} + \hat{V}^{\text{MF}} - \mu \hat{N} \right)} c_i^\dagger c_j\right),
+\rho_{ij} \equiv \braket{c_i^\dagger c_j } = \text{Tr}\left(e^{-\beta \left(\hat{H_0} + \hat{V}_{\text{MF}} - \mu \hat{N} \right)} c_i^\dagger c_j\right),
 :::
 where $\beta = 1/ (k_B T)$ is the inverse temperature, $\mu$ is the chemical potential, and $\hat{N} = \sum_i c_i^\dagger c_i$ is the number operator.
 
-### Finite tight-binding grid
+## Finite tight-binding grid
 
 To simplify the mean-field Hamiltonian, we assume a finite, normalised orthogonal tight-binding grid defined by the single-particle basis states:
 
@@ -49,11 +47,11 @@ We project our mean-field interaction in {eq}`mf_approx` onto the tight-binding
 
 :::{math}
 :label: mf_finite
-V^\text{MF}_{nm} = \braket{n | \hat{V}^{\text{MF}} | m} =  \sum_{i} \rho_{ii} v_{in} \delta_{nm} - \rho_{mn} v_{mn},
+V_{\text{MF}, nm} = \braket{n | \hat{V}_{\text{MF}} | m} =  \sum_{i} \rho_{ii} v_{in} \delta_{nm} - \rho_{mn} v_{mn},
 :::
 where $\delta_{nm}$ is the Kronecker delta function.
 
-### Infinite tight-binding grid
+## Infinite tight-binding grid
 
 In the limit of a translationally invariant system, the index $n$ that labels the basis states partitions into two independent variables: the unit cell internal degrees of freedom (spin, orbital, sublattice, etc.) and the position of the unit cell $R_n$:
 
@@ -71,51 +69,7 @@ That allows us to re-write the mean-field interaction in {eq}`mf_finite` as:
 
 :::{math}
 :label: mf_infinite
-V^\text{MF}_{nm} (R) =  \sum_{i} \rho_{ii} (0) v_{in} (0) \delta_{nm} \delta(R) - \rho_{mn}(R) v_{mn}(R),
+V_{\text{MF}, nm} (R) =  \sum_{i} \rho_{ii} (0) v_{in} (0) \delta_{nm} \delta(R) - \rho_{mn}(R) v_{mn}(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 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, 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.