move Kostas' notes from deprecated notebook to file

notes/README.md

-Put your written notes here.
+# mean-field solver 
+
+## Some background
+
+### What is a mean field solver and why do we need it?
+
+Solving many-body systems such as a hellium atom in chemistry ([reference I used here](https://adambaskerville.github.io/posts/HartreeFockGuide/)):
+
+<p style="text-align:center;"><img src="https://adambaskerville.github.io/assets/img/HartreeFockInfographic.png" width="400"></p>
+
+Why is it useful?
+* If we tried to solve $N$ particles with $M$ distinct states exactly, the full Hilbert space is $M^N$. Mean-field maps into a much smaller non-interacting system of size $M N$
+* Quantitative accuracy is either hit or miss, but gives a good understanding of the qualitative behaviour of the system.
+
+### General idea through an example
+
+#### The Ising model
+
+Consider an Ising model:
+
+<p style="text-align:center;"><img src="https://pbs.twimg.com/media/DKu31zYXcAECSDL.jpg" width="400"></p>
+
+with the Hamiltonian (straight from [wikipedia](https://en.wikipedia.org/wiki/Mean-field_theory#Ising_model)):
+
+$$
+H=-J\sum _{\langle i,j\rangle }s_{i}s_{j}-h\sum _{i}s_{i}
+$$
+
+where $s_i$ is the spin of the $i$th particle, $J$ is the exchange coupling, and $h$ is the external magnetic field. The sum is over nearest neighbours.
+
+#### Identifying and applying mean field
+
+It is an interacting problem and thus a headache to solve. Rather than consider all the particles, lets consider a single particle in an **effective field** of other spins. To do that, lets define the average spin $m_{i}\equiv \langle s_{i} \rangle$ and re-write the $s_i s_j$ product into:
+
+
+$$
+s_i s_j = m_i m_j + m_i \delta s_j + m_j \delta s_i + \delta s_i \delta s_j 
+$$
+
+if we assume the fluctuations are small, then at the very least the last term can be neglected:
+
+
+$$
+s_i s_j \approx m_i m_j + m_i \delta s_j + m_j \delta s_i = m_i  s_j + m_j s_i - m_i m_j
+$$
+
+and thus the Hamiltonian becomes:
+
+$$
+H\approx H^{\text{MF}}\equiv -J m \sum _{\langle i,j\rangle }(2 s_{i} - m)-h\sum _{i}s_{i}
+$$
+
+where for simplicity I also assumed that the average spin $m_i$ is constant for all particles.
+
+#### Self-consistency
+
+The average mean-field $m$ here acts as a variable, or better yet, **an initial guess**. After solving $H_{\text{MF}}$, we need to make sure that the $m$ is **self-consistent** with the new $s_i$:
+
+$$
+m = \frac{1}{N} \sum_i^N \langle s_i \rangle.
+$$
+
+So we re-calculate $m$, plug it back into the equation of $H_{\text{MF}}$, and repeat until $m$ converges.
+
+#### Summary
+
+1. Identify the mean-field variables and construct the mean-field Hamiltonian.
+2. Guess the initial mean-field.
+3. Self-consistency loop:
+    1. Solve the mean-field Hamiltonian $H_{\text{MF}}$ for the given mean-field.
+    2. Calculate the new mean-field.
+    3. Check convergence. If not converged, go back to step 3.1. with the new mean-field.
+4. ???
+5. Profit.
+
+
+## Why waste time with this?
+
+Interacting systems have become quite hot research field in condensed matter physics (I mean, take a look at graphene). However, numerical packages to solve them on tight-binding systems lack the following:
+* Not many well-maintained packages.
+* Code is needlessly complex and documentation is lacking.
+* Lack generality.
+
+## Idea behind our implementation
+
+### Identifying mean-fields
+
+#### Real Space
+
+You can find the whole theory [here](https://hackmd.io/@-DUiWUyjQXei-EsdckYO-w/HyEbQhIjo).
+
+Here the the main points. A general particle number preserving interaction with all mean-fields can be written as a sort of [Wick's contraction](https://en.wikipedia.org/wiki/Wick%27s_theorem):
+
+$$
+V = \frac{1}{2}\sum_{ijkl} v_{ijkl} c_i^{\dagger} c_j^{\dagger} c_l c_k
+\approx
+\frac12 \sum_{ijkl} v_{ijkl} \left[ \langle c_i^{\dagger} c_k \rangle c_j^{\dagger} c_l - \langle c_j^{\dagger} c_k \rangle c_i^{\dagger} c_l - \langle c_i^{\dagger} c_l \rangle c_j^{\dagger} c_k + \langle c_j^{\dagger} c_l \rangle c_i^{\dagger} c_k \right]
+$$
+(we neglect superconductivity)
+
+here $i,j,k,l$ label any degree of freedom written on a tight-binding grid, so we maintain full generality. 
+
+The mean-fields are in terms of second quantization operators, but how do we translate the problem to a tight-binding grid/matrix problem?
+
+$$
+\langle c_i^{\dagger} c_j\rangle = \langle \Psi_F|c_i^{\dagger} c_j | \Psi_F \rangle
+$$
+
+whereas $|\Psi_F \rangle = \Pi_{i=0}^{N_F} b_i^\dagger |0\rangle$. To make sense of things, we need to transform between $c_i$ basis (position + internal dof basis) into the $b_i$ basis (eigenfunction of a given mean-field Hamiltonian):
+
+$$
+c_i^\dagger = \sum_{k} U_{ik} b_k^\dagger
+$$
+
+where $U$ is the matrix of eigenvectors of the mean-field Hamiltonian:
+
+$$
+U_{ik} = \langle{i|\psi_k} \rangle.
+$$
+
+That gives us:
+
+$$
+c_i^{\dagger} c_j = \sum_{k, l} U_{ik} U_{lj}^* b_k^\dagger b_{l}
+$$
+
+and its expectation value gives us the mean-field ... field $F_{ij}$:
+
+$$
+F_{ij} = \langle c_i^{\dagger} c_j\rangle =  \sum_{k, l} U_{ik} U_{lj}^* \langle \Psi_F| b_k^\dagger b_{l}| \Psi_F \rangle =  \sum_{k} U_{ik} U_{kj}^{*}
+$$
+
+Coming back to the interaction, under mean-field and on our chosen tight-binding grid it reads:
+
+$$
+V_{nm} \approx -\sum_{ij} F_{ij} \left(v_{inmj} - v_{injm} \right)
+$$
+
+In the simple case of a Coulomb interaction, the potential reads:
+
+$$
+V_{nm} = -F_{mn} v_{mn} + \sum_{i} F_{ii} v_{in} \delta_{nm}
+$$
+
+where the second term is the Direct Coulomb interaction, and the first term is the exchange interaction.
+
+#### k-space or translational invariance case
+
+The above works for a finite sized tight-binding model, but what about a periodic system? In that case, we can use the Fourier transform to write the mean-field Hamiltonian in k-space. I'll spare you the details, the final result reads:
+
+$$
+V_{nm}(k) =-F_{mn}(k) \circledast v_{mn}(k) + \sum_{p} \rho_{p} v_{pn}(0) \delta_{nm}
+$$
+
+where $\rho_{p}$ is the particle density at unit cell site $p$, averaged over a k-grid:
+
+$$
+\rho_{p} = \int F_{pp}(k) dk 
+$$
+
+Once again, the first term (exchange) is purely responsible for the hopping whereas the second term (direct) is a potential term coming from the mean-field. 
+
+## Scaling of the algorithm
+
+Lets say $M$ is the number of degrees of freedom within the unit cell, and $N$ is the number of k-points along a direction (we consider 2D problem here). Then the following steps limit the scaling of the algorithm:
+* Eigenvalue problem for each k-point: $O(N^2 M^3)$.
+* Convolution in k-space: $O(N^4 M^2)$. In this case, this is the most expensive step.
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