clean up notes

docs/source/conf.py

@@ -71,6 +71,8 @@ intersphinx_mapping = {
 
 default_role = "autolink"
 
+latex_elements = {"extrapackages": r"\usepackage{braket}"}
+
 # Add any paths that contain templates here, relative to this directory.
 templates_path = ["_templates"]
 

docs/source/mf_notes.md

@@ -3,87 +3,56 @@
 ## Interacting problems
 
 In physics, one often encounters problems where a system of multiple particles interact with each other.
-By using the second quantization notation, a general Hamiltonian of such system reads:
+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_{ijkl} v_{ijkl} c_i^\dagger c_j^\dagger c_l c_k
+\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 second term $\hat{V}$ is the interaction term between two particles.
+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 often needs to resort to approximations.
+Such a task is often infeasible for large systems and one needs to resort to approximations.
 
-## Mean-field approximation
+## Mean-field Hamiltonian
 
-In many interacting systems, there exist constant order parameters $\langle A \rangle$ that describe the phase of the system.
-Here we define $\hat{A}$ as some operator and $\langle \rangle$ denotes the expectation value with respect to the ground state of the system.
-Famous examples of such order parameter is the magnetization in a ferromagnet and the superconducting order parameter in a superconductor.
-If we are interested in properties of the system close to the ground state, we can re-write the operator $\hat{A}$ around the order parameter:
+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:
 
 $$
-\hat{A} = \langle A \rangle + \delta \hat{A},
+\hat{V} \approx \hat{V}^{\text{MF}} \equiv \sum_{ij} v_{ij} \left[
+\braket{c_i^\dagger c_i} c_j^\dagger c_j - \braket{c_i^\dagger c_j} c_j^\dagger c_i \right]
 $$
 
-where operator $\delta \hat{A}$ describes the fluctuations around the order parameter.
-Let us consider an additional operator $\hat{B}$ and say we are interested in the product of the two operators $\hat{A}\hat{B}$.
-If we assume that the fluctuations $\delta$ are small, we can approximate the product of operators into a sum of single operators and the product of the expectation values:
+we neglect the superconducting pairing and constant offset terms.
+The expectation value terms  $\langle c_i^\dagger c_j \rangle$ are due to the ground-state density matrix:
 
 $$
-\hat{A}\hat{B} \approx \langle A \rangle \hat{B} + \hat{A} \langle B \rangle - \langle A \rangle \langle B \rangle
+\rho_{ij} \equiv \langle c_i^\dagger c_j \rangle,
 $$
 
-where we neglect $\delta^2$ terms.
-This approximation is known as the mean-field approximation.
-
-## Mean-field Hamiltonian
+and therefore act as an effective field acting on the system.
 
-We apply the mean-field approximation to the quartic interaction term in {eq}`hamiltonian`:
-
-$$
-V \approx \frac12 \sum_{ijkl} v_{ijkl} \left[
-\langle c_i^\dagger c_k \rangle c_j^\dagger c_l - \langle c_i^\dagger c_l \rangle c_j^\dagger c_k - \langle c_j^\dagger c_k \rangle c_i^\dagger c_l + \langle c_j^\dagger c_l \rangle c_i^\dagger c_k \right]
-$$
-
-where we make use of Wicks theorem to simplify the expression and neglect the superconducting pairing and constant offset terms.
-
-:::{admonition} Derivation of the mean-field Hamiltonian with Wicks theorem
+<!-- :::{admonition} Derivation of the mean-field Hamiltonian with Wicks theorem
 :class: dropdown info
 ```{include} mf_details.md
 ```
-:::
-
-The expectation value terms $\langle c_i^\dagger c_j \rangle$ are the density matrix elements of the ground state of the system:
-$$
-\rho_{ij} = \langle c_i^\dagger c_j \rangle.
-$$
+::: -->
 
 ### Tight-binding grid
 
-We project $\hat{V}$ onto a tight-binding grid:
+To simplify the mean-field Hamiltonian, we assume a normalised orthogonal tight-binding grid defined by the single-particle basis states:
 
 $$
-V_{nm} = \langle n | \hat{V} | m \rangle = \\
-\frac12 \left[ \sum_{ik} v_{inkm} F_{ik} - \sum_{jk} v_{njkm} F_{jk} - \sum_{il} v_{inml} F_{il} + \sum_{jl} v_{njml} F_{jl} \right] = \\
--\sum_{ij} F_{ij} \left(v_{inmj} - v_{injm} \right)
+\ket{n} = c^\dagger_n\ket{\text{vac}}
 $$
 
-where I used the $v_{ijkl} = v_{jilk}$ symmetry from Coulomb.
-
-For density-density interactions (like Coulomb repulsion) the interaction tensor reads:
+where $\ket{\text{vac}}$ is the vacuum state.
+We project our mean-field interaction onto the tight-binding grid:
 
 $$
-v_{ijkl} = v_{ij} \delta_{ik} \delta_{jl},
+V^\text{MF}_{nm} = \langle n | \hat{V}^{\text{MF}} | m \rangle =  \sum_{i} \rho_{ii} v_{in} \delta_{nm} - \rho_{mn} v_{mn},
 $$
-
-which simplifies the interaction to:
-
-$$
-V_{nm} = - \sum_{ij} F_{ij} \left(v_{inmj} - v_{injm} \right) = \\
--\sum_{ij}F_{ij} v_{in} \delta_{im} \delta_{nj} + \sum_{ij}F_{ij} v_{in} \delta_{ij} \delta_{nm} = \\
--F_{mn} v_{mn} + \sum_{i} F_{ii} v_{in} \delta_{nm}
-$$
-
-the first term is the exchange interaction whereas the second one is the direct interaction.
+where $\delta_{nm}$ is the Kronecker delta function.

pyproject.toml

@@ -47,5 +47,5 @@ include = [
 ]
 
 [tool.codespell]
-skip = "*.ipynb,"
-ignore-words-list = "multline,"
+skip = "*.ipynb"
+ignore-words-list = "multline, ket, bra, braket"