Merge remote-tracking branch 'origin/main' into explore-fix-hamiltonian-phase-v2

Author: thchr

docs/make.jl

@@ -29,7 +29,7 @@ makedocs(;
         "Internal API" => "internal-api.md",
         "Theory" => "theory.md",
         "Developer notes" => [
-            "devdocs/README.md",
+            "Overview" => "devdocs/README.md",
             "devdocs/trs_notes.md",
             "devdocs/fourier.md",
             "devdocs/1d_example.md",

docs/src/devdocs/1d_example.md

@@ -4,35 +4,36 @@ Assume we have two sites in a one dimensional lattice of parameter $a=1$ where w
 inversion-even orbital at the origin denoted by (1a|A); and an inversion-odd orbital at $x=1/2$
 denoted by (1b|B).
 
-**Notation:** for simplicity we will denote orbitals (1a|A) as $a$ and (1b|B) as $b$.
-Additionally, we will indicate with a subscript the unit cell it belongs to. For example,
-$a_0$ will be placed at $x=0$, while $b_1$ will be placed at $x=3/2$ or $a_{-1}$ at $x=-1$.
+!!! note "Notation"
+    For brevity we will denote orbitals (1a|A) as $a$ and (1b|B) as $b$.
+    Additionally, we will indicate with a subscript the unit cell it belongs to.
+    For example, $a_0$ will be placed at $x=0$, while $b_1$ will be placed at $x=3/2$ or $a_{-1}$ at $x=-1$.
 
-## Deduction by inspection
+## Derivation by inspection
 
 These orbitals will transform under inversion symmetry in the following way:
 
-$$
-\mathcal{I} a_n = a_{-n}; \quad \mathcal{I} b_n = -b_{-n-1}
-$$
+```math
+\mathcal{I} a_n = a_{-n}; \quad \mathcal{I} b_n = -b_{-n-1}.
+```
 
 Then the most general inversion-symmetric Hamiltonian for first nearest neighbors is:
 
-$$
-\mathcal{H} = \sum_n t (a_n^\dagger b_n - a_n^\dagger b_{n-1}) + \text{c.c.}
-$$
+```math
+\mathcal{H} = \sum_n t (a_n^\dagger b_n - a_n^\dagger b_{n-1}) + \text{c.c.}.
+```
 
 It is easy to check that this Hamiltonian is inversion symmetric.
 
 If we consider the following Fourier transform: $a_n = \frac{1}{\sqrt{N}} \sum_n e^{-ikn} a_k$,
 and $b_n = \frac{1}{\sqrt{N}} \sum_n e^{-ik(n+1/2)} b_k$
 then the Hamiltonian in $k$-space will look like:
 
-$$
-\mathcal{H} = \sum_k 2it \sin(k/2) a_k^\dagger b_k + \text{c.c}
-$$
+```math
+\mathcal{H} = \sum_k 2it \sin(k/2) a_k^\dagger b_k + \text{c.c}.
+```
 
-## Deduction from our method
+## Derivation from our method
 
 First, consider the translation $t=0$. Then, remember that $\Delta_{\alpha\to\beta+R} =
 \mathbf{q}_\beta + \mathbf{R} - \mathbf{q}_\alpha$.
@@ -44,25 +45,24 @@ First, consider the translation $t=0$. Then, remember that $\Delta_{\alpha\to\be
 
 Then the non-symmetrized Hamiltonian for this translation will be:
 
-$$
+```math
 H_{k,0} = \begin{pmatrix} t_{a\to a} & t_{b\to a}^1e^{ik/2}+t_{b\to a}^2e^{-ik/2} \\
-t_{a\to b}^1e^{ik/2}+t_{a\to b}^2e^{-ik/2} & t_{b\to b} \end{pmatrix}
-$$
+t_{a\to b}^1e^{ik/2}+t_{a\to b}^2e^{-ik/2} & t_{b\to b} \end{pmatrix}.
+```
 
 Let's proceed now to symmetrize this Hamiltonian. Since we only have inversion we only need
 to check that: $H_{k,0} = \rho(\mathcal{I})H_{k,0}\rho^{-1}(\mathcal{I})$, where:
 
-$$
-\rho(\mathcal{I}) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
-$$
+```math
+\rho(\mathcal{I}) = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.
+```
 
-Then, that constraint imposes that: $\left\{ \begin{matrix} t_{b\to a}^1 = -t_{b\to a}^2 \\
-t_{a\to b}^1 = -t_{a\to b}^2 \end{matrix} \right.$, so the Hamiltonian will look like:
+Then, that constraint imposes that: $\begin{cases} t_{b\to a}^1 = -t_{b\to a}^2 \\
+t_{a\to b}^1 = -t_{a\to b}^2 \end{cases}$, so the Hamiltonian will look like:
 
-$$
+```math
 H_{k,0} = \begin{pmatrix} t_{a\to a} & 2it_{b\to a} \sin(k/2) \\
-2it_{a\to b} \sin(k/2) & t_{b\to b} \end{pmatrix}
-$$
+2it_{a\to b} \sin(k/2) & t_{b\to b} \end{pmatrix},
+```
 
-Which is exactly the Hamiltonian deduced with the previous method but with onsite terms and
-without hermiticity imposed.
+which is exactly the Hamiltonian deduced with the previous method but with onsite terms and without hermiticity imposed.

docs/src/devdocs/README.md

@@ -1,4 +1,4 @@
-# Developer Documentation
+# Developer notes overview
 
 These notes are for developers working on the SymmetricTightBinding.jl internals. For the
 user-facing theory exposition, see [`theory.md`](../theory.md).

docs/src/devdocs/fourier.md

@@ -1,6 +1,6 @@
-# Notes on Fourier Transforms and Different Conventions
+# Fourier transforms conventions
 
-This document presents a brief overview of Fourier transforms, focusing on two conventions that have been used during the development of this codebase.
+This document presents a brief overview of Fourier transforms in the tight-binding setting, focusing on two conventions that have been used during the development of this codebase.
 
 Specifically, we distinguish between two common conventions for the Fourier transform. For lack of standard terminology, we will refer to them as: