1 files changed, 41 insertions, 41 deletions
diff --git a/source/know/concept/random-phase-approximation/index.md b/source/know/concept/random-phase-approximation/index.md
index ac007eb..0f53136 100644
--- a/source/know/concept/random-phase-approximation/index.md
+++ b/source/know/concept/random-phase-approximation/index.md
@@ -8,19 +8,19 @@ categories:
 layout: "concept"
 ---
 
-Recall that the [self-energy](/know/concept/self-energy/) $\Sigma$
+Recall that the [self-energy](/know/concept/self-energy/) $$\Sigma$$
 is defined as a sum of [Feynman diagrams](/know/concept/feynman-diagram/),
-which each have an order $n$ equal to the number of interaction lines.
+which each have an order $$n$$ equal to the number of interaction lines.
 We consider the self-energy in the context of [jellium](/know/concept/jellium/),
-so the interaction lines $W$ represent Coulomb repulsion,
+so the interaction lines $$W$$ represent Coulomb repulsion,
 and we use [imaginary time](/know/concept/imaginary-time/).
 
 Let us non-dimensionalize the Feynman diagrams in the self-energy,
-by measuring momenta in units of $\hbar k_F$,
-and energies in $\epsilon_F = \hbar^2 k_F^2 / (2 m)$.
-Each internal variable then gives a factor $k_F^5$,
-where $k_F^3$ comes from the 3D momentum integral,
-and $k_F^2$ from the energy $1 / \beta$:
+by measuring momenta in units of $$\hbar k_F$$,
+and energies in $$\epsilon_F = \hbar^2 k_F^2 / (2 m)$$.
+Each internal variable then gives a factor $$k_F^5$$,
+where $$k_F^3$$ comes from the 3D momentum integral,
+and $$k_F^2$$ from the energy $$1 / \beta$$:
 
 $$\begin{aligned}
     \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty \cdots \:\dd{\vb{k}}
@@ -28,8 +28,8 @@ $$\begin{aligned}
     k_F^5
 \end{aligned}$$
 
-Meanwhile, every line gives a factor $1 / k_F^2$.
-The [Matsubara Green's function](/know/concept/matsubara-greens-function/) $G^0$
+Meanwhile, every line gives a factor $$1 / k_F^2$$.
+The [Matsubara Green's function](/know/concept/matsubara-greens-function/) $$G^0$$
 for a system with continuous translational symmetry
 is found from [equation-of-motion theory](/know/concept/equation-of-motion-theory/):
 
@@ -44,38 +44,38 @@ $$\begin{aligned}
     \frac{1}{k_F^2}
 \end{aligned}$$
 
-An $n$th-order diagram in $\Sigma$ contains $n$ interaction lines,
-$2n\!-\!1$ fermion lines, and $n$ integrals,
-so in total it evolves as $1 / k_F^{n-2}$.
-In jellium, we know that the electron density is proportional to $k_F^3$,
-so for high densities we can rest assured that higher-order terms in $\Sigma$
+An $$n$$th-order diagram in $$\Sigma$$ contains $$n$$ interaction lines,
+$$2n\!-\!1$$ fermion lines, and $$n$$ integrals,
+so in total it evolves as $$1 / k_F^{n-2}$$.
+In jellium, we know that the electron density is proportional to $$k_F^3$$,
+so for high densities we can rest assured that higher-order terms in $$\Sigma$$
 converge to zero faster than lower-order terms.
 
-However, at a given order $n$, not all diagrams are equally important.
+However, at a given order $$n$$, not all diagrams are equally important.
 In a given diagram, due to momentum conservation,
 some interaction lines carry the same momentum variable.
-Because $W(\vb{k}) \propto 1 / |\vb{k}|^2$,
-small $\vb{k}$ make a large contribution,
-and the more interaction lines depend on the same $\vb{k}$,
+Because $$W(\vb{k}) \propto 1 / |\vb{k}|^2$$,
+small $$\vb{k}$$ make a large contribution,
+and the more interaction lines depend on the same $$\vb{k}$$,
 the larger the contribution becomes.
 
 In other words, each diagram is dominated by contributions
 from the momentum carried by the largest number of interactions.
-At order $n$, there is one diagram
-where all $n$ interactions carry the same momentum,
+At order $$n$$, there is one diagram
+where all $$n$$ interactions carry the same momentum,
 and this one dominates all others at this order.
 
 The **random phase approximation** consists of removing most diagrams
-from the defintion of the full self-energy $\Sigma$,
-leaving only the single most divergent one at each order $n$,
-i.e. the ones where all $n$ interaction lines
+from the defintion of the full self-energy $$\Sigma$$,
+leaving only the single most divergent one at each order $$n$$,
+i.e. the ones where all $$n$$ interaction lines
 carry the same momentum and energy:
 
 <a href="rpasigma.png">
 <img src="rpasigma.png" style="width:92%">
 </a>
 
-Where we have defined the **screened interaction** $W^\mathrm{RPA}$,
+Where we have defined the **screened interaction** $$W^\mathrm{RPA}$$,
 denoted by a double wavy line:
 
 <a href="screened.png">
@@ -99,19 +99,19 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where we have defined the **pair-bubble** $\Pi_0$ as follows,
-with an internal wavevector $\vb{q}$, fermionic frequency $i \omega_m^F$, and spin $s$.
-Abbreviating $\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)$
-and $\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$:
+Where we have defined the **pair-bubble** $$\Pi_0$$ as follows,
+with an internal wavevector $$\vb{q}$$, fermionic frequency $$i \omega_m^F$$, and spin $$s$$.
+Abbreviating $$\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)$$
+and $$\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$$:
 
 <a href="pairbubble.png">
 <img src="pairbubble.png" style="width:45%">
 </a>
 
-We isolate the Dyson equation for $W^\mathrm{RPA}$,
+We isolate the Dyson equation for $$W^\mathrm{RPA}$$,
 which reveals its physical interpretation as a *screened* interaction:
-the "raw" interaction $W \!=\! e^2 / (\varepsilon_0 |\vb{k}|^2)$
-is weakened by a term containing $\Pi_0$:
+the "raw" interaction $$W \!=\! e^2 / (\varepsilon_0 |\vb{k}|^2)$$
+is weakened by a term containing $$\Pi_0$$:
 
 $$\begin{aligned}
     W^\mathrm{RPA}(\vb{k}, i \omega_n^B)
@@ -119,7 +119,7 @@ $$\begin{aligned}
     = \frac{e^2}{\varepsilon_0 |\vb{k}|^2 - e^2 \Pi_0(\vb{k}, i \omega_n^B)}
 \end{aligned}$$
 
-Let us evaluate the pair-bubble $\Pi_0$ more concretely.
+Let us evaluate the pair-bubble $$\Pi_0$$ more concretely.
 The Feynman diagram translates to:
 
 $$\begin{aligned}
@@ -133,9 +133,9 @@ $$\begin{aligned}
 
 Here we recognize a [Matsubara sum](/know/concept/matsubara-sum/),
 and rewrite accordingly.
-Note that the residues of $n_F$ are $1 / (\hbar \beta)$
+Note that the residues of $$n_F$$ are $$1 / (\hbar \beta)$$
 when it is a function of frequency,
-and $1 / \beta$ when it is a function of energy, so:
+and $$1 / \beta$$ when it is a function of energy, so:
 
 $$\begin{aligned}
     \Pi_0(\vb{k}, i \omega_n^B)
@@ -147,12 +147,12 @@ $$\begin{aligned}
     {i \hbar \omega_n^B + \varepsilon_{\vb{q}} - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}}
 \end{aligned}$$
 
-Where we have used that $n_F(\varepsilon \!+\! i \hbar \omega_n^B) = n_F(\varepsilon)$.
-Analogously to extracting the retarded Green's function $G^R(\omega)$
-from the Matsubara Green's function $G^0(i \omega_n^F)$,
-we replace $i \omega_n^F \to \omega \!+\! i \eta$,
-where $\eta \to 0^+$ is a positive infinitesimal,
-yielding the retarded pair-bubble $\Pi_0^R$:
+Where we have used that $$n_F(\varepsilon \!+\! i \hbar \omega_n^B) = n_F(\varepsilon)$$.
+Analogously to extracting the retarded Green's function $$G^R(\omega)$$
+from the Matsubara Green's function $$G^0(i \omega_n^F)$$,
+we replace $$i \omega_n^F \to \omega \!+\! i \eta$$,
+where $$\eta \to 0^+$$ is a positive infinitesimal,
+yielding the retarded pair-bubble $$\Pi_0^R$$:
 
 $$\begin{aligned}
     \boxed{