Expand knowledge base

11 files changed, 419 insertions, 14 deletions

diff --git a/content/know/concept/random-phase-approximation/dyson.png b/content/know/concept/random-phase-approximation/dyson.png
new file mode 100644
index 0000000..e92dbf8
--- /dev/null
+++ b/content/know/concept/random-phase-approximation/dyson.png
diff --git a/content/know/concept/random-phase-approximation/index.pdc b/content/know/concept/random-phase-approximation/index.pdc
new file mode 100644
index 0000000..970a884
--- /dev/null
+++ b/content/know/concept/random-phase-approximation/index.pdc
@@ -0,0 +1,185 @@
+---
+title: "Random phase approximation"
+firstLetter: "R"
+publishDate: 2021-12-01
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-11-15T21:01:34+01:00
+draft: false
+markup: pandoc
+---
+
+# Random phase approximation
+
+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.
+We consider the self-energy in the context of [jellium](/know/concept/jellium/),
+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 $1 / \beta$:
+
+$$\begin{aligned}
+    \frac{1}{(2 \pi)^3} \int_{-\infty}^\infty \frac{1}{\hbar \beta} \sum_{n = -\infty}^\infty \cdots \:\dd{\vb{k}}
+    \:\:\sim\:\:
+    k_F^5
+\end{aligned}$$
+
+Meanwhile, every line gives a factor $1 / k_F^2$.
+The [Matsubara Green's function](/know/concept/matsubara-greens-function/) $G^0$
+for a system with continuous translational symmetry
+is found from [equation-of-motion theory](/know/concept/equation-of-motion-theory/):
+
+$$\begin{aligned}
+    W(\vb{k}) = \frac{e^2}{\varepsilon_0 |\vb{k}|^2}
+    \:\:\sim\:\:
+    \frac{1}{k_F^2}
+    \qquad \qquad
+    G_s^0(\vb{k}, i \omega_n^F)
+    = \frac{1}{i \hbar \omega_n^F - \varepsilon_\vb{k}}
+    \:\:\sim\:\:
+    \frac{1}{k_F^2}
+\end{aligned}$$
+
+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 $\Sigma$
+converge to zero faster than lower-order terms.
+
+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 $\vb{k}$,
+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,
+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
+carry the same momentum and energy:
+
+<a href="rpasigma.png">
+<img src="rpasigma.png" style="width:91%;display:block;margin:auto;">
+</a>
+
+Where we have defined the **screened interaction** $W^\mathrm{RPA}$,
+denoted by a double wavy line:
+
+<a href="screened.png">
+<img src="screened.png" style="width:95%;display:block;margin:auto;">
+</a>
+
+Rearranging the above sequence of diagrams quickly leads to the following
+[Dyson equation](/know/concept/dyson-equation/):
+
+<a href="dyson.png">
+<img src="dyson.png" style="width:55%;display:block;margin:auto;">
+</a>
+
+In Fourier space, this equation's linear shape
+means it is algebraic, so we can write it out:
+
+$$\begin{aligned}
+    \boxed{
+        W^\mathrm{RPA}
+        = W + W \Pi_0 W^\mathrm{RPA}
+    }
+\end{aligned}$$
+
+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 $\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$:
+
+<a href="pairbubble.png">
+<img src="pairbubble.png" style="width:45%;display:block;margin:auto;">
+</a>
+
+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 $\Pi_0$:
+
+$$\begin{aligned}
+    W^\mathrm{RPA}(\vb{k}, i \omega_n^B)
+    = \frac{W(\vb{k})}{1 - W(\vb{k}) \: \Pi_0(\vb{k}, i \omega_n^B)}
+    = \frac{e^2}{\varepsilon_0 |\vb{k}|^2 - e^2 \Pi_0(\vb{k}, i \omega_n^B)}
+\end{aligned}$$
+
+Let us evaluate the pair-bubble $\Pi_0$ more concretely.
+The Feynman diagram translates to:
+
+$$\begin{aligned}
+    -\hbar \Pi_0(\vb{k}, i \omega_n^B)
+    &= - \sum_{s} \frac{1}{(2 \pi)^3} \int \frac{1}{\hbar \beta} \sum_{m = -\infty}^\infty
+    \hbar G_s(\vb{k} \!+\! \vb{q}, i \omega_n^B \!+\! i \omega_m^F) \: \hbar G_s(\vb{q}, i \omega_m^F) \dd{\vb{q}}
+    \\
+    &= - \frac{2 \hbar}{(2 \pi)^3} \int \frac{1}{\beta} \sum_{m = -\infty}^\infty
+    \frac{1}{i \hbar \omega_n^B + i \hbar \omega_m^F - \varepsilon_{\vb{k}+\vb{q}}} \: \frac{1}{i \hbar \omega_m^F - \varepsilon_{\vb{q}}} \dd{\vb{q}}
+\end{aligned}$$
+
+Here we recognize a [Matsubara sum](/know/concept/matsubara-sum/),
+and rewrite accordingly.
+Note that the residues of $n_F$ are $1 / (\hbar \beta)$
+when it is a function of frequency,
+and $1 / \beta$ when it is a function of energy, so:
+
+$$\begin{aligned}
+    \Pi_0(\vb{k}, i \omega_n^B)
+    &= \frac{2}{(2 \pi)^3} \int
+    \frac{n_F(\varepsilon_{\vb{k}+\vb{q}} - i \hbar \omega_n^B)}{(\varepsilon_{\vb{k}+\vb{q}} - i \hbar \omega_n^B) - \varepsilon_{\vb{q}}}
+    + \frac{n_F(\varepsilon_{\vb{q}})}{i \hbar \omega_n^B + (\varepsilon_{\vb{q}}) - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}}
+    \\
+    &= \frac{2}{(2 \pi)^3} \int \frac{n_F(\varepsilon_{\vb{q}}) - n_F(\varepsilon_{\vb{k}+\vb{q}})}
+    {i \hbar \omega_n^B + \varepsilon_{\vb{q}} - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}}
+\end{aligned}$$
+
+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 $\Pi_0^R$:
+
+$$\begin{aligned}
+    \boxed{
+        \Pi_0^R(\vb{k}, \omega)
+        = \frac{2}{(2 \pi)^3} \int \frac{n_F(\varepsilon_{\vb{q}}) - n_F(\varepsilon_{\vb{k}+\vb{q}})}
+        {\hbar (\omega + i \eta) + \varepsilon_{\vb{q}} - \varepsilon_{\vb{k}+\vb{q}}} \dd{\vb{q}}
+    }
+\end{aligned}$$
+
+This is as far as we can go before making simplifying assumptions.
+Therefore, we leave it at:
+
+$$\begin{aligned}
+    \boxed{
+        W^\mathrm{RPA}(\vb{k}, \omega)
+        = \frac{e^2}{\varepsilon_0 |\vb{k}|^2 - e^2 \Pi_0(\vb{k}, \omega)}
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Bruus, K. Flensberg,
+    *Many-body quantum theory in condensed matter physics*,
+    2016, Oxford.
diff --git a/content/know/concept/random-phase-approximation/pairbubble.png b/content/know/concept/random-phase-approximation/pairbubble.png
new file mode 100644
index 0000000..c082d19
--- /dev/null
+++ b/content/know/concept/random-phase-approximation/pairbubble.png
diff --git a/content/know/concept/random-phase-approximation/rpasigma.png b/content/know/concept/random-phase-approximation/rpasigma.png
new file mode 100644
index 0000000..87ba3cc
--- /dev/null
+++ b/content/know/concept/random-phase-approximation/rpasigma.png
diff --git a/content/know/concept/random-phase-approximation/screened.png b/content/know/concept/random-phase-approximation/screened.png
new file mode 100644
index 0000000..e8b56e1
--- /dev/null
+++ b/content/know/concept/random-phase-approximation/screened.png
diff --git a/content/know/concept/self-energy/dyson.png b/content/know/concept/self-energy/dyson.png
index efa7a63..f576632 100644
--- a/content/know/concept/self-energy/dyson.png
+++ b/content/know/concept/self-energy/dyson.png
diff --git a/content/know/concept/self-energy/fullgf.png b/content/know/concept/self-energy/fullgf.png
index 3c88c6a..5767dba 100644
--- a/content/know/concept/self-energy/fullgf.png
+++ b/content/know/concept/self-energy/fullgf.png
diff --git a/content/know/concept/self-energy/index.pdc b/content/know/concept/self-energy/index.pdc
index c86f8c5..7e67143 100644
--- a/content/know/concept/self-energy/index.pdc
+++ b/content/know/concept/self-energy/index.pdc
@@ -172,7 +172,7 @@ $$\begin{aligned}
     &= \frac{\displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!}
     \bigg[ \sum_{m = 0}^{n} \frac{n!}{m! (n \!-\! m)!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}}
     \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; (n \!-\! m)}_{\!\Sigma\mathrm{all}} \bigg]}
-    {\displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}}
+    {-\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}}
 \end{aligned}$$
 
 Where the total order is the sum of the orders of all considered diagrams,
@@ -186,8 +186,7 @@ $$\begin{aligned}
     &= \frac{\displaystyle\sum_{m = 0}^{\infty} \frac{1}{2^m m!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}}
     \bigg[ \sum_{n = 0}^\infty \frac{1}{2^{n-m} (n \!-\! m)!}
     \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; (n \!-\! m)}_{\!\Sigma\mathrm{all}} \bigg]}
-    {\displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!}
-    \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}}
+    {-\hbar \displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}}
 \end{aligned}$$
 
 Since both $n$ and $m$ start at zero,
@@ -195,7 +194,7 @@ and the sums include all possible diagrams,
 we see that the second sum in the numerator does not actually depend on $m$:
 
 $$\begin{aligned}
-    G_{ba}
+    -\hbar G_{ba}
     &= \frac{\displaystyle\sum_{m = 0}^{\infty} \frac{1}{2^m m!} \binom{1 \; \mathrm{external}}{\mathrm{order} \; m}_{\!\Sigma\mathrm{all}}
     \bigg[ \sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}} \bigg]}
     {\displaystyle\sum_{n = 0}^\infty \frac{1}{2^n n!} \binom{\mathrm{0\;or\;more\;internal}}{\mathrm{total\;order} \; n}_{\!\Sigma\mathrm{all}}}
@@ -245,7 +244,7 @@ you can convince youself that $G(b,a)$ obeys
 a [Dyson equation](/know/concept/dyson-equation/) involving $\Sigma(y, x)$:
 
 <a href="dyson.png">
-<img src="dyson.png" style="width:90%;display:block;margin:auto;">
+<img src="dyson.png" style="width:95%;display:block;margin:auto;">
 </a>
 
 This makes sense: in the "normal" Dyson equation
diff --git a/content/know/concept/self-energy/selfenergy.png b/content/know/concept/self-energy/selfenergy.png
index 8eaffff..55e182e 100644
--- a/content/know/concept/self-energy/selfenergy.png
+++ b/content/know/concept/self-energy/selfenergy.png
diff --git a/sources/know/concept/random-phase-approximation/main.tex b/sources/know/concept/random-phase-approximation/main.tex
new file mode 100644
index 0000000..11c9bde
--- /dev/null
+++ b/sources/know/concept/random-phase-approximation/main.tex
@@ -0,0 +1,221 @@
+\documentclass[11pt]{article}
+\usepackage[utf8]{inputenc}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{physics}
+
+\usepackage{feynmp}
+\DeclareGraphicsRule{*}{mps}{*}{}
+
+\begin{document}
+
+\begin{center}
+	{\Huge \sc Feynman diagrams}
+\end{center}
+
+\noindent
+Definition of RPA self-energy
+\begin{align*}
+\frac{\Sigma_s^\mathrm{RPA}(\vb{k}, i \omega_n^F)}{-\hbar}
+&\equiv \:\:
+\begin{fmffile}{sigma1order}
+	\begin{fmfgraph*}(30,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- o}
+		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{sigma2order}
+	\begin{fmfgraph*}(55,20)
+		\fmfipair{i,a,b,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.33w,0h)}
+		\fmfiequ{b}{(0.67w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{boson}{b -- o}
+		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{sigma3order}
+	\begin{fmfgraph*}(90,20)
+		\fmfipair{i,a,b,c,d,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.2w,0h)}
+		\fmfiequ{b}{(0.4w,0h)}
+		\fmfiequ{c}{(0.6w,0h)}
+		\fmfiequ{d}{(0.8w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{boson}{b -- c}
+		\fmfi{fermion}{c{up} .. tension 1.5 .. {down}d}
+		\fmfi{fermion}{d{down} .. tension 1.5 .. {up}c}
+		\fmfi{boson}{d -- o}
+		\fmfi{fermion}{i{up} .. tension 3 .. {down}o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{c}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{d}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \: \cdots
+\\
+&= \:\:
+\begin{fmffile}{sigmashort}
+	\begin{fmfgraph*}(40,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{dbl_wiggly}{i -- o}
+		\fmfi{fermion}{i{up} .. tension 2 .. {down}o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\end{align*}
+Definition of screened interaction
+\begin{align*}
+\frac{W^\mathrm{RPA}(\vb{k}, i \omega_n^B)}{-\hbar}
+= \:\:
+\begin{fmffile}{intshort}
+	\begin{fmfgraph*}(25,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{dbl_wiggly}{i -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: &\equiv \:\:
+\begin{fmffile}{int1order}
+	\begin{fmfgraph*}(20,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{int2order}
+	\begin{fmfgraph*}(48,20)
+		\fmfipair{i,a,b,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.33w,0h)}
+		\fmfiequ{b}{(0.67w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{boson}{b -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{int3order}
+	\begin{fmfgraph*}(80,20)
+		\fmfipair{i,a,b,c,d,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.2w,0h)}
+		\fmfiequ{b}{(0.4w,0h)}
+		\fmfiequ{c}{(0.6w,0h)}
+		\fmfiequ{d}{(0.8w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{boson}{b -- c}
+		\fmfi{fermion}{c{up} .. tension 1.5 .. {down}d}
+		\fmfi{fermion}{d{down} .. tension 1.5 .. {up}c}
+		\fmfi{boson}{d -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{c}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{d}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \: \cdots
+\end{align*}
+Dyson equation for screened interaction:
+\begin{align*}
+\begin{fmffile}{dysonlhs}
+	\begin{fmfgraph*}(30,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{dbl_wiggly}{i -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: &= \:\:
+\begin{fmffile}{dyson1rhs}
+	\begin{fmfgraph*}(30,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\:\: + \:\:
+\begin{fmffile}{dyson2rhs}
+	\begin{fmfgraph*}(80,20)
+		\fmfipair{i,a,b,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{a}{(0.33w,0h)}
+		\fmfiequ{b}{(0.67w,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{boson}{i -- a}
+		\fmfi{fermion}{a{up} .. tension 1.5 .. {down}b}
+		\fmfi{fermion}{b{down} .. tension 1.5 .. {up}a}
+		\fmfi{dbl_wiggly}{b -- o}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{a}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{b}
+		\fmfiv{decor.shape=circle,decor.size=4,label.angle=-90,label.dist=3}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\end{align*}
+Pair bubble definition:
+\begin{align*}
+-\hbar \Pi_0(\vb{k}, i \omega_n^B)
+&= \qquad\:\:
+\begin{fmffile}{pairbubble}
+	\begin{fmfgraph*}(30,20)
+		\fmfipair{i,o}
+		\fmfiequ{i}{(0,0h)}
+		\fmfiequ{o}{(w,0h)}
+		\fmfi{fermion,label=$\tilde{\vb{k}}\!+\!\tilde{\vb{q}}$,label.dist=3}{i{up} .. tension 1.5 .. {down}o}
+		\fmfi{fermion,label=$\tilde{\vb{q}}$,label.dist=3}{o{down} .. tension 1.5 .. {up}i}
+		\fmfiv{decor.shape=circle,decor.size=4,label=$\tilde{\mathbf{k}}\!\leadsto$,label.angle=160,label.dist=5}{i}
+		\fmfiv{decor.shape=circle,decor.size=4,label=$\leadsto\!\tilde{\mathbf{k}}$,label.angle=20,label.dist=5}{o}
+	\end{fmfgraph*}
+\end{fmffile}
+\end{align*}
+
+\end{document}
diff --git a/sources/know/concept/self-energy/main.tex b/sources/know/concept/self-energy/main.tex
index 114b433..e359053 100644
--- a/sources/know/concept/self-energy/main.tex
+++ b/sources/know/concept/self-energy/main.tex
@@ -15,8 +15,8 @@
 
 \noindent
 Full Green's function
-\begin{align}
-	G(b,a)
+\begin{align*}
+-\hbar G(b,a)
 &= \quad
 \begin{fmffile}{0order1}
 	\begin{fmfgraph*}(30,20)
@@ -67,10 +67,10 @@ Full Green's function
 	\end{fmfgraph*}
 \end{fmffile}
 \quad + \:\: \cdots
-\end{align}
+\end{align*}
 Self-energy definition
-\begin{align}
-\Sigma(y,x)
+\begin{align*}
+\frac{\Sigma(y,x)}{-\hbar}
 =\!\!\!\!
 \begin{fmffile}{selfenergy}
 	\begin{fmfgraph*}(20,20)
@@ -123,10 +123,10 @@ Self-energy definition
 	\end{fmfgraph*}
 \end{fmffile}
 \:\:\: + \: \cdots
-\end{align}
+\end{align*}
 Dyson equation
-\begin{align}
-G(b,a)
+\begin{align*}
+-\hbar G(b,a)
 = \quad
 \begin{fmffile}{fullgf}
 	\begin{fmfgraph*}(50,20)
@@ -166,6 +166,6 @@ G(b,a)
 		\fmfblob{15}{v}
 	\end{fmfgraph*}
 \end{fmffile}
-\end{align}
+\end{align*}
 
 \end{document}