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+---
+title: "Random phase approximation"
+date: 2021-12-01
+categories:
+- Physics
+- Quantum mechanics
+layout: "concept"
+---
+
+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:92%">
+</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%">
+</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%">
+</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%">
+</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.
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