From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- .../concept/random-phase-approximation/index.md | 82 +++++++++++----------- 1 file changed, 41 insertions(+), 41 deletions(-) (limited to 'source/know/concept/random-phase-approximation') 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$$ interaction lines carry the same momentum and energy: -Where we have defined the **screened interaction** $W^\mathrm{RPA}$, +Where we have defined the **screened interaction** $$W^\mathrm{RPA}$$, denoted by a double wavy line: @@ -99,19 +99,19 @@ $$\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)$$: -We isolate the Dyson equation for $W^\mathrm{RPA}$, +We isolate the Dyson equation for $$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{ -- cgit v1.2.3