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authorPrefetch2023-01-01 16:40:56 +0100
committerPrefetch2023-01-01 17:02:29 +0100
commitb1a9b1b9b2f04efd6dc39bd2a02c544d34d1259c (patch)
tree1fd87919deee17e58f8ad19c09abd54bd4a70886 /source/know/concept/random-phase-approximation
parent1d700ab734aa9b6711eb31796beb25cb7659d8e0 (diff)
Change license, add Makefile, add image caching control
Diffstat (limited to 'source/know/concept/random-phase-approximation')
-rw-r--r--source/know/concept/random-phase-approximation/index.md12
1 files changed, 8 insertions, 4 deletions
diff --git a/source/know/concept/random-phase-approximation/index.md b/source/know/concept/random-phase-approximation/index.md
index 0d0b428..03fd302 100644
--- a/source/know/concept/random-phase-approximation/index.md
+++ b/source/know/concept/random-phase-approximation/index.md
@@ -71,17 +71,20 @@ 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:
-{% include image.html file="self-energy.png" width="92%" alt="RPA self-energy definition" %}
+{% include image.html file="self-energy.png" width="92%"
+ alt="RPA self-energy definition" %}
Where we have defined the **screened interaction** $$W^\mathrm{RPA}$$,
denoted by a double wavy line:
-{% include image.html file="interaction.png" width="95%" alt="RPA screened interaction definition" %}
+{% include image.html file="interaction.png" width="95%"
+ alt="RPA screened interaction definition" %}
Rearranging the above sequence of diagrams quickly leads to the following
[Dyson equation](/know/concept/dyson-equation/):
-{% include image.html file="dyson.png" width="55%" alt="Dyson equation for screened interaction" %}
+{% include image.html file="dyson.png" width="55%"
+ alt="Dyson equation for screened interaction" %}
In Fourier space, this equation's linear shape
means it is algebraic, so we can write it out:
@@ -98,7 +101,8 @@ with an internal wavevector $$\vb{q}$$, fermionic frequency $$i \omega_m^F$$, an
Abbreviating $$\tilde{\vb{k}} \equiv (\vb{k}, i \omega_n^B)$$
and $$\tilde{\vb{q}} \equiv (\vb{q}, i \omega_n^F)$$:
-{% include image.html file="pairbubble.png" width="45%" alt="Internal variables of pair-bubble diagram" %}
+{% include image.html file="pairbubble.png" width="45%"
+ alt="Internal variables of pair-bubble diagram" %}
We isolate the Dyson equation for $$W^\mathrm{RPA}$$,
which reveals its physical interpretation as a *screened* interaction: