summaryrefslogtreecommitdiff
path: root/source/know/concept/matsubara-greens-function
diff options
context:
space:
mode:
Diffstat (limited to 'source/know/concept/matsubara-greens-function')
-rw-r--r--source/know/concept/matsubara-greens-function/index.md41
1 files changed, 13 insertions, 28 deletions
diff --git a/source/know/concept/matsubara-greens-function/index.md b/source/know/concept/matsubara-greens-function/index.md
index fdcadb3..fd46abf 100644
--- a/source/know/concept/matsubara-greens-function/index.md
+++ b/source/know/concept/matsubara-greens-function/index.md
@@ -83,11 +83,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-period"/>
-<label for="proof-period">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-period">Proof.</label>
+
+{% include proof/start.html id="proof-period" -%}
First $$\tau \!-\! \tau' < 0$$.
We insert the argument $$\tau \!-\! \tau' \!+\! \hbar \beta$$,
and use the cyclic property:
@@ -133,9 +130,8 @@ $$\begin{aligned}
\\
&= \pm C_{AB}(\tau \!-\! \tau')
\end{aligned}$$
+{% include proof/end.html id="proof-period" %}
-</div>
-</div>
Due to this limited domain $$\tau \in [-\hbar \beta, \hbar \beta]$$,
the [Fourier transform](/know/concept/fourier-transform/)
@@ -157,11 +153,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-FT-def"/>
-<label for="proof-FT-def">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-FT-def">Proof.</label>
+
+{% include proof/start.html id="proof-fourier-def" -%}
We will prove that one is indeed the inverse of the other.
We demand that the inverse FT of the forward FT of $$C_{AB}(\tau)$$
is simply $$C_{AB}(\tau)$$ again:
@@ -198,9 +191,8 @@ $$\begin{aligned}
\\
&= C_{AB}(\tau)
\end{aligned}$$
+{% include proof/end.html id="proof-fourier-def" %}
-</div>
-</div>
Let us now define the **Matsubara frequencies** $$\omega_n$$
as a species-dependent subset of $$k_n$$:
@@ -228,11 +220,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-FT-alt"/>
-<label for="proof-FT-alt">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-FT-alt">Proof.</label>
+
+{% include proof/start.html id="proof-fourier-alt" -%}
We split the integral, shift its limits,
and use the (anti)periodicity of $$C_{AB}$$:
@@ -265,9 +254,8 @@ $$\begin{aligned}
\\
&= \frac{1}{2} \big( 1 \pm e^{-i k_n \hbar \beta} \big) \int_{-\hbar \beta}^0 C_{AB}(\tau) \: e^{i k_n \tau} \dd{\tau}
\end{aligned}$$
+{% include proof/end.html id="proof-fourier-alt" %}
-</div>
-</div>
If we actually evaluate this,
we obtain the following form of $$C_{AB}$$,
@@ -283,11 +271,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-Lehmann"/>
-<label for="proof-Lehmann">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-Lehmann">Proof.</label>
+
+{% include proof/start.html id="proof-lehmann" -%}
For $$\tau \!-\! \tau' > 0$$, we start by expanding
in the many-particle eigenstates $$\Ket{n}$$:
@@ -363,8 +348,8 @@ $$\begin{aligned}
\end{aligned}$$
Where swapping $$n$$ and $$n'$$ gives the desired result.
-</div>
-</div>
+{% include proof/end.html id="proof-lehmann" %}
+
This gives us the primary use of the Matsubara Green's function $$C_{AB}$$:
calculating the retarded $$C_{AB}^R$$ and advanced $$C_{AB}^A$$.