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Diffstat (limited to 'source/know/concept/matsubara-greens-function')
| -rw-r--r-- | source/know/concept/matsubara-greens-function/index.md | 41 |
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 $$C_{AB}^A$$. |