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diff --git a/content/know/concept/greens-functions/index.pdc b/content/know/concept/greens-functions/index.pdc new file mode 100644 index 0000000..10ab09b --- /dev/null +++ b/content/know/concept/greens-functions/index.pdc @@ -0,0 +1,152 @@ +--- +title: "Green's functions" +firstLetter: "G" +publishDate: 2021-11-03 +categories: +- Physics +- Quantum mechanics + +date: 2021-11-01T09:46:27+01:00 +draft: false +markup: pandoc +--- + +# Green's functions + +In many-body quantum theory, **Green's functions** +are correlation functions between particle creation/annihilation operators. +They are somewhat related to +[fundamental solution](/know/concept/fundamental-solution/) functions, +which are also often called *Green's functions*. + +The **retarded Green's function** $G_{\nu \nu'}^R$ +and the **advanced Green's function** $G_{\nu \nu'}^A$ +are defined like so, +where the expectation value $\expval{}$ is +with respect to thermal equilibrium, +$\nu$ and $\nu'$ are labels of single-particle states that may include spin, +and $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$ are annihilation/creation operators +from the [second quantization](/know/concept/second-quantization/): + +$$\begin{aligned} + \boxed{ + \begin{aligned} + G_{\nu \nu'}^R(t, t') + &\equiv -\frac{i}{\hbar} \Theta(t - t') \expval{\comm{\hat{c}_{\nu}(t)}{\hat{c}_{\nu'}^\dagger(t')}} + \\ + G_{\nu \nu'}^A(t, t') + &\equiv \frac{i}{\hbar} \Theta(t' - t) \expval{\comm{\hat{c}_{\nu}(t)}{\hat{c}_{\nu'}^\dagger(t')}} + \end{aligned} + } +\end{aligned}$$ + +Where $\Theta$ is the [Heaviside step function](/know/concept/heaviside-step-function/). +This is for bosons; for fermions the commutator +must be replaced by an anticommutator, as usual. +Notice that $G^R_{\nu \nu'}$ has the same form as the correlation function +from the [Kubo formula](/know/concept/kubo-formula/). + +Furthermore, the **greater Green's function** $G_{\nu \nu'}^>$ +and **lesser Green's function** $G_{\nu \nu'}^<$ are: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + G_{\nu \nu'}^>(t, t') + &\equiv -\frac{i}{\hbar} \expval{\hat{c}_{\nu}(t) \hat{c}_{\nu'}^\dagger(t')} + \\ + G_{\nu \nu'}^<(t, t') + &\equiv \mp \frac{i}{\hbar} \expval{\hat{c}_{\nu'}^\dagger(t') \hat{c}_{\nu}(t)} + \end{aligned} + } +\end{aligned}$$ + +Where $-$ is for bosons, and $+$ is for fermions. +The retarded and advanced Green's functions can thus be expressed as follows: + +$$\begin{aligned} + G_{\nu \nu'}^R(t, t') + &= \Theta(t - t') \Big( G_{\nu \nu'}^>(t, t') - G_{\nu \nu'}^<(t, t') \Big) + \\ + G_{\nu \nu'}^A(t, t') + &= \Theta(t' - t) \Big( G_{\nu \nu'}^<(t, t') - G_{\nu \nu'}^>(t, t') \Big) +\end{aligned}$$ + +If the Hamiltonian involves interactions, +it might be more natural to use quantum field operators $\hat{\Psi}(\vb{r}, t)$ +instead of choosing a basis of single-particle states $\psi_\nu$. +In that case, instead of a label $\nu$, +we use the spin $s$ and position $\vb{r}$, leading to: + +$$\begin{aligned} + G_{ss'}^R(\vb{r}, t; \vb{r}', t') + &= -\frac{i}{\hbar} \Theta(t - t') \expval{\comm{\hat{\Psi}_{s}(\vb{r}, t)}{\hat{\Psi}_{s'}^\dagger(\vb{r}', t')}} + \\ + &= \sum_{\nu \nu'} \psi_\nu(\vb{r}) \: \psi^*_{\nu'}(\vb{r}') \: G_{\nu \nu'}^R(t, t') +\end{aligned}$$ + +And analogously for $G_{ss'}^A$, $G_{ss'}^>$ and $G_{ss'}^<$. +Note that the time-dependence is given to the old $G_{\nu \nu'}^R$, +i.e. to $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$. +In other words, we are using the +[Heisenberg picture](/know/concept/heisenberg-picture/). + +If the Hamiltonian is time-independent, +then it can be shown that all the Green's functions +only depend on the time-difference $t - t'$ +(for a proof, see [Kubo formula](/know/concept/kubo-formula/)): + +$$\begin{aligned} + G_{\nu \nu'}^>(t, t') = G_{\nu \nu'}^>(t - t') + \qquad \quad + G_{\nu \nu'}^<(t, t') = G_{\nu \nu'}^<(t - t') +\end{aligned}$$ + + +If the Hamiltonian is both time-independent and non-interacting, +then the time-dependence of $\hat{c}_\nu$ +can simply be factored out as follows: + +$$\begin{aligned} + \hat{c}_\nu(t) + = \hat{c}_\nu \exp\!(- i \varepsilon_\nu t / \hbar) +\end{aligned}$$ + +Then the diagonal ($\nu = \nu'$) greater and lesser Green's functions +can be written in the form below, where $f_\nu$ is either +the [Fermi-Dirac distribution](/know/concept/fermi-dirac-distribution/) +or the [Bose-Einstein distribution](/know/concept/bose-einstein-distribution/). +Note that the off-diagonal ($\nu \neq \nu'$) functions vanish, +because $\expval*{\hat{c}_{\nu} \hat{c}_{\nu'}^\dagger} = 0$ there, +since the many-particle states are simply orthogonal +[Slater determinants](/know/concept/slater-determinant/)/permanents: + +$$\begin{aligned} + G_{\nu \nu}^>(t, t') + &= -\frac{i}{\hbar} \expval{\hat{c}_{\nu} \hat{c}_{\nu}^\dagger} \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big) + \\ + &= -\frac{i}{\hbar} (1 - f_\nu) \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big) + \\ + G_{\nu \nu}^<(t, t') + &= \mp \frac{i}{\hbar} \expval{\hat{c}_{\nu}^\dagger \hat{c}_{\nu}} \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big) + \\ + &= \mp \frac{i}{\hbar} f_\nu \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big) +\end{aligned}$$ + +The diagonal retarded and advanced Green's functions then reduce to +the following, where $+$ applies to fermions, and $-$ to bosons: + +$$\begin{aligned} + G_{\nu \nu}^R(t, t') + &= - \frac{i}{\hbar} \Theta(t - t') \big( 1 - f_\nu \pm f_\nu \big) \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big) + \\ + G_{\nu \nu}^A(t, t') + &= \frac{i}{\hbar} \Theta(t - t') \big( 1 - f_\nu \pm f_\nu \big) \exp\!\big(\!-\! i \varepsilon_\nu (t \!-\! t') / \hbar \big) +\end{aligned}$$ + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. |