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diff --git a/source/know/concept/lehmann-representation/index.md b/source/know/concept/lehmann-representation/index.md new file mode 100644 index 0000000..dd8c112 --- /dev/null +++ b/source/know/concept/lehmann-representation/index.md @@ -0,0 +1,228 @@ +--- +title: "Lehmann representation" +date: 2021-11-03 +categories: +- Physics +- Quantum mechanics +layout: "concept" +--- + +In many-body quantum theory, the **Lehmann representation** +is an alternative way to write the [Green's functions](/know/concept/greens-functions/), +obtained by expanding in the many-particle eigenstates +under the assumption of a time-independent Hamiltonian $\hat{H}$. + +First, we write out the greater Green's function $G_{\nu \nu'}^>(t, t')$, +and then expand its expected value $\Expval{}$ (at thermodynamic equilibrium) +into a sum of many-particle basis states $\Ket{n}$: + +$$\begin{aligned} + G_{\nu \nu'}^>(t, t') + = - \frac{i}{\hbar} \Expval{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t')} + &= - \frac{i}{\hbar Z} \sum_{n} \Matrixel{n}{\hat{c}_\nu(t) \hat{c}_{\nu'}^\dagger(t') e^{-\beta \hat{H}}}{n} +\end{aligned}$$ + +Where $\beta = 1 / (k_B T)$, and $Z$ is the grand partition function +(see [grand canonical ensemble](/know/concept/grand-canonical-ensemble/)); +the operator $e^{\beta \hat{H}}$ gives the weight of each term at equilibrium. +Since $\Ket{n}$ is an eigenstate of $\hat{H}$ with energy $E_n$, +this gives us a factor of $e^{\beta E_n}$. +Furthermore, we are in the [Heisenberg picture](/know/concept/heisenberg-picture/), +so we write out the time-dependence of $\hat{c}_\nu$ and $\hat{c}_{\nu'}^\dagger$: + +$$\begin{aligned} + G_{\nu \nu'}^>(t, t') + &= - \frac{i}{\hbar Z} \sum_{n} e^{-\beta E_n} \Matrixel{n}{e^{i \hat{H} t / \hbar} \hat{c}_\nu e^{- i \hat{H} t / \hbar} + e^{i \hat{H} t' / \hbar} \hat{c}_{\nu'}^\dagger e^{- i \hat{H} t' / \hbar}}{n} + \\ + &= - \frac{i}{\hbar Z} \sum_{n} e^{-\beta E_n} + \Matrixel{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar} \hat{c}_{\nu'}^\dagger}{n} +\end{aligned}$$ + +Where we used that the trace $\Tr\!(x) = \sum_{n} \matrixel{n}{x}{n}$ +is invariant under cyclic permutations of $x$. +The $\Ket{n}$ form a basis of eigenstates of $\hat{H}$, +so we insert an identity operator $\sum_{n'} \Ket{n'} \Bra{n'}$: + +$$\begin{aligned} + G_{\nu \nu'}^>(t - t') + &= - \frac{i}{\hbar Z} \sum_{n n'} e^{- \beta E_n} + \Matrixel{n}{e^{i \hat{H} (t - t') / \hbar} \hat{c}_\nu e^{- i \hat{H} (t - t') / \hbar}}{n'} \Matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} + \\ + &= - \frac{i}{\hbar Z} \sum_{n n'} e^{-\beta E_n} + \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} e^{i (E_n - E_{n'}) (t - t') / \hbar} +\end{aligned}$$ + +Note that $G_{\nu \nu'}^>$ now only depends on the time difference $t - t'$, +because $\hat{H}$ is time-independent. +Next, we take the [Fourier transform](/know/concept/fourier-transform/) +$t \to \omega$ (with $t' = 0$): + +$$\begin{aligned} + G_{\nu \nu'}^>(\omega) + &= - \frac{i}{\hbar Z} \sum_{n n'} e^{-\beta E_n} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} + \int_{-\infty}^\infty e^{i (E_n - E_{n'}) t / \hbar} \: e^{i \omega t} \dd{t} +\end{aligned}$$ + +Here, we recognize the integral +as a [Dirac delta function](/know/concept/dirac-delta-function/) $\delta$, +thereby introducing a factor of $2 \pi$, +and arriving at the Lehmann representation of $G_{\nu \nu'}^>$: + +$$\begin{aligned} + \boxed{ + G_{\nu \nu'}^>(\omega) + = - \frac{2 \pi i}{Z} \sum_{n n'} e^{-\beta E_n} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} + \: \delta(E_n - E_{n'} + \hbar \omega) + } +\end{aligned}$$ + +We now go through the same process for the lesser Green's function $G_{\nu \nu'}^<(t, t')$: + +$$\begin{aligned} + G_{\nu \nu'}^<(t - t') + &= \mp \frac{i}{\hbar Z} \sum_{n} \matrixel{n}{\hat{c}_{\nu'}^\dagger(t') \hat{c}_\nu(t) e^{-\beta \hat{H}}}{n} + \\ + &= \mp \frac{i}{\hbar Z} e^{-\beta E_n} \sum_{n n'} \matrixel{n}{\hat{c}_{\nu'}^\dagger}{n'} \matrixel{n'}{\hat{c}_\nu}{n} + e^{i (E_{n'} - E_n) (t - t') / \hbar} +\end{aligned}$$ + +Where $-$ is for bosons, and $+$ for fermions. +Fourier transforming yields the following: + +$$\begin{aligned} + G_{\nu \nu'}^<(\omega) + &= \mp \frac{2 \pi i}{\hbar Z} \sum_{n n'} e^{-\beta E_n} \matrixel{n}{\hat{c}_{\nu'}^\dagger}{n'} \matrixel{n'}{\hat{c}_\nu}{n} + \: \delta(E_{n'} - E_n + \hbar \omega) +\end{aligned}$$ + +We swap $n$ and $n'$, leading to the following +Lehmann representation of $G_{\nu \nu'}^<$: + +$$\begin{aligned} + \boxed{ + G_{\nu \nu'}^<(\omega) + = \mp \frac{2 \pi i}{Z} \sum_{n n'} e^{-\beta E_{n'}} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} + \: \delta(E_n - E_{n'} + \hbar \omega) + } +\end{aligned}$$ + +Due to the delta function $\delta$, +each term is only nonzero for $E_n' = E_n + \hbar \omega$, +so we write: + +$$\begin{aligned} + G_{\nu \nu'}^<(\omega) + = \mp \frac{2 \pi i}{\hbar Z} \sum_{n n'} e^{-\beta (E_n + \hbar \omega)} + \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \: \delta(E_n - E_{n'} + \hbar \omega) +\end{aligned}$$ + +Therefore, we arrive at the following useful relation +between $G_{\nu \nu'}^<$ and $G_{\nu \nu'}^>$: + +$$\begin{aligned} + \boxed{ + G_{\nu \nu'}^<(\omega) + = \pm e^{-\beta \hbar \omega} G_{\nu \nu'}^>(\omega) + } +\end{aligned}$$ + +Moving on, let us do the same for +the retarded Green's function $G_{\nu \nu'}^R(t, t')$, given by: + +$$\begin{aligned} + G_{\nu \nu'}^R(t \!-\! t') + &= \Theta(t \!-\! t') \Big( G_{\nu \nu'}^>(t - t') - G_{\nu \nu'}^<(t - t') \Big) + \\ + &= - \frac{i}{\hbar Z} \Theta(t \!-\! t') \sum_{n n'} + \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar} +\end{aligned}$$ + +We take the Fourier transform, but to ensure convergence, +we must introduce an infinitesimal positive $\eta \to 0^+$ to the exponent +(and eventually take the limit): + +$$\begin{aligned} + G_{\nu \nu'}^R(\omega) + &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^\infty \Theta(t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t} + \\ + &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_0^\infty e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t} + \\ + &= - \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) + \bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{- \eta t}}{i (\hbar \omega + E_n - E_{n'}) - \hbar \eta} \bigg]_0^\infty +\end{aligned}$$ + +Leading us to the following Lehmann representation +of the retarded Green's function $G_{\nu \nu'}^R$: + +$$\begin{aligned} + \boxed{ + G_{\nu \nu'}^R(\omega) + = \frac{1}{Z} \sum_{n n'} + \frac{\matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}}{\hbar (\omega + i \eta) + E_n - E_{n'}} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) + } +\end{aligned}$$ + +Finally, we go through the same steps for the advanced Green's function $G_{\nu \nu'}^A(t, t')$: + +$$\begin{aligned} + G_{\nu \nu'}^A(t \!-\! t') + &= \Theta(t' \!-\! t) \Big( G_{\nu \nu'}^<(t - t') - G_{\nu \nu'}^>(t - t') \Big) + \\ + &= \frac{i}{\hbar Z} \Theta(t' \!-\! t) \sum_{n n'} + \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) e^{i (E_n - E_{n'}) (t - t') / \hbar} +\end{aligned}$$ + +For the Fourier transform, we must again introduce $\eta \to 0^+$ +(although note the sign): + +$$\begin{aligned} + G_{\nu \nu'}^A(\omega) + &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^\infty \Theta(-t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t} + \\ + &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) \int_{-\infty}^0 e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t} + \\ + &= \frac{i}{\hbar Z} \sum_{n n'} \Big( ... \Big) + \bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{\eta t}}{i (\hbar \omega + E_n - E_{n'}) + \hbar \eta} \bigg]_{-\infty}^0 +\end{aligned}$$ + +Therefore, the Lehmann representation of +the advanced Green's function $G_{\nu \nu'}^A$ is as follows: + +$$\begin{aligned} + \boxed{ + G_{\nu \nu'}^A(\omega) + = \frac{1}{Z} \sum_{n n'} + \frac{\matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n}}{\hbar (\omega - i \eta) + E_n - E_{n'}} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) + } +\end{aligned}$$ + +As a final note, let us take the complex conjugate of this expression: + +$$\begin{aligned} + \big( G_{\nu \nu'}^A(\omega) \big)^* + = \frac{1}{Z} \sum_{n n'} + \frac{\matrixel{n}{\hat{c}_{\nu'}}{n'} \matrixel{n'}{\hat{c}_\nu^\dagger}{n}}{\hbar (\omega + i \eta) + E_n - E_{n'}} + \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) +\end{aligned}$$ + +Note the subscripts $\nu$ and $\nu'$. +Comparing this to $G_{\nu \nu'}^R$ gives us another useful relation: + +$$\begin{aligned} + \boxed{ + G^R_{\nu \nu'}(\omega) + = \big( G^A_{\nu' \nu}(\omega) \big)^* + } +\end{aligned}$$ + + + +## References +1. H. Bruus, K. Flensberg, + *Many-body quantum theory in condensed matter physics*, + 2016, Oxford. |