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+---
+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.