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Categories: Physics, Quantum mechanics.

Lehmann representation

In many-body quantum theory, the Lehmann representation is an alternative way to write the Green’s 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); 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, 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 $t \to \omega$ (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 $\delta$, 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 introduce an infinitesimal positive $\eta \to 0^+$ to the exponent (and eventually take the limit):

$$\begin{aligned} G_{\nu \nu'}^R(\omega) &= - \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \int_{-\infty}^\infty \Theta(t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t} \\ &= - \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \int_0^\infty e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega + i \eta) t} \dd{t} \\ &= - \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{- \eta t}}{i (\hbar \omega + E_n - E_{n'}) - \hbar \eta} \bigg]_0^\infty \end{aligned}$$

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):

$$\begin{aligned} G_{\nu \nu'}^A(\omega) &= \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \int_{-\infty}^\infty \Theta(-t) e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t} \\ &= \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \int_{-\infty}^0 e^{i (E_n - E_{n'}) t / \hbar} e^{i (\omega - i \eta) t} \dd{t} \\ &= \frac{i}{\hbar Z} \sum_{n n'} \matrixel{n}{\hat{c}_\nu}{n'} \matrixel{n'}{\hat{c}_{\nu'}^\dagger}{n} \Big( e^{-\beta E_n} \mp e^{- \beta E_{n'}} \Big) \bigg[ \frac{\hbar e^{i (\hbar \omega + E_n - E_{n'}) t / \hbar} e^{\eta t}}{i (\hbar \omega + E_n - E_{n'}) + \hbar \eta} \bigg]_{-\infty}^0 \end{aligned}$$

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.

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