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 H^\hat{H}.

First, we write out the greater Green’s function Gνν>(t,t)G_{\nu \nu'}^>(t, t'), and then expand its expected value \Expval{} (at thermodynamic equilibrium) into a sum of many-particle basis states n\Ket{n}:

Gνν>(t,t)=ic^ν(t)c^ν(t)=iZnn|c^ν(t)c^ν(t)eβH^|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 β=1/(kBT)\beta = 1 / (k_B T), and ZZ is the grand partition function (see grand canonical ensemble); the operator eβH^e^{\beta \hat{H}} gives the weight of each term at equilibrium. Since n\Ket{n} is an eigenstate of H^\hat{H} with energy EnE_n, this gives us a factor of eβEne^{\beta E_n}. Furthermore, we are in the Heisenberg picture, so we write out the time-dependence of c^ν\hat{c}_\nu and c^ν\hat{c}_{\nu'}^\dagger:

Gνν>(t,t)=iZneβEnn|eiH^t/c^νeiH^t/eiH^t/c^νeiH^t/|n=iZneβEnn|eiH^(tt)/c^νeiH^(tt)/c^ν|n\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)=nnxn\Tr(x) = \sum_{n} \matrixel{n}{x}{n} is invariant under cyclic permutations of xx. The n\Ket{n} form a basis of eigenstates of H^\hat{H}, so we insert an identity operator nnn\sum_{n'} \Ket{n'} \Bra{n'}:

Gνν>(tt)=iZnneβEnn|eiH^(tt)/c^νeiH^(tt)/|nn|c^ν|n=iZnneβEnnc^νnnc^νnei(EnEn)(tt)/\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 Gνν>G_{\nu \nu'}^> now only depends on the time difference ttt - t', because H^\hat{H} is time-independent. Next, we take the Fourier transform tωt \to \omega (with t=0t' = 0):

Gνν>(ω)=iZnneβEnnc^νnnc^νnei(EnEn)t/eiωtdt\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 δ\delta, thereby introducing a factor of 2π2 \pi, and arriving at the Lehmann representation of Gνν>G_{\nu \nu'}^>:

Gνν>(ω)=2πiZnneβEnnc^νnnc^νnδ(EnEn+ω)\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 Gνν<(t,t)G_{\nu \nu'}^<(t, t'):

Gνν<(tt)=iZnnc^ν(t)c^ν(t)eβH^n=iZeβEnnnnc^νnnc^νnei(EnEn)(tt)/\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:

Gνν<(ω)=2πiZnneβEnnc^νnnc^νnδ(EnEn+ω)\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 nn and nn', leading to the following Lehmann representation of Gνν<G_{\nu \nu'}^<:

Gνν<(ω)=2πiZnneβEnnc^νnnc^νnδ(EnEn+ω)\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 δ\delta, each term is only nonzero for En=En+ωE_n' = E_n + \hbar \omega, so we write:

Gνν<(ω)=2πiZnneβ(En+ω)nc^νnnc^νnδ(EnEn+ω)\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 Gνν<G_{\nu \nu'}^< and Gνν>G_{\nu \nu'}^>:

Gνν<(ω)=±eβωGνν>(ω)\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 GννR(t,t)G_{\nu \nu'}^R(t, t'), given by:

GννR(t ⁣ ⁣t)=Θ(t ⁣ ⁣t)(Gνν>(tt)Gνν<(tt))=iZΘ(t ⁣ ⁣t)nnnc^νnnc^νn(eβEneβEn)ei(EnEn)(tt)/\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 η0+\eta \to 0^+ to the exponent (and eventually take the limit):

GννR(ω)=iZnn(...)Θ(t)ei(EnEn)t/ei(ω+iη)tdt=iZnn(...)0ei(EnEn)t/ei(ω+iη)tdt=iZnn(...)[ei(ω+EnEn)t/eηti(ω+EnEn)η]0\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 GννRG_{\nu \nu'}^R:

GννR(ω)=1Znnnc^νnnc^νn(ω+iη)+EnEn(eβEneβEn)\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 GννA(t,t)G_{\nu \nu'}^A(t, t'):

GννA(t ⁣ ⁣t)=Θ(t ⁣ ⁣t)(Gνν<(tt)Gνν>(tt))=iZΘ(t ⁣ ⁣t)nnnc^νnnc^νn(eβEneβEn)ei(EnEn)(tt)/\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 η0+\eta \to 0^+ (although note the sign):

GννA(ω)=iZnn(...)Θ(t)ei(EnEn)t/ei(ωiη)tdt=iZnn(...)0ei(EnEn)t/ei(ωiη)tdt=iZnn(...)[ei(ω+EnEn)t/eηti(ω+EnEn)+η]0\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 GννAG_{\nu \nu'}^A is as follows:

GννA(ω)=1Znnnc^νnnc^νn(ωiη)+EnEn(eβEneβEn)\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:

(GννA(ω))=1Znnnc^νnnc^νn(ω+iη)+EnEn(eβEneβEn)\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 ν\nu and ν\nu'. Comparing this to GννRG_{\nu \nu'}^R gives us another useful relation:

GννR(ω)=(GννA(ω))\begin{aligned} \boxed{ G^R_{\nu \nu'}(\omega) = \big( G^A_{\nu' \nu}(\omega) \big)^* } \end{aligned}


  1. H. Bruus, K. Flensberg, Many-body quantum theory in condensed matter physics, 2016, Oxford.