The Bethe Ansatz

After having introduced this zoology of correlation functions, let us take a step back and introduce the building blocks which we will

For a given specific state \(\alpha\) and operator \({\cal O}\), we will define the dynamical correlator

\begin{equation*} S^{{\cal O}, {\cal O}^\dagger}_\alpha (j, j'; t, t') \equiv \langle \alpha | O_j (t) O^\dagger_{j'} (t') | \alpha \rangle. \end{equation*}

If the system exhibits translational invariance, and if the Hamiltonian is time-independent, then the correlator becomes a function of the position and time differences only,

\begin{equation*} S^{{\cal O}, {\cal O}^\dagger}_\alpha (j, j'; t, t') = S^{{\cal O}, {\cal O}^\dagger}_\alpha (j - j', t - t'). \end{equation*}

Let us introduce the Fourier representation

\begin{equation*} {\cal O}_j = \frac{1}{N} \sum_k e^{ikj} {\cal O}_k, \hspace{5mm} {\cal O}_k = \sum_j e^{-ikj} {\cal O}_j \end{equation*}

for the operator, so we can write the momentum space correlator as

\begin{equation*} S^{{\cal O}, {\cal O}^\dagger}_\alpha (k, t) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k (j-j')} S^{{\cal O}, {\cal O}^\dagger}_\alpha (j-j', t) = \frac{1}{N} \langle \alpha | {\cal O}_k(t) {\cal O}^\dagger_k(0) | \alpha \rangle. \end{equation*}

One can also take the Fourier transform to frequency, giving

\begin{equation*} S^{{\cal O}, {\cal O}^\dagger}_\alpha (k, \omega) = \int_{-\infty}^\infty dt e^{i \omega t} S^{{\cal O}, {\cal O}^\dagger}_\alpha (k, t) = \frac{1}{N} \int_{-\infty}^{\infty} dt e^{i \omega t} \langle \alpha | {\cal O}_k(t) {\cal O}^\dagger_k(0) | \alpha \rangle. \end{equation*}

Introducing a resolution of the identity \({\bf 1} = \sum_{\alpha} | \alpha \rangle \langle \alpha |\) in terms of eigenstates of the Hamiltonian, we have

\begin{align*} S^{{\cal O}, {\cal O}^\dagger}_\alpha (k, \omega) &= \frac{1}{N} \int_{-\infty}^\infty dt e^{i \omega t} \langle \alpha | e^{iHt} {\cal O}_k e^{-iHt} \sum_{\alpha'} | \alpha' \rangle \langle \alpha' | {\cal O}^\dagger_k | \alpha \rangle \\ &= \frac{1}{N} \sum_{\alpha'} \int_{-\infty}^\infty dt e^{i (\omega + E_\alpha - E_{\alpha'}) t} \langle \alpha | {\cal O}_k | \alpha' \rangle \langle \alpha' | O^\dagger_{k} | \alpha \rangle. \end{align*}

This can be rewritten using the identity \(\int_{-\infty}^\infty dt e^{i (\omega - \omega') t} = 2\pi \delta (\omega - \omega')\), yielding the Lehmann representation

\begin{equation} S^{{\cal O}, {\cal O}^\dagger}_\alpha (k, \omega) = \frac{2\pi}{N} \sum_{\alpha'} |\langle \alpha | O_k | \alpha' \rangle|^2 \delta(\omega - (E_{\alpha'} - E_\alpha)). \tag{Lr}\label{Lr} \end{equation}

The physical content of the Lehmann representation is made clear by Fermi's Golden rule Fgr: starting from the state \(\alpha\), one entry of the operator induces a transition to the excited state \(\alpha'\) which lives at momentum/energy \(k, \omega\) above \(\alpha\), and acts as a mediator of correlations. The full correlator is given by the sum over all accessible intermediate states \(\alpha'\).

Note the very important fact that through the Lehmann representation, the dynamical correlator is given by a sum of strictly non-negative terms. This is of immense importance for the practical utility of sum rules.

The corresponding real-time retarded Cret and advanced Cadv functions can easily be rewritten in terms of this building block:

\begin{align*} S^{{\cal O}, {\cal O}^\dagger}_{ret, \alpha} (k, t) &= -i \theta(t) \left( S^{{\cal O}, {\cal O}^\dagger}_\alpha (k,t) - S^{{\cal O}^\dagger, {\cal O}}_\alpha (-k,-t) \right), \\ S^{{\cal O}, {\cal O}^\dagger}_{adv, \alpha} (k, t) &= i \theta(-t) \left( S^{{\cal O}, {\cal O}^\dagger}_\alpha (k,t) - S^{{\cal O}^\dagger, {\cal O}}_\alpha (-k,-t) \right). \end{align*}

Introducing the spectral function

\begin{align*} A^{{\cal O}, {\cal O}^\dagger}_\alpha (j-j', t-t') &\equiv \langle \alpha | \left[ O_j (t), O^\dagger_{j'} (t') \right] | \alpha \rangle \\ &= S^{{\cal O}, {\cal O}^\dagger}_\alpha (j - j', t - t') - S^{{\cal O}^\dagger, {\cal O}}_\alpha (j' - j, t' - t) \tag{specfun}\label{specfun} \end{align*}

or equivalently

\[ A^{{\cal O}, {\cal O}^\dagger}_\alpha (k, t) \equiv S^{{\cal O}, {\cal O}^\dagger}_\alpha (k,t) - S^{{\cal O}^\dagger, {\cal O}}_\alpha (-k,-t) \]

this means

\[ S^{{\cal O}, {\cal O}^\dagger}_{ret, \alpha} (k, t) = -i \theta(t) A^{{\cal O}, {\cal O}^\dagger}_\alpha (k, t) \hspace{10mm} S^{{\cal O}, {\cal O}^\dagger}_{adv, \alpha} (k, t) = i \theta(-t) A^{{\cal O}, {\cal O}^\dagger}_\alpha (k, t). \]

To Fourier transform from time to frequency, we use the convolution theorem

\[ A(t) = B(t) C(t) \rightarrow A(\omega) = \int_{-\infty}^\infty \frac{d\omega'}{2\pi} B(\omega - \omega') C(\omega') \]

together with the regularized integrals

\[ \mp i \int_{-\infty}^\infty dt e^{i \omega t} \theta(\pm t) = \frac{1}{\omega \pm i \epsilon} \]

to obtain

\[ S^{{\cal O}, {\cal O}^\dagger}_{ret, \alpha} (k, \omega) = \int_{-\infty}^\infty \frac{d\omega'}{2\pi} \frac{A^{{\cal O}, {\cal O}^\dagger}_\alpha (k, \omega')}{\omega - \omega' + i\epsilon}, \hspace{10mm} S^{{\cal O}, {\cal O}^\dagger}_{adv, \alpha} (k, t) = \int_{-\infty}^\infty \frac{d\omega'}{2\pi} \frac{A^{{\cal O}, {\cal O}^\dagger}_\alpha (k, \omega')}{\omega - \omega' - i\epsilon} \]

making manifest that the momentum/frequency advanced function is just the complex conjugate of the retarded one, and that knowing the spectral function is sufficient to completely determine all other forms of the correlator.

Substituting the Lehmann representation Lr gives (with \(\omega_{\alpha' \alpha} \equiv E_{\alpha'} - E_\alpha\))

\begin{align*} S^{{\cal O}, {\cal O}^\dagger}_{ret, \alpha} (k, \omega) &= \frac{1}{N} \sum_{\alpha'} \left( \frac{|\langle \alpha | {\cal O}_k | \alpha' \rangle |^2}{\omega - \omega_{\alpha' \alpha} + i\epsilon} - \frac{|\langle \alpha' | {\cal O}_{k} | \alpha \rangle|^2}{\omega + \omega_{\alpha' \alpha} + i\epsilon} \right) \\ S^{{\cal O}, {\cal O}^\dagger}_{adv, \alpha} (k, \omega) &= \frac{1}{N} \sum_{\alpha'} \left( \frac{|\langle \alpha | {\cal O}_k | \alpha' \rangle |^2}{\omega - \omega_{\alpha' \alpha} - i\epsilon} - \frac{|\langle \alpha' | {\cal O}_{k} | \alpha \rangle|^2}{\omega + \omega_{\alpha' \alpha} - i\epsilon} \right). \end{align*}

The spectral function, from which all dynamical two-point correlations of \({\cal O}\) can be obtained, can thus be represented in the following equivalent forms:

\begin{align*} A^{{\cal O}, {\cal O}^\dagger}_\alpha (k, \omega) &= -2~\mbox{Im}~ S^{{\cal O}, {\cal O}^\dagger}_{ret, \alpha} (k, \omega) \\ &= S^{{\cal O}, {\cal O}^\dagger}_{\alpha} (k, \omega) - S^{{\cal O}, {\cal O}^\dagger}_{\alpha} (-k, -\omega) \\ &= \frac{2\pi}{N} \sum_{\alpha'} \left( |\langle \alpha | {\cal O}_k | \alpha' \rangle |^2 \delta(\omega - \omega_{\alpha' \alpha}) - |\langle \alpha' | {\cal O}_{k} | \alpha \rangle|^2 \delta(\omega + \omega_{\alpha' \alpha}) \right) \end{align*}