The Bethe Ansatz

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

\begin{equation*} S_\alpha (j, j'; t, t') \equiv \langle \alpha | O_j (t) O_{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_\alpha (j, j'; t, t') = S_\alpha (j - j', t - t'). \end{equation*}

Let us introduce the Fourier representation

\begin{equation*} S_\alpha (k, \omega) \equiv \frac{1}{N} \sum_{j, j'} e^{-i k (j-j')} \int_{-\infty}^\infty dt e^{i \omega t} S_\alpha (j-j', t) \end{equation*}

for the dynamical correlation, and

\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. 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_\alpha (k, \omega) &= \frac{1}{N} \int_{-\infty}^\infty dt e^{i \omega t} \langle \alpha | e^{iHt} \sum_j e^{-ikj} {\cal O}_j e^{-iHt} \sum_{\alpha'} | \alpha' \rangle \langle \alpha' | \sum_{j'} e^{i k j'} {\cal O}_{j'} | \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_{-k} | \alpha \rangle. \end{align*}

This can be rewritten using the identity \(\int_{-\infty}^\infty dt e^{i (\omega - \omega') t} = 2\pi \delta (\omega - \omega')\) (and \((O_k)^\dagger = O_{-k}\) for a Hermitian operator), yielding the Lehmann representation

\begin{equation} S_\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.