The Bethe Ansatz

Besides the definitions of correlations in real time provided above, it is also convenient to define the imaginary-time (thermal equilibrium) correlation function as

\[ {\cal C}_{\tau}^{\hat{O}, \hat{P}} (\tau_1 - \tau_2) \equiv - \langle T_{\tau} (\hat{O} (\tau_1) \hat{P} (\tau_2)) \rangle \]

in which \(\langle (...) \rangle\) means thermal averaging and \(T_\tau\) is the imaginary-time ordering operator (acting in a similar manner to the real-time ordering operator \(T_t\)), and in which we have used the imaginary-time interaction (in other words Heisenberg for the Hamiltonian \(H_0\)) representation \[ \hat{O} (\tau) \equiv e^{\tau \hat{H}_0} \hat{O} e^{-\tau \hat{H}_0}. \]

In the Lehmann representation, this becomes \[ {\cal C}_{\tau}^{\hat{O}, \hat{P}} (\tau) = - \frac{1}{\cal Z} \sum_{\alpha, \alpha'} O_{\alpha \alpha'} P_{\alpha' \alpha} e^{ (E_\alpha - E_{\alpha'}) \tau} \left(\theta (\tau) e^{-\beta E_\alpha} + ~\theta (-\tau) e^{-\beta E_{\alpha'}} \right), \hspace{3mm} \tau \in ]-\beta, \beta[ \]

(the argument's restriction to \(]-\beta, \beta[\) coming from the fact that it's really \(\tau_1 - \tau_2\) with \(\tau_i \in [0, \beta[\)). By inspection, this has the periodicity property

\[ {\cal C}_{\tau} ^{\hat{O}, \hat{P}} (\tau) = {\cal C}_{\tau} ^{\hat{O}, \hat{P}} (\tau + \beta), \hspace{10mm} -\beta < \tau < 0 \]

so this correlation function has a Fourier transformation in imaginary-time with (bosonic) Matsubara frequencies \(i\omega_n\) according to \({\cal C} (i\omega_n) = \int_0^\beta d\tau {\cal C} (\tau) e^{i \omega_n \tau}\). Performing this Fourier transform gives

\[ {\cal C}_{\tau}^{\hat{O}, \hat{P}} (i\omega_n) = \sum_{\alpha, \alpha'} O_{\alpha \alpha'} P_{\alpha' \alpha} \frac{e^{-\beta E_{\alpha}} - e^{-\beta E_{\alpha'}}}{i\omega_n + E_\alpha - E_{\alpha'}}. \]

Comparing with Creteq shows that the imaginary-time and (real-time) (thermal equilibrium) retarded correlation functions are related by the formal analytic continuation

\[ {\cal C}^{\hat{O}, \hat{P}}_{ret} (\omega) = {\cal C}_{\tau}^{\hat{O}, \hat{P}} (i\omega_n) {\large |}_{i\omega_n \rightarrow \omega + i \eta}. \]

Defining the master function

\[ {\cal C}_{\tau}^{\hat{O}, \hat{P}} (z) = \sum_{\alpha, \alpha'} O_{\alpha \alpha'} P_{\alpha' \alpha} \frac{e^{-\beta E_{\alpha}} - e^{-\beta E_{\alpha'}}}{z + E_\alpha - E_{\alpha'}} \]

for generic complex argument \(z\), we have that \({\cal C}_{ret}, {\cal C}_{adv}, {\cal C}_\tau\) are respectively given by taking \(z \rightarrow \omega + i\eta\), \(\omega - i\eta\), \(i\omega_n\). By inspection, \(C(z)\) is analytic everywhere except on the real axis. Suppose that we somehow have managed to compute \({\cal C}_\tau (i\omega_n)\) for all positive Matsubara frequencies \(i\omega_{n > 0}\), and that we can find the analytic continuation of \({\cal C}(z)\) to the upper half-plane \(\Im (z) > 0\). The retarded correlation function would then be given by the evaluation of this function on the shifted real axis \(z = \omega + i\eta\).