The Bethe Ansatz

Up to now, we have written out correlation functions for a given specific initial state. More generally, if the initial condition is represented by an ensemble, the correlation function can be written as (here the retarded function for a Gibbs ensemble; advanced and real-time correlations are defined in a similar manner)

\[ {\cal C}^{\hat{O}, \hat{P}}_{ret} (t) = \frac{1}{\cal Z} \sum_\alpha {\cal C}^{\hat{O}, \hat{P}}_{ret, \psi_\alpha} (t ) e^{-\beta E_\alpha} = -i \frac{\theta(t)}{\cal Z} \sum_{\alpha} \langle \psi_\alpha | [ \hat{O}^I (t), \hat{P}^I (0) ] | \psi_\alpha \rangle e^{-\beta E_\alpha}, \]

with partition function (note: if you are working in a grand canonical ensemble, you can view the chemical potential as being included in the definition of \(E_\alpha\))

\[ {\cal Z} = \sum_\alpha e^{-\beta E_\alpha}. \]

Performing the same computations as above (using the Lehmann representation), and interchanging indices in the second sum, we obtain

\[ {\cal C}^{\hat{O}, \hat{P}}_{ret} (\omega) = \frac{1}{\cal Z} \sum_{\alpha, \alpha'} O_{\alpha \alpha'} P_{\alpha' \alpha} \frac{e^{-\beta E_\alpha} - e^{-\beta E_{\alpha'}}}{\omega + E_\alpha - E_{\alpha'} + i\eta} \tag{Creteq}\label{Creteq} \]

\[ {\cal C}^{\hat{O}, \hat{P}}_{adv} (\omega) = \frac{1}{\cal Z} \sum_{\alpha, \alpha'} O_{\alpha \alpha'} P_{\alpha' \alpha} \frac{e^{-\beta E_\alpha} - e^{-\beta E_{\alpha'}}}{\omega + E_\alpha - E_{\alpha'} - i\eta} \]

\[ {\cal C}^{\hat{O}, \hat{P}} (\omega) = \frac{1}{\cal Z} \sum_{\alpha, \alpha'} O_{\alpha \alpha'} P_{\alpha' \alpha} \left( \frac{e^{-\beta E_\alpha}}{\omega + E_\alpha - E_{\alpha'} + i\eta} - \frac{e^{-\beta E_{\alpha'}}}{\omega + E_\alpha - E_{\alpha'} - i\eta} \right) \]

The analytic structure thus remains the same at finite temperature: the retarded (advanced) function has singularities in the lower (upper) half-plane, whereas the real-time correlation function has singularities in both half-planes.