Have the correlation functions of the XY spin chain model been calculated using a functional partition function with source terms?

Question

Have the correlation functions of the XY spin chain model,

\begin{equation} H=-\sum_l (J_x \sigma_l^x \sigma_{l+1}^x+J_x \sigma_l^y \sigma_{l+1}^y)-B\sum_l \sigma_l^z \end{equation}

been calculated using a functional partition function with source terms?

By functional partition function I mean the partition function plus source terms (or generating functional) which allow correlation functions such as $\langle \sigma_l^x \sigma_{l+1}^x \rangle$ to be calculated analytically:

\begin{equation} Z[\bar{\omega},\omega] = Tr \left[Exp \left[-\int_0^\beta d\tau H(a^{\dagger},a) - \int_0^\beta d\tau \sum_l (\bar{\omega} a + a^{\dagger} \omega) \right] \right] \end{equation}

where $H(a^{\dagger},a)$ is the XY Hamiltonian expressed in terms of spinless fermions (i.e. after a Jordan-Wigner transformation).

Edit: Anyone? An Ising chain would be fine too!

Answer 1

Some times it is not an efficient way to calculate the partition function using the functional integral. Even for a system of free fermions, we must have to evalute some complex Matsubara sums