I am currently working on studying how to diagonalize the spin-1/2 XY model using the method included in " Annals of Physics 16.3 (1961): 407-466" by Lieb et al.

In fact, I'd like someone to help me understand how the authors used Wick's theorem to put the correlation functions:

$$\rho _{lm}^x = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}\left\langle {{\Psi _0}} \right|{B_l}{A_{l + 1}}{B_{l + 1}}.....{A_{m - 1}}{B_{m - 1}}{A_m}\left| {{\Psi _0}} \right\rangle $$

to take the following form

$$\rho _{lm}^x = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}{\sum\limits_p {( - 1)} ^p}{G_{l,P(l + 1)}}....{G_{m - 1,P(m)}}$$

There are two things I cannot understand:

1- The authors used the following relationship to express Wick's theorem,

$$\left\langle {{\Psi _0}} \right|{O_1}.....{O_{2n}}\left| {{\Psi _0}} \right\rangle = {\sum\limits_{all\;pairings} {[( - 1)} ^p}\prod\limits_{all\,pairs} {(contractions \;of \;pairs)]} $$ but according to my knowledge, Wick's theorem does not contain this ${( - 1)^p}$ coefficient!. If the existence of theis coefficient here is because both A and B obey fermionic anticommutation relations, and consequently $\left\langle {{B_i}{A_j}} \right\rangle = - \left\langle {{A_j}{B_i}} \right\rangle $, then how do I determine whether ${( - 1)^p}$ is positive or negative in case of N spins?

2- What is the mathematical basis used by the author to convert correlation functions into determinants?

$$\rho _{lm}^x = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 4$}}\left| \matrix{ {G_{l,l + 1}}\,\,\,{G_{l,l + 2}}\,\,\,....\,\,\,{G_{l,m}} \hfill \cr \vdots \hfill \cr {G_{m - 1,l + 1}}\,\,\,.....\,\,\,\,\,\,\,\,\,{G_{m - 1,m}} \hfill \cr} \right|$$

Finally, I would be very grateful if anyone could provide me with some resources that explain this issue in an easy way.



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