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Title says it all really.. Why is the XX spin chain a free fermion (non-interacting) model, and the XXZ chain not?

Is it right that $\sum_l a_l^\dagger a_{l+1}$ isn't an interaction between fermions because it's creating a fermion on one site and destroying it on another? But why is $\sum_l a_l^\dagger a_l a_{l+1}^\dagger a_{l+1}$ an interaction term?

Is something like

\begin{equation} H_1 = -\sum_l (J+(-1)^lK) ( \sigma_l^x \sigma_{l+1}^x +\sigma_l^y \sigma_{l+1}^y) \end{equation}

a free fermion model? If not, why not?

Edit I don't have enough reputation to set a bounty, but if anyone could answer this question, I'd be very grateful!

Edit 2 Anyone?

user6050
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1 Answers1

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To expand on LubošMotl's comment, see the following classic paper by Lieb, Schultz and Mattis. For one-dimensional systems and nearest neighbor interactions, the spin chain that you mention as an example in the comment can be converted into a free fermionic model. See section II in the above paper for details.

Vijay Murthy
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