What is the Out-of-time-ordered correlator for free fermion system? The Out-of-time-ordered correlator (OTOC) is defined as $\langle[W(t),V(0)]^2\rangle$, and can be considered as a new way to extract quantum chaos. However, the understanding of this special correlator is not clear. For a simple example, what the free fermion OTOC behaves? Both lattice or continuum case are welcome. 
 A: First of all, your definition of the OTOC is wrong. It is supposed to be composed of the static and time-evolved versions of the same operator, so it should read $C(t) = - \langle [\hat{V}(t),\hat{V}(0)]^2 \rangle$. The minus sign has become standard in the field.
The understanding of this function is not that hard to grasp: if $\hat{V}(t)$ and $\hat{V}(0)$ commute for all times, nothing happens. Your question, regarding the free fermion case, can be generalised for all cases where $\hat{V}$ commutes with the Hamiltonian: nothing happens. This makes sense, since if you're looking for something that could be called chaos, then it shouldn't arise with such a trivial time evolution. But if your time evolution is interesting enough to provide operators that stop commuting, then your OTOC starts giving off meaningful stuff. It was postulated that a special (momentum) OTOC version growth rate should be closely tied with Lyapunov exponents, something that didn't turn out to be exactly true, but close enough in some aspects. The reason behind this postulate regarding the momentum-OTOC is that, using a test function $f$ and the static position basis,
\begin{align}
[\hat{p}(t), \hat{p}(0) ]f  &= -i \hbar \left[ \hat{p}(t) \frac{\partial f}{\partial q(0)} - \frac{\partial}{\partial q(0)}(\hat{p}(t) f) \right] \\
&= (i \hbar f) \frac{\partial \hat{p}(t)}{\partial q(0)} \, ,
\end{align}
so that
\begin{align}
C(t) &= -\langle [\hat{p}(t), \hat{p}(0) ]^2 \rangle \\
&= \hbar^2 \bigg\langle \left( \frac{\partial \hat{p}(t)}{\partial q(0)} \right)^2 \bigg\rangle \\
&\approx \bigg\langle \left( \frac{\Delta{p}(t)}{\Delta q(0)} \right)^2 \bigg\rangle_{\text{phase space}} \quad (\equiv C_{cl}(t))\, ,
\end{align}
that is, for times smaller than Ehrenfest's time, this approximation should be quite reasonable. Now, if we're talking about a chaotic map, it means that for small deviations on position we'll have very large (exponential) deviations on momentum. This suggests that, after a little analysis,
\begin{align}
C_{cl} (t) &\approx (e^{ \Lambda t})^2 \\
\Rightarrow \Lambda &= \lim_{t \to \infty} \lim_{\Delta q(0) \to 0} \frac{1}{2t} \ln \left[ \frac{C_{cl}(t+1)}{C_{cl}(1)} \right] \, .
\end{align}
The classical expression for the Lyapunov exponent is given by
\begin{align}
\lambda = \lim_{t \to \infty} \lim_{\delta Z(0) \to 0} \frac{1}{t} \ln \left( \frac{|\delta Z(t)|}{|\delta Z(0)|} \right) \, ,
\end{align}
where $\delta Z$ represents the separation between trajectories and $|.|$ the norm. Notice how $\Lambda$ and $\lambda$ are very similar. This motivates the study of OTOCs in the context of quantum chaos.
