How to understand the idea of functional renormalization group? I have been looking at how to use the functional RG method in many-body systems, but I don't quite get the idea of it, it looks different from Wilson's RG approach (eg. why shall we integrate out the field of all energy level?).  Hope somebody can give a nice explanation.
 A: The FRG can be thought of as a modern version of Wilson RG, although the technical details are of course very different. But all in all, if one could do all calculations exactly, these different versions would all be the same.
Now, about these technical differences. In Wilson RG (and in Polchinski's functional version) one works with a low energy action for low energy modes, that is, one studies the flow of $S_k[\varphi_{q<k}]$ defined as
$$e^{-S_k[\varphi_{q<k}]}=\int D \varphi_{q>k}e^{-S[\varphi_{q<k}+\varphi_{q>k}]},$$
where $S[\phi]$ is the microscopic action at a cut-off scale $\Lambda$ (before any fluctuations have been integrated out). Wilson (and Polchinski) give us a flow equation $\partial_k S_k=\ldots$ which has to be solved one way or another.
In the FRG (Wetterich's version at least), one does not work with $S_k$, which is not a very useful object in itself. Indeed, to compute correlation functions, one would need to follow the flow of (non-local) source terms, which is not really an option. It is then better to work with an object which has a physical interpretation when the momentum scale $k$ goes to $0$, which will be the effective action $\Gamma[\phi]$, or Gibbs free energy, the Legendre transform of the generating functional of connected correlation functions $W = \ln Z$ with respect to linear sources. It depends on the order parameter $\phi=\langle \varphi\rangle$.
In order to do so, one introduces a regulator term in the path integral $\Delta S_k$, which will reproduce (in a smooth fashion) the decoupling of Wilson (modes with $q>k$ are integrated out, and not the others). The partition function is then
$$Z_k[J]=\int D\varphi \, e^{-S[\varphi]-\Delta S_k[\phi]+\int J\varphi}.$$
Due to the regulator term, $Z_k[J]$ is pretty much the exact partition function for the mode with $q>k$, and a mean-field partition function for the modes with $q<k$. The scale dependent effective action is then defined as a modified Legendre transform of $W_k = \ln Z_k$ by
$$ \Gamma_k[\phi]=-W_k[J_k[\phi]]+\int J_k[\phi] \phi -\Delta S_k[\phi],\quad \frac{\delta \Gamma_k}{\delta\phi}=J_k[\phi].$$
The subtraction of $\Delta S_k[\phi]$ is just technical. In the limit $k\to\Lambda$, we want to suppress all fluctuations, and $\Delta S_{k=\Lambda}[\varphi]\to\infty$, so that
$$\Gamma_\Lambda[\phi]=S[\phi],$$
is the mean-field (microscopic) action. Furthermore, when $k=0$, $\Delta S_{k=0}[\varphi]=0$, and $$\Gamma_{k=0}[\phi]=\Gamma[\phi],$$ becomes the exact (quantum) effective action.
By differentiating with respect to $k$, we obtain a flow equation for $\Gamma_k[\phi]$, which interpolates between the mean-field (microscopic) and the exact (quantum) effective action.
This flow equation cannot be solved exactly, and there are several approximation schemes to simplify this task. The advantage of this method is that it allows for non-perturbative approximations (in the sense of the $\epsilon$ expansion). In particular, the simplest approximations are exact at one-loop, recover the first order of the $4-\epsilon$ and $2+\epsilon$ expansion, as well as the large N expansion. There are also schemes that allows to keep most of the microscopic physics, and allow to compute phase diagrams of (for example) many-body quantum systems.  Finally, the advantage of working with $\Gamma$ is that we can extract physical information from it (not just fixed points of the flow, though one can of course do that too). In particular, one can compute the correlation functions for all momenta, close to a critical point, a rather difficult task.
To me, the best introduction is that of B. Delamotte : arxiv:0702.365
