How to understand the idea of functional renormalization group?

Question

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.

Answer 1

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

Answer 2

In this answer we derive all the results mentioned in Adam's nice answer.

1. Recall that an Euclidean path integral $Z[J]$ for real scalar fields $\phi$ satisfies $$\begin{align} \exp&\left\{-\frac{1}{\hbar}W[J]\right\}\cr ~=~~~&Z[J]\cr ~=~~~& \int \! {\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{1}{\hbar} \left(-\frac{1}{2}\phi G^{-1}\phi -S_{\rm int}[\phi]+J \phi\right)\right\}\cr \stackrel{\text{Gauss. int.}}{=}&~{\rm Det}(G)^{1/2}\exp\left\{-\frac{1}{\hbar}S_{\rm int}[\frac{\hbar\delta}{\delta J}]\right\} \exp\left\{ \frac{1}{2\hbar}J G J\right\} ,\end{align} \tag{1}$$ where $W[J]$ is the generator of connected Feynman diagrams, cf. e.g. my Phys.SE answer here. Here the bare Greens function $G$ and the bare action $S_{\rm int}[\phi]$ may depend on a UV cut-off/regulator $\Lambda_H$. Also recall that the 1PI effective action$^1$ $$ \Gamma[\phi_{\rm cl}]~=~W[J]+J\phi_{\rm cl}\tag{2}$$ is the Legendre transform.

2. In the functional renormalization group (FRG) method we rewrite the Greens function $$G ~=~G_L+G_H\tag{3}$$ into a low-pass and high-pass momentum filter that depends on a scale $\Lambda\equiv\Lambda_L$. We double the number of fields $$ \phi_{H/L}~=~\frac{\phi}{2}\pm \varphi\quad\Leftrightarrow\quad\left\{\begin{array}{lrc} \phi&=&\phi_H+\phi_L,\cr \varphi&=&\frac{\phi_H-\phi_L}{2}.\end{array}\right.\tag{4}$$ Then the path integral (1) becomes$^2$ $$\begin{align} Z[J] ~\stackrel{(1)}{=}~&{\cal N} \int \! {\cal D}\frac{\phi_L}{\sqrt{\hbar}}~{\cal D}\frac{\phi_H}{\sqrt{\hbar}}~\cr &\exp\left\{ \frac{1}{\hbar} \left(-\frac{1}{2}\phi_L G_L^{-1}\phi_L -\frac{1}{2}\phi_H G_H^{-1}\phi_H -S_{\rm int}[\phi_L\!+\!\phi_H]+J (\phi_L\!+\!\phi_H)\right)\right\} \cr \stackrel{(6)+(7)}{=}&{\cal N} \int \! {\cal D}\frac{\phi_L}{\sqrt{\hbar}}~\exp\left\{ \frac{1}{\hbar} \left(-\frac{1}{2}\phi_L G_L^{-1}\phi_L -W_{\rm int}[J,\phi_L]\right)\right\},\end{align} \tag{5}$$ where $$\begin{align} {\cal N}^{-1}~:=~&\int \! {\cal D}\frac{\varphi}{\sqrt{\hbar}}~\exp\left\{-\frac{1}{2\hbar}\varphi (G_L^{-1}\!+\!G_H^{-1})\varphi \right\}\cr ~=~&{\rm Det}(G_L^{-1}\!+\!G_H^{-1})^{-1/2}.\end{align}\tag{6}$$

3. In eq. (5) we introduced (the Polchinski version of) the Wilsonian effective action $$\begin{align} \exp&\left\{-\frac{1}{\hbar}W_{\rm int}[J,\phi_L] \right\}\cr ~:=~~~&\int \! {\cal D}\frac{\phi_H}{\sqrt{\hbar}}~ \exp\left\{ \frac{1}{\hbar} \left( -\frac{1}{2}\phi_H G_H^{-1}\phi_H -S_{\rm int}[\phi_L\!+\!\phi_H]+J (\phi_L\!+\!\phi_H)\right)\right\}\cr \stackrel{\text{Gauss. int.}}{=}&~{\cal N}_H\exp\left\{-\frac{1}{\hbar}S_{\rm int}[\frac{\hbar\delta}{\delta J}]\right\} \exp\left\{ \frac{1}{\hbar} \left( \frac{1}{2}J G_HJ +J \phi_L\right)\right\}\cr ~=~~~&{\cal N}_H\exp\left\{ \frac{1}{\hbar} J \phi_L \right\}\cr &\exp\left\{-\frac{1}{\hbar}S_{\rm int}[\phi_L+\frac{\hbar\delta}{\delta J}]\right\}\exp\left\{ \frac{1}{2\hbar} J G_HJ \right\} ,\end{align} \tag{7}$$ where $$ {\cal N}_H~:=~{\rm Det}(G_H)^{1/2}.\tag{8}$$ The Wilsonian$^3$ effective action (7) is the generator of connected Feynman diagrams with the propagator $G_H$ in a background $\phi_L$.

4. The Wilsonian effective action decomposes into $$ W_{\rm int}[J,\phi_L]~=~W_H[J^H]+\frac{1}{2}\phi_L G_H^{-1}\phi_L, \tag{9}$$ where $$J^H~:=~J+G_H^{-1}\phi_L,\tag{10}$$ and where $$\begin{align} \exp&\left\{-\frac{1}{\hbar}W_H[J^H] \right\}\cr ~:=~~~& \int \! {\cal D}\frac{\phi}{\sqrt{\hbar}}~\exp\left\{ \frac{1}{\hbar} \left(-\frac{1}{2}\phi G_H^{-1}\phi -S_{\rm int}[\phi]+J^H \phi\right)\right\}\cr \stackrel{\text{Gauss. int.}}{=}&~{\cal N}_H\exp\left\{-\frac{1}{\hbar}S_{\rm int}[\frac{\hbar\delta}{\delta J^H}]\right\} \exp\left\{\frac{1}{2\hbar}J^H G_HJ^H\right\} \end{align} \tag{11}$$ is the generator of connected Feynman diagrams with the propagator $G_H$.

5. With the above definitions we can now derive the Polchinski exact renormalization group flow equation (ERGE) [1] $$\begin{align} \hbar\frac{d}{d\Lambda}\exp&\left\{-\frac{1}{\hbar}W_{\rm int}[J,\phi_L] \right\}\cr ~\stackrel{(7)}{=}~&-\frac{1}{2}\int \! {\cal D}\frac{\phi_H}{\sqrt{\hbar}}~ \phi_H \frac{\partial G_H^{-1}}{\partial\Lambda}\phi_H~\cr &\exp\left\{ \frac{1}{\hbar} \left( -\frac{1}{2}\phi_H G_H^{-1}\phi_H -S_{\rm int}[\phi_L\!+\!\phi_H]+J (\phi_L\!+\!\phi_H)\right)\right\}\cr ~\stackrel{(7)}{=}~&-\frac{1}{2}\left(\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta J}-\phi_L\right) \frac{\partial G_H^{-1}}{\partial\Lambda}\left(\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta J}-\phi_L\right)\exp\left\{-\frac{1}{\hbar}W_{\rm int}[J,\phi_L] \right\}\cr ~\stackrel{(9)}{=}~&-\frac{1}{2} \exp\left\{-\frac{1}{2\hbar}\phi_L G_H^{-1}\phi_L \right\} \left(\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta J^H}-\phi_L\right)\cr &\frac{\partial G_H^{-1}}{\partial\Lambda}\left(\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta J^H}-\phi_L\right) \exp\left\{-\frac{1}{\hbar}W_H[J^H] \right\} \cr ~\stackrel{(10)}{=}~&\frac{1}{2} \exp\left\{-\frac{1}{2\hbar}\phi_L G_H^{-1}\phi_L\right\} \left(\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta \phi_L}-\phi_LG_H^{-1}\right)\cr &\frac{\partial G_H}{\partial\Lambda}\left(\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta \phi_L}-G_H^{-1}\phi_L\right) \exp\left\{-\frac{1}{\hbar}W_H[J^H] \right\} \cr ~\stackrel{(9)}{=}~&\frac{1}{2}\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta\phi_L}\frac{\partial G_H}{\partial\Lambda}\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta\phi_L} \exp\left\{-\frac{1}{\hbar}W_{\rm int}[J,\phi_L] \right\}\cr ~\stackrel{(3)}{=}~&-\frac{1}{2}\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta \phi_L}\frac{\partial G_L}{\partial\Lambda}\frac{\hbar\stackrel{\rightarrow}{\delta}}{\delta \phi_L} \exp\left\{-\frac{1}{\hbar}W_{\rm int}[J,\phi_L] \right\} .\end{align} \tag{12}$$

6. The corresponding ERGE for the $W_H[J^H]$ action (11) reads $$\begin{align} \left(\frac{\partial W_H[J^H]}{\partial \Lambda}\right)_{J^H} ~\equiv~~~&\frac{dW_H[J^H]}{d\Lambda}-\frac{\partial J^H}{\partial \Lambda}\frac{\delta W_H[J^H]}{\delta J^H}\cr ~\stackrel{(10)+(12)}{=}&\frac{1}{2} \frac{\delta W_H[J^H]}{\delta J^H}\frac{\partial G_H^{-1}}{\partial\Lambda}\frac{\delta W_H[J^H]}{\delta J^H}\cr &-\frac{\hbar}{2} {\rm Tr}\left(\frac{\partial G_H^{-1}}{\partial\Lambda} \frac{\delta^2 W_H[J^H]}{\delta J^H\delta J^H} \right) .\end{align}\tag{13}$$

7. The $\Lambda$-dependent 1PI effective action is defined via a Legendre transformation [3] $$\begin{align} \frac{1}{2}\phi_{\rm cl} G_H^{-1}\phi_{\rm cl} +\Gamma_{\rm int}[\phi_{\rm cl}]~\equiv~&\Gamma_H[\phi_{\rm cl}]\cr ~=~&W_H[J^H]+J^H\phi_{\rm cl},\end{align}\tag{14}$$ or equivalently, $$\begin{align} \frac{1}{2}(\phi_{\rm cl}\!-\!\phi_L) G_H^{-1}(\phi_{\rm cl}\!-\!\phi_L) +\Gamma_{\rm int}[\phi_{\rm cl}]~\equiv~~~~~&\Gamma_{\rm int}[\phi_{\rm cl},\phi_L]\cr ~\stackrel{(9)+(10)+(14)}{=}&W_{\rm int}[J,\phi_L] +J\phi_{\rm cl}.\end{align}\tag{15}$$

8. The corresponding ERGE for $\Gamma_{\rm int}[\phi_{\rm cl}]$ is Wetterich's ERGE [2] $$\begin{align} \left(\frac{\partial \Gamma_{\rm int}[\phi_{\rm cl}]}{\partial \Lambda}\right)_{\phi_{\rm cl}} ~\equiv~~~&\frac{d\Gamma_{\rm int}[\phi_{\rm cl}]}{d\Lambda}-\frac{\partial \phi_{\rm cl}}{\partial \Lambda}\frac{\delta \Gamma_{\rm int}[\phi_{\rm cl}]}{\delta \phi_{\rm cl}}\cr ~\stackrel{(13)+(14)}{=}&\frac{\hbar}{2} {\rm Tr}\left(\frac{\partial G_H^{-1}}{\partial\Lambda} \left(\frac{\delta^2 \Gamma_H[\phi_{\rm cl}]}{\delta \phi_{\rm cl}\delta \phi_{\rm cl}} \right)^{-1}\right) .\end{align}\tag{16}$$

9. Limit $\Lambda\to 0$: Then $G_H\to G$. Then $$W_H[J_H]~\stackrel{(1)+(11)}{\longrightarrow}~ W[J_H]\tag{17}$$ and $$\Gamma_H[\phi_{\rm cl}]~\stackrel{(2)+(14)+(17)}{\longrightarrow}~ \Gamma[\phi_{\rm cl}].\tag{18}$$

10. Limit $\Lambda\to \Lambda_H$: Then $G_H\to 0$. Then $$W_{\rm int}[J,\phi_L]+\hbar\ln{\cal N}_H ~\stackrel{(7)}{\longrightarrow} ~S_{\rm int}[\phi_L]-J \phi_L,\tag{19}$$ so that $$\phi_{\rm cl}~\stackrel{(15)}{=} -\frac{\delta W_{\rm int}[J,\phi_L]}{\delta J}~\stackrel{(19)}{\longrightarrow} ~\phi_L,\tag{20} $$ and $$ \begin{align}\Gamma_{\rm int}[\phi_L]+\hbar\ln{\cal N}_H ~\stackrel{(15)+(20)}{\approx}~~& \Gamma_{\rm int}[\phi_{\rm cl},\phi_L]+\hbar\ln{\cal N}_H\cr ~\stackrel{(15)+(19)+(20)}{\longrightarrow}& ~S_{\rm int}[\phi_L] .\end{align} \tag{21} $$

References:

1. J. Polchinski, Renormalization and effective lagrangians, Nucl. Phys. B231 (1984) 269; eqs. (18).

2. C. Wetterich, Exact evolution equation for the effective potential, arXiv:1710.05815; eq. (3).

3. T. Morris, The Exact Renormalisation Group and Approximate Solutions, arXiv:hep-ph/9308265; eq. (3.17).

4. B. Delamotte, An Introduction to the Nonperturbative Renormalization Group, arXiv:cond-mat/0702365; section 2.1.1 + appendix 3.3.1. (Hat tip: Adam.)

5. ncatlab.

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$^1$ Eq. (2) is an Euclidean version of the Legendre transform. We use DeWitt condensed notation to not clutter the notation, e.g. spacetime integration symbols are suppressed.

$^2$ It is straightforward to check that when performing the Gaussian integral over $\varphi$ and completing the square of the free action $\frac{1}{2}\phi_L G_L^{-1}\phi_L +\frac{1}{2}\phi_H G_H^{-1}\phi_H$ in eq. (5), the surviving free action is indeed $\frac{1}{2}\phi G^{-1}\phi$ that we started with in eq. (1). One may show that the Greens functions $G_H$ and $G_L$ in principle don't need to commute (as infinite-dimensional matrices), although this probably doesn't have any practical relevance, given that most renormalization is performed multiplicatively.

$^3$ Often what authors call the Wilsonian effective action is $W_{\rm int}[J\!=\!0,\phi_L]$, i.e. the corresponding vacuum bubbles with no sources $J=0$. Note that the Polchiski ERGE (12) continues to makes sense if we restrict to the $J=0$ sector.