Connection between the $\beta$-function and critical exponents In my recent readings in QFT, I came across the fact that there is a connection between critical exponents in thermodynamics and the $\beta$-function of the renormalization group flow. Does anybody know how that works, or maybe a good resource where to read up on that?
 A: The $\beta$ function alongside with the flow equations for other couplings in the action determine how the theory behaves at different energy/length scales.
Flow equations have fixed points, there is almost always a Gaussian (free) fixed point and sometimes depending on the theory and the dimensions of space , there are other (non-trivial) fixed points as well. Each fixed point is considered by its stability. If a fixed point has only one unstable directions for perturbations around it in the parameter space, then it corresponds to a critical point.
After linearizing flow equations around a fixed point, you see that each perturbation close to the critical point scales with some power of length scale, and only one of them has a positive power, call this coupling $r$ and its scaling dimension $\lambda_r$. Now, you can write for the correlation length:
$$\xi=f(r)$$
going to another scale $x'=bx$ results in:
$$\xi'=b^{-1}\xi \to f(b^{\lambda_r}r)=b^{-1}f(r)$$
Putting $b=r^{-1/\lambda_r}$ gives:
$$f(r)\propto r^{-1/\lambda_r}$$
Assuming that $r$ changes linearly with temperature near the critical point $r\propto |T-T_c|$ finally gives:
$$\xi \propto |T-T_c|^{-1/\lambda_r}$$
This way, we found one of the critical exponents of the theory. Similar arguments can be applied to find other critical exponents as well.
You can look at Kardar's "Statistical Physics of Fields" or Zinn-Justin's "Quantum field theory and critical phenomena" for a detailed description.
