In the book, I read some remarks about the criticality:

Iterations of the renormalization (group) map generate a sequence of points in the space of couplings, which we call a renormalization group trajectory. Since the correlation length is reduced by a factor $b<1$ at each step, a typical renormalization-group trajectory tends to take the system away from criticality. Because the correlation length is infinite at the critical point, it takes an infinite of iterations to leave that point. In general, a system is critical not only at a given point in the coupling space but on a whole "hypersurface", which we call the critical surface, or sometimes the critical line.

I think these remarks are fine. But then the authors give a statement:

The critical surface is the set of points in coupling space whose renormalization-group trajectories end up at the fixed point: $$\lim_{n\rightarrow\infty} T^n(J)=J_c,$$ where $T$ is the renormalization-group map and $J$ represents the couplings in general.

The question is, how to properly understand this statement? For this, I particularly have two small questions. Why the critical line need to hit the fixed point? And why the critical line need to end up at the fixed point rather leave the fixed point under the renormalization group transformation?


I'll try to explain why there could be a critical line and not just a critical point, and hopefully that will answer your question. If you think about the Ising model, we have the standard Hamiltonian:

\begin{equation} -\beta H = J_1\sum_{<i,j>}s_i s_j + h\sum_{i}s_i \end{equation} where $\sum_{<i,j>}$ is a sum over nearest neighbors. This model has a fixed point at a critical value of $J_1$ and $h$. Simple generalizations of the Ising model might include interactions between next nearest neighbors as well as nearest neighbors

\begin{equation} -\beta H = J_1\sum_{<i,j>}s_i s_j + J_2\sum_{(i,j)}s_i s_j + h\sum_{i}s_i \end{equation}

where $(i,j)$ indicates next nearest neighbor pairs. This model can also be critical for certain values of $J_1$ and $J_2$ and these points lie on a critical line. This makes intuitive sense because $J_1$ and $J_2$ are clearly capturing similar physical effects (they both bias spins towards alignment), and therefore they are somewhat redundant.

It's important to recognize that the critical point actually exists in a higher dimensional space $\{J\}$. Points on the critical line (or surface) all end up at the critical point under iterated RG transformations. The reason this is important is because points which are arbitrarily close to the critical surface end up arbitrarily close to the critical point under RG transformation, and they are then all driven away from the critical point in the same way. This is the origin of universality.

  • $\begingroup$ Do you mean for the lower dimensional space $\{J\}$ where there is only one critical point, that the critical point is exactly the same thing of the fixed point? $\endgroup$ – Wein Eld Jul 18 '16 at 14:35
  • $\begingroup$ yes, the critical point is a fixed point when you just have $J_1$, however there are other, noncritical, fixed points $\endgroup$ – Patrick Shaffer Jul 18 '16 at 14:39
  • $\begingroup$ How can we prove that the critical point is a fixed point when we only have $J_1$? $\endgroup$ – Wein Eld Jul 18 '16 at 15:00
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    $\begingroup$ At a fixed point the system is invariant to scale transformations. This means that the correlation length at a fixed point is either zero or infinity. A fixed point with infinite correlation length is a critical fixed point. This is the best definition of critical point that I know. $\endgroup$ – Patrick Shaffer Jul 19 '16 at 8:14

See Critical 2d Ising Model for clarifications of the difference between critical point and fixed point. What Patrick said is not correct. The Ising (nearest couplings only) Hamiltonian at the special value of $J_1$ is a critical point but not a fixed point. The critical surface is a huge infinite-dimensional manifold. In it lies the critical line (less standard terminology) which starts at the Gausian fixed point in the UV and end at the Wilson-Fisher fixed point in the IR. This is the line denoted by "F" in this related question: $\phi^4$-theory: interpreting the RG flow


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