We know when $D>4$, i.e. $D$ larger than upper critical dimension, then critical exponents are exactly same as the ones of mean field . When $D<4$, critical exponents are not given correctly by the Landau theory. Many books will list criticial exponents like Ising model in $D=2$ and $D=3$. But it seems that textbooks don't talk about or list the critical exponents in $D=4$.

Besides what's about RG flow in $D=4$. In class I've learnt the perturbative RG flow of Guassian model in $D<4$ and $D>4$ and we see their RG have totally different properties. But what's about $D=4$?

My questions:

  1. For Ising model, Heisenberg model and $O(n)$ model, what are their critical exponents in $D=4$ (There should be numerical result). You could directly give me the reference.

  2. What's special when $D=4$? There should be some reasons why normal textbooks avoid discussion of $D=4$.


In $D=4$ there is no interacting CFT in the universality class of the action

$$S[M] = \int\!d^4x\, (\partial M)^2 + r M^2 + g M^4$$

or similarly if $M$ is an $N$-vector. That's the whole point of the RG analysis in $D=4$. The free (Gaussian) theory is completely consistent; if you try to turn on the $g$-coupling above, you see that it blows up at short distances, hence you conclude that only $g=0$ describes a healthy QFT. This is known as the triviality problem of 4d QFT. (Of course, it's very well possible that the above action has a non-trivial UV completion, which will probably look a bit more complicated.) The above story is consistent with your expectations: at $D>4$ you only have the Gaussian theory, at $D=4-\epsilon$ the interaction strength at the fixed point is of order $\epsilon$, and it vanishes as $D \to 4$.

  • $\begingroup$ You didn't actually answer all of OP's questions. $\endgroup$
    – KF Gauss
    Aug 2 '17 at 23:30
  • $\begingroup$ Well, to be explicit: the critical exponents are those of the Gaussian theory, and there's nothing special about $D=4$ compared to $D=4.0001$ or $D=3.99999$ (except that the running of the coupling is purely driven by loops). $\endgroup$
    – user159249
    Aug 3 '17 at 5:43

What happens at the critical dimension $D=4$ is that critical exponents are equal to their mean field values but one has logarithmic corrections which can be seen as some kind of remnant of anomalous values as one takes the limit $D\rightarrow 4$ from below. Take for example the universality class corresponding to the Ising model as well as the $\phi^4$ model and consider the susceptibility exponent $\gamma$. The latter is usually defined by $$ \chi\sim |T-T_{\rm c}|^{-\gamma}\ . $$ For $D>4$, one has $\gamma=1$ which is the mean field value. For $D<4$, one has $\gamma>1$ which therefore is a non-classical or anomalous value. At $D=4$, what happens is that $$ \chi\sim |T-T_{\rm c}|^{-1}\times (\log |T-T_{\rm c}|)^{\frac{1}{3}}\ . $$ This kind of logarithmic correction has been understood using Wilson's RG, in a non mathematically rigorous way, by Wegner and Riedel in the article "Logarithmic Corrections to the Molecular-Field Behavior of Critical and Tricritical Systems". However, this is now a rigorous mathematical result. See the recent work of Bauerschmidt, Brydges and Slade "Scaling Limits and Critical Behaviour of the 4-Dimensional $n$-Component $|\varphi|^{4}$ Spin Model" (or here for the arXiv version).

To make the connection with Marty's answer, I should add that when one takes the continuum or scaling limit these logarithmic corrections get wiped out and one ends up with the massless Gaussian field. This has only been proven on a finite volume torus (see the above article by Bauerschmidt, Brydges and Slade).

  • 1
    $\begingroup$ Thanks, this is a cool addition to my post, and the last ref is also interesting. $\endgroup$
    – user159249
    Aug 16 '17 at 19:08
  • $\begingroup$ How do you make dimensionless argument of log ? $\endgroup$
    – Nikita
    Apr 3 '20 at 10:27
  • $\begingroup$ @Nikita: You don't have to. In any case for any positive constant $\alpha$, $\log(\alpha |x|)\sim\log |x|$ when $x\rightarrow 0$, so the asymptotics and exponents do not change if you multiply $|T-T_{\rm c}|$ by a constant that reflects a change of units in a dimensionful quantity. $\endgroup$ Apr 3 '20 at 14:07

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