# Scale invariant thermal fluctuations

There's something that has been bugging me for over a year now and I seem to be unable to find the answer. I would appreciate it very much if somebody could help me out.

The thing is that I don't understand how it is possible that in second order phase transitions the correlation legth
becomes infinite. I don't understand the scale invariance of thermal fluctuations in these critical phenomena. How is this fact consistent with special relativity? After all a local perturbation should never be able to propagate faster than the speed of light, right?

haruspex
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There's something that has been bugging me for over a year now and I seem to be unable to find the answer. I would appreciate it very much if somebody could help me out.

The thing is that I don't understand how it is possible that in second order phase transitions the correlation legth
becomes infinite. I don't understand the scale invariance of thermal fluctuations in these critical phenomena. How is this fact consistent with special relativity? After all a local perturbation should never be able to propagate faster than the speed of light, right?

In almost all practical circumstances, you might as well treat the speed of light as infinite for this kind of problem.
The same approximation arises in heat diffusion. The standard differential equation implies a raised temperature at one point instantly leads to heat flows at all distances, but extremely small at any great distance.

Carlos L. Janer
Thank you very much for your post.

I wonder if you could help me further?

In cosmology, as far as I know, the EW phase transition seems to be treated as a true thermodynamical phase transition (which seems very strange to me since it describes a change of physical laws at a given temperature, not a change in the state of the existing elementary particles). Leaving this fact aside, the fact that Higgs field VEV seems to be the same everywhere in the Universe seems to indicate that this "phase transition" was critical (second order). However this seems to contradict special relativity for the very same reason pointed out in my previous post.

I guess I must be wrong but I don't know why.

Thanks again for your kind help.

haruspex
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Thank you very much for your post.

I wonder if you could help me further?

In cosmology, as far as I know, the EW phase transition seems to be treated as a true thermodynamical phase transition (which seems very strange to me since it describes a change of physical laws at a given temperature, not a change in the state of the existing elementary particles). Leaving this fact aside, the fact that Higgs field VEV seems to be the same everywhere in the Universe seems to indicate that this "phase transition" was critical (second order). However this seems to contradict special relativity for the very same reason pointed out in my previous post.

I guess I must be wrong but I don't know why.

Thanks again for your kind help.
That does sound like a situation where the speed limit may be important, but it also means it is beyond my fields of expertise.
That said...
If the phase transition is a result of the expanding universe, would it not have multiple independent origins? If there are choices for the final state, could lead to some interesting domain boundaries.

Carlos L. Janer
That does sound like a situation where the speed limit may be important, but it also means it is beyond my fields of expertise.
That said...
If the phase transition is a result of the expanding universe, would it not have multiple independent origins? If there are choices for the final state, could lead to some interesting domain boundaries.
Well, in my opinion (and I don't think that my opinion is of much worth), that's exactly the problem: we seem to be unable to detect them.

haruspex
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Well, in my opinion (and I don't think that my opinion is of much worth), that's exactly the problem: we seem to be unable to detect them.
This is before the end of inflation, right? So as long as the domains were large (i.e. not too great a density of independent origins) any such boundary could be beyond our visible universe now?

This is before the end of inflation, right? So as long as the domains were large (i.e. not too great a density of independent origins) any such boundary could be beyond our visible universe now?
What I'm about to write is just my understanding of this subject which is very limited. I am neither a cosmologist nor a high energy physicist.

The EW phase transition happened (assuming it really did, which I find dificult to believe) around 10^(-12) seconds after the initial singularity and I think that the general consensus is that this time is many orders of magnitude larger than the time it took for inflation to finish.

haruspex
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What I'm about to write is just my understanding of this subject which is very limited. I am neither a cosmologist nor a high energy physicist.

The EW phase transition happened (assuming it really did, which I find dificult to believe) around 10^(-12) seconds after the initial singularity and I think that the general consensus is that this time is many orders of magnitude larger than the time it took for inflation to finish.
Then I have no further suggestions. I'm no cosmologist either.

Then I have no further suggestions. I'm no cosmologist either.
Thanks, again, for your time and attention to this subject.

When studying phase transitions, you are always taking the thermodynamic limit. Phase transitions, by definition don't happen in finite systems. You have finite size effects which show up as corrections to the free energy etc since you technically do have a length scale. These can be predicted in the Ising model for example using conformal field theory since it is a CFT at the critical point. These corrections go as 1/L and are proportional to the central charge.

In addition to this, when talking about phase transitions and universalitt, you are thinking of a low energy limit which corresponds to low frequencies and long time scales. So in classical phase transitions you don't think about time dependence. However, if you have a quantum phase transition, you do have time dependence of correlation functions at finite temperature that you did not have in the classical problem. Many times you go from classical to quantum by Wick rotation where the quantum system is in one less spatial dimension. This new time scale is the phase decoherence time which is finite at finite temperatures. This corresponds to the time the wavefunction takes to forget it's phase. This becomes very short near the critical point which makes this analytical continuation break down.

You can also have emergent Lorentz invariant corresponding to the dynamical critical exponent at quantum critical points. Another thing you have is the Lieb Robinson velocity, which is the maximum speed a local perturbation/information can travel in a system. This exists even in theories without Lorentz invariance. You can observe it in the decay of correlation functions/growth of commutators.

When studying phase transitions, you are always taking the thermodynamic limit. Phase transitions, by definition don't happen in finite systems. You have finite size effects which show up as corrections to the free energy etc since you technically do have a length scale. These can be predicted in the Ising model for example using conformal field theory since it is a CFT at the critical point. These corrections go as 1/L and are proportional to the central charge.

In addition to this, when talking about phase transitions and universalitt, you are thinking of a low energy limit which corresponds to low frequencies and long time scales. So in classical phase transitions you don't think about time dependence. However, if you have a quantum phase transition, you do have time dependence of correlation functions at finite temperature that you did not have in the classical problem. Many times you go from classical to quantum by Wick rotation where the quantum system is in one less spatial dimension. This new time scale is the phase decoherence time which is finite at finite temperatures. This corresponds to the time the wavefunction takes to forget it's phase. This becomes very short near the critical point which makes this analytical continuation break down.

You can also have emergent Lorentz invariant corresponding to the dynamical critical exponent at quantum critical points. Another thing you have is the Lieb Robinson velocity, which is the maximum speed a local perturbation/information can travel in a system. This exists even in theories without Lorentz invariance. You can observe it in the decay of correlation functions/growth of commutators.
And your point being what, exactly?

This answers your questions about the correlation length and causality. This is an explanation of why you can say that it diverges (only in the thermodynamic limit) and how this is consistent with causality when you are talking about conventional classical phase transitions.

This answers your questions about the correlation length and causality. This is an explanation of why you can say that it diverges (only in the thermodynamic limit) and how this is consistent with causality when you are talking about conventional classical phase transitions.
The physical problem I was wondering about is the EW phase transition in the early Universe and why the Higg's field vev (phase included) seems to be the same everywhere in our observable Universe. It does not make any sense to me because if it really was a thermodynamical phase transition and even if it was a critical phase transition we should be able to "see" domain boundaries in regions of our obsevable Universe that were causally disconected at t=10^(-12)s. This idea has been haunting me for over a year now. I cannot make any sense of it.

I would assume it is because we are looking at long time behavior/the low energy effective theory when we consider phase transitions. You are not thinking in terms of the spreading of some perturbation (you could also calculate that through a correlation function but not the standard one) you are thinking of the collective/local fluctuations in the system. And of course in addition the real system has a finite size. All of these quantities are analytic in a finite size system, the correlation length does not diverge, this comes from taking the thermodynamic limit.

Carlos L. Janer
I would assume it is because we are looking at long time behavior/the low energy effective theory when we consider phase transitions. You are not thinking in terms of the spreading of some perturbation (you could also calculate that through a correlation function but not the standard one) you are thinking of the collective/local fluctuations in the system. And of course in addition the real system has a finite size. All of these quantities are analytic in a finite size system, the correlation length does not diverge, this comes from taking the thermodynamic limit.
Thanks for your post. Our observable universe was supposed to be rather small at that time (sub cm range) but it was expanding incredibly fast. Only tiny subregions of it were causally connected. How could possibly the phase transition in causally disconnented parts lead to the same Higss field? As far as I can tell any phase was as good as another. Moreover, what would the consequences of breaking the symmetry locally instead of globally be? What I mean is a spacetime dependent phase of the Higgs field. Most probably this is pure nonsense, but I'd like to know why so that I can stop thinking about this.

Demystifier
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In cosmology, as far as I know, the EW phase transition seems to be treated as a true thermodynamical phase transition (which seems very strange to me since it describes a change of physical laws at a given temperature, not a change in the state of the existing elementary particles).
Why do you think that it describes a change of physical laws? It certainly does not describe a change of fundamental physical laws.

Because before the phase transition there were many different ways in which the symmetry could have been broken. Yet, for non understandable reasons (for me, at least), our whole observable universe deciced to break it everywhere in exactly the same way.

Demystifier