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My question is somehow related to: The relation between critical surface and the (renormalization) fixed point but there is another problem:

The problem is that if we accept that all points on the critical surface are critical in the manner that their corresponding correlation length is infinite, then according to the scaling hypothesis a system whose parameters lie on the critical surface should be scale invariant. And therefore its parameters shouldn't change under the RG transformation. So each point on the critical surface should be a fixed point and hence there is not any RG flow over the critical surface. It implies that the RG transformation doesn't push any point on the critical surface to our first fixed point.

From the above statements one can deduce that the RG method is inconsistent. Surely there should be something which I'm missing but what's that?

Karim Chahine
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Hossein
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1 Answers1

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I'll attempt to give an answer to this old post.

I think the statement:

[...] a system whose parameters lie on the critical surface should be scale-invariant. And therefore its parameters shouldn't change under the RG transformation.

Is wrong. Infinite correlation length does not imply being on a fixed point of the RG transformation, it just means that we are on a critical manifold of some critical fixed point. If a system has a divergent correlation length then yes, we expect it to be scale-invariant, but that does not mean that the parameters don't change. It means rather that, by coarse-graining and rescaling, the symmetries of the system are preserved and the free energy doesn't change. Only the fixed points have, by definition, the property of being invariant under the RG transformation.

Karim Chahine
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