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What is the relation between the critical point and the fixed point?

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Is the critical point in critical phenomena the same thing of the fixed point of renormalization group flow? If yes, why? And if no, then what is the relation between the critical point and the renormalization group flow?

asked Jul 18, 2016 in Theoretical Physics by Wein Eld (195 points) [ no revision ]

My understanding is the following. Consider the space, $\mathcal{H}$ of Hamilitonians spanned by several operators $O_i$ (that is $H$ ($\in \mathcal{H}$) = $\sum c_i O_i$).  When we do RG, some operators (so called irrelevant operators) become less and less important as we go to large length scales. Some operators (so called relevant operators) becomes more and more important as we go to large length scales and it is coefficients of these operators  that determine what phase we are in. One can get to the critical point by tuning the coefficients of these (relevant) operators. At the critical point, $\mathcal{H}$ is spanned only by the irrelevant operators. The subspace of $\mathcal{H}$, that is spanned by only the irrelevant operators is called the critical surface. Fixed point is a special point in this critical surface.

@Thhin Subhra  Why do you say at the critical point, $\mathcal{H}$ is spanned only by the irrelevant operators? And at the fixed point, only relevant (and marginal) operators survive, how can the fixed point belong to the critical surface by your definition?
 

@Wein Eld  I do not really understand the inconsistency that you are trying to point out. What I have in mind is the following. Suppose the space $\mathcal{H}$ is spanned by $(O_1,O_2,O_3)$. Let $O_1$ be relevant and $O_2, O_3$ irrelevant. Any point $H$ in $\mathcal{H}$ can be specified by three numbers $(x_1, x_2, x_3) \in R^3 (\sim \mathcal{H})$: $H = x_1 O_1+ x_2 O_2 +x_3 O_3$.  Critical point is given by a particular value of $x_1$ (the coefficient of relevant operator), say $x_1 =1$ which describes an $x_2-x_3$ plane (the critical surface). Any point in this plane will remain in this plane under RG flow. The fix point lies somewhere in this plane where the RG flow stops.

1 Answer

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The short answer is yes: critical points are effectively described by scale invariant field theories, which are by definition fixed points of the renormalization group flow. A longer answer would probably depend on the precise choice of definition of a critical point.

answered Jul 18, 2016 by 40227 (4,690 points) [ no revision ]

Could you please make the answer somewhat longer?

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