Renormalization group and universality

[tom.stoer]

I remember an argument which says that closed to critical points all systems are universal in the sense that their behavior is described by the critical exponents and that these critical exponents depend only on the dimension of the system and the dimension of the order parameter.

I remember a diagram with space-dimension on the abscissa and order-parameter-dimension on the ordinate showing curves of constant critical exponent and several physical systems.

Does anybody know a reference or web resource for such a diagram?

 

[jrlaguna]

It's not that easy. You must be thinking of a particular model, for example, the \lambda\phi^4. Then you can get the diagram. In that precise case, if d\geq 4 all critical exponents correspond to mean-field (that's the upper critical dimension). But other theories have different behaviours.

 

[tom.stoer]

It's not that easy. You must be thinking of a particular model

No, universality means that a huge class of models show identical behavior closed to the critical point

 

[jrlaguna]

I think you're overestimating the universality concept. There are many "classes of universality", and many new of them appear every year in the scientific literature.

Imagine that you have a microscopic model, characterized by a series of "operators" O_i. When you renormalize (i.e.: see things from far away, you blur the details) some of them increase their importance and some of them decrease. The first are called relevant, and the second irrelevant. There are even "marginal" operators, which neither increase or decrease. Fixed points of the renormalization group, or universality classes, are characterized by the set of relevant operators. It's not like you have a single all-encompassing universality class. No, it depends on the operators, so it depends on your theory.

 

[tom.stoer]

I think you're overestimating the universality concept. There are many "classes of universality" ... No, it depends on the operators, so it depends on your theory.

OK; nevertheless there is a kind of 'labelling' of universality classes related to the critical exponents,the dimensions of the model and the order parameter. Have seen something like that?

 

[jrlaguna]

Yes, that's true. Sorry, I have never seen that pic. I agree it would be very interesting! :)

 

[tom.stoer]

!

Look at this PDF: http://www.spektrum.de/alias/physik/die-renormierungsgruppe/944646

Unfortunately it's in German, but there should be an article in Scientific American as well.

 

[Dickfore]

So, when you say "dimensionality of the field", do you mean an SO(N) theory with an N-dimensional vector field (or N copies of a scalar field) that obeys that symmetry?

 

[tom.stoer]

Whith 'dimension' I mean 'dimension of space' and 'number of independent components of the order parameter'.

 

[Dickfore]

Whith 'dimension' I mean 'dimension of space' and 'number of independent components of the order parameter'.

But, does the Hamiltonian for the order parameter field obey some symmetry, like SO(N)? If yes, then this gives a huge constraint on the possible forms of the Hamiltonian and the universality is not surprising.

 

[atyy]

O(N) models
N dependence, d=3: http://arxiv.org/abs/cond-mat/9803240
A formula estimating critical exponents as a function of N and d=4-ε is given in http://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2008/lecture-notes/lec11.pdf

 

[tom.stoer]

How can one relate the dimension of the order parameter to the dimension of the fields? In QCD the chiral condensate is decsribed by the order parameter

(\langle\bar{q}q\rangle,(\langle\bar{q}\gamma_5 q\rangle)

which is two-dim. but where spacetime is d-dim and SU(N), N = number of flavours, has not been specified

 

[atyy]

How can one relate the dimension of the order parameter to the dimension of the fields? In QCD the chiral condensate is decsribed by the order parameter

(\langle\bar{q}q\rangle,(\langle\bar{q}\gamma_5 q\rangle)

which is two-dim. but where spacetime is d-dim and SU(N), N = number of flavours, has not been specified

Doesn't that mean there's no relation, ie. holds for all d and N?

In some models, even what the appropriate order parameter is is still researched, so I'd be surprised if there's a general algorithm for finding the order parameter. For example, http://arxiv.org/abs/0704.1650 asks "do we consider <Z> or <r> to be the order parameter?".

A similar sentiment is found at http://www.lassp.cornell.edu/sethna/OrderParameters/OrderParameter.html "Finally, let's mention that guessing the order parameter (or the broken symmetry) isn't always so straightforward. For example, it took many years before anyone figured out that the order parameter for superconductors and superfluid Helium 4 is a complex number ψ."

 

[tom.stoer]

Doesn't that mean there's no relation, ie. holds for all d and N?

No, according to the classification of Wilson's classification (d=4, n=2) the critical exponents should not depend on N but are sensitive to the spacetime dimension d=4.

 

[atyy]

No, according to the classification of Wilson's classification (d=4, n=2) the critical exponents should not depend on N but are sensitive to the spacetime dimension d=4.

I meant the order parameter is different for different systems,and I don't think there is a rule for finding the order parameter.

The critical exponents do depend on N (and d) for O(N) models.