Renormalization group and universality

In summary, there is a concept of universality in physics that states that many different systems exhibit the same behavior near critical points, described by critical exponents that only depend on the dimension of the system and the dimension of the order parameter. There is a diagram that shows this relationship, but it is specific to a particular model. There are many different classes of universality, each determined by the relevant operators in a renormalization group fixed point. There is also a labeling system for universality classes based on critical exponents and dimensions, but it is not a single, all-encompassing class. The dimension of the order parameter is related to the dimension of space and the number of independent components. However, there is no general rule for
  • #1
tom.stoer
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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?
 
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  • #2
It's not that easy. You must be thinking of a particular model, for example, the [itex]\lambda\phi^4[/itex]. Then you can get the diagram. In that precise case, if [itex]d\geq 4[/itex] all critical exponents correspond to mean-field (that's the upper critical dimension). But other theories have different behaviours.
 
  • #3
jrlaguna said:
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
 
  • #4
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" [itex]O_i[/itex]. 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.
 
  • #5
jrlaguna said:
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?
 
  • #6
Yes, that's true. Sorry, I have never seen that pic. I agree it would be very interesting! :)
 
  • #8
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?
 
  • #9
Whith 'dimension' I mean 'dimension of space' and 'number of independent components of the order parameter'.
 
  • #10
tom.stoer said:
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.
 
  • #11
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 [Broken]
 
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  • #12
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

[tex](\langle\bar{q}q\rangle,(\langle\bar{q}\gamma_5 q\rangle)[/tex]

which is two-dim. but where spacetime is d-dim and SU(N), N = number of flavours, has not been specified
 
  • #13
tom.stoer said:
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

[tex](\langle\bar{q}q\rangle,(\langle\bar{q}\gamma_5 q\rangle)[/tex]

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."
 
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  • #14
atyy said:
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.
 
  • #15
tom.stoer said:
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.
 

1. What is the renormalization group?

The renormalization group is a mathematical framework used in theoretical physics to study the behavior of physical systems at different length scales. It involves systematically changing the parameters of a system and observing how its properties change, allowing us to understand how a system behaves at both large and small scales.

2. What is universality in the context of the renormalization group?

In the context of the renormalization group, universality refers to the phenomenon where different physical systems can exhibit similar behavior at a critical point, despite having different microscopic details. This allows us to understand the behavior of different complex systems using the same mathematical tools.

3. How does the renormalization group help us understand phase transitions?

The renormalization group allows us to understand phase transitions by studying how the behavior of a system changes as it approaches a critical point. By analyzing the behavior at different length scales, we can determine the critical exponents that characterize the phase transition and predict the behavior of the system near the critical point.

4. What is the relationship between the renormalization group and statistical mechanics?

The renormalization group is closely related to statistical mechanics, as both fields study the behavior of systems with many interacting particles. The renormalization group provides a way to analyze the behavior of these systems at different length scales, while statistical mechanics uses probability and thermodynamics to describe the behavior of these systems as a whole.

5. Can the renormalization group be applied to all physical systems?

Yes, the renormalization group can be applied to a wide range of physical systems, from simple models in statistical mechanics to complex phenomena in condensed matter physics and quantum field theory. It is a powerful tool for understanding the behavior of physical systems at different length scales, and has applications in many areas of physics.

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