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Is there field theory which describe a second order phase transition without upper critical dimension ? Mermin-Wagner says something about lower critical dimension but nothing about upper dimension.

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You need a way to declare that two field theories in different dimensions are "the same". There are natural ways to do this for simple examples, but it is easy to make up a model where there is no upper critical dimension because you extrapolate wrong. –  Ron Maimon Apr 28 '12 at 5:32
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@PanAkry : Do you know that you can "accept" an answer by clicking the green tick next to it? It's not necessary if you're not satisfied with the answer, but please go through your questions and check if there are any that have correct answers in need of an accept. Thanks! –  Manishearth May 7 '12 at 6:09
    
Thanks @Manishearth for pointing that out to PanAkry, I was just about to leave the same 'citizenship patrol' comment ;) –  New Alexandria Dec 29 '12 at 23:54

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The upper critical dimension is the dimension where the statistical field theory is well described by a mean field theory. It is also the dimension where the fluctuation theory turns into a free field theory. You can avoid having an upper critical dimension by tuning the kinetic terms properly:

Consider the Euclidean action:

$$S= \int |q|^{2n} |\phi|^2 + \lambda \phi^4 d^n x$$

This field theory never has an upper critical dimension. But this is because the dimensional extrapolation is wrong. For any fixed power of q, there is an upper critical dimension.

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Does this action correspond to some physical model? It is not clear : "This field theory never has an upper critical dimension... there is an upper critical dimension"! –  PanAkry Apr 28 '12 at 12:25
    
@PanAkry: Not really physical. The point of this example is that I continued the theory wrong into higher dimensions. The right continuation holds the power of q fixed (the exponent doesn't change with n), and this continuation does have an upper critical d. –  Ron Maimon Apr 28 '12 at 14:26
    
Why the downvote? It answers the question--- this is a silly mathematical continuation without upper critical dimension. –  Ron Maimon May 9 '12 at 22:12

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