Mean Field Indices in D>4: Modern Proofs

[tirrel]

Ehi u... It’s a lot of time since visited this site for the first time ... unluckily I haven’t got much time to enter: I hope I’ll have time in the future...

Anyway I’ve got a problem... I’m studying fase transitions from an elementary point of view and in particular the mean field approximation of the ising model...

I know that the critical indices calculated from this model should be correct in D>4, where D is the spatial dimension of the model. I also know that the modern approach to the calculation of the critical indeces is through field theory and the renormalization group. I’ve tried to learn something about these topics from the book of Cardy but I couldn’t find a proof that these indices are correct in D>4 using this modern approach. I’ve understood that D<4 is a mess, but very few about D>=4.

Does anyone know the logic steps necessary to verify the validity of mean field using this modern approaches ? (I want them to be used!)

 

[tirrel]

mmm... I'm on holiday (or somehing like that) and I cannot access to the site of my university... is there somebody who could send me one of these two articles at my e-mail ar_ma86@libero.it...

M.E.Fisher and D.S.Gaunt, Phys.Rev.133,A224(1964)

R.Abe, Prog. Theor.Phys. 47,62 (1972)

there should be the proof that mean field is correct for D tending to infinity... which is a not exactly what I was asking but closely related...

 

[Zacku]

Ehi u... It’s a lot of time since visited this site for the first time ... unluckily I haven’t got much time to enter: I hope I’ll have time in the future...

Anyway I’ve got a problem... I’m studying fase transitions from an elementary point of view and in particular the mean field approximation of the ising model...

I know that the critical indices calculated from this model should be correct in D>4, where D is the spatial dimension of the model. I also know that the modern approach to the calculation of the critical indeces is through field theory and the renormalization group. I’ve tried to learn something about these topics from the book of Cardy but I couldn’t find a proof that these indices are correct in D>4 using this modern approach. I’ve understood that D<4 is a mess, but very few about D>=4.

Does anyone know the logic steps necessary to verify the validity of mean field using this modern approaches ? (I want them to be used!)

Hello,

If you want to make a field theory from the Ising model (as a $$\phi^4$$ model for example), then the mean field theory is based on the Ginsburg-Landau approximation.
Actually this is an approximation to find the phenomenological Landau free energy for the phase transition. You don't need the renormalisation group to prove that the predictions of this model are correct if D>4 (I would say that as far as I know). Actually in the Ginsburg-Landau approximation, the Ginsburg contribution has been to prove that this mean field method (i.e. a saddle point method) is correct for D>4. So I recommand that you search for tags as "Ginsburg criteria" or something like that and that would be enough.
Of course if you want to know what happens for D < 4 you will need the RG but you will need a lot of practice and time to understand plainly what it is all about.

 

[tirrel]

hi zacko thanks for the answer!

I tried to follow your advice looking in google for ginsburg criteria! and I fuond this page that looks promising

http://www.tcm.phy.cam.ac.uk/~bds10/phase.html

but there is the final discussion I've got problems in following... I'd like to ask u something...

If u go to chapter2 (G-L theory)... not in the last paragraph but in the previous one there are two formulas concerning corrections to specific heat... from these the writer would like to see (or at least to see by intuition) that the model has got problems for d<4 but not for d>4. To see that it's observed (it's just few lines to read!) that for d>4 integral is divergent and so is dominated but the cutoff and so adds simply a costant to the specific heats not changing the critical indices. I don't understand this... in d<4 the integral is convergent but why the cutoff doesn't play any role? furthermore the first integral is divergent if d<4 for t=0... this is not important?

well... I've understood the mathematical passages to that two formulas but the final discussion seems obscure to me...