A The relation between ferromagnets, Phi4 and non-linear sigma model

Aethermimicus
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It is possible to describe the long-distance behavior of the Heisenberg ferromagnets in two different ways:the phi4 theory which corresponds to an expansion around mean-field thoery and the nonlinear sigma model obtained by low-temperature expansion.
I'm struggling to understand the relation between phi4 theory,non-linear sigma model and ferromagnets.
I've read this in a paper(Phys.Rev.B14(1976)3110):'It is possible to describe the long-distance behavior of the Heisenberg ferromagnets in two different ways:the phi4 theory which corresponds to an expansion around mean-field thoery and the nonlinear sigma model obtained by low-temperature expansion.'
I do understand that phi4 field theory is a coarse graining version of ferromagnets (for example,the Ising model),and achieve better results than mean field theory,but why is it 'an expansion around mean field theory'?
Also I fail to understand non-linear sigma model as a low-temperature exapansion of ferromagnets.
 

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From the BCS theory of superconductivity is well known that the superfluid density smoothly decreases with increasing temperature. Annihilated superfluid carriers become normal and lose their momenta on lattice atoms. So if we induce a persistent supercurrent in a ring below Tc and after that slowly increase the temperature, we must observe a decrease in the actual supercurrent, because the density of electron pairs and total supercurrent momentum decrease. However, this supercurrent...

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