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

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The discussion centers on the relationship between phi4 theory, the non-linear sigma model, and ferromagnets, particularly in the context of Heisenberg ferromagnets. Phi4 theory is described as a coarse-graining approach that improves upon mean-field theory by providing a more accurate representation of ferromagnetic behavior. The term "expansion around mean-field theory" refers to how phi4 theory builds on mean-field approximations to capture fluctuations. The non-linear sigma model is identified as a low-temperature expansion that effectively describes the dynamics of ferromagnets at reduced temperatures. Understanding these connections is crucial for grasping the theoretical underpinnings of ferromagnetic systems.
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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