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

In summary, the relationship between ferromagnets, the Phi4 model, and the non-linear sigma model lies in their shared foundation in statistical mechanics and quantum field theory. Ferromagnets exhibit spontaneous magnetization, which can be described using the Phi4 model that accounts for interactions and symmetry breaking. The non-linear sigma model provides a framework to analyze low-energy excitations and critical phenomena in ferromagnetic systems, linking their behavior to topological aspects and collective modes. Together, these models enhance the understanding of phase transitions and magnetic properties in ferromagnetic materials.
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Aethermimicus
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TL;DR Summary
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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