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

Dec 11, 2023

Aethermimicus

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.

What is the relationship between ferromagnets and the Phi4 model?

The Phi4 model is a scalar field theory that describes a system of interacting scalar fields. In the context of ferromagnets, the Phi4 model can be used to represent the order parameter associated with magnetic ordering. The potential term in the Phi4 model captures the interactions that lead to spontaneous magnetization, making it a useful tool for studying phase transitions in ferromagnetic materials.

How does the non-linear sigma model relate to ferromagnets?

The non-linear sigma model is another theoretical framework used to describe systems with a continuous symmetry, such as ferromagnets. In this model, the magnetic spins are represented as points on a sphere, and the dynamics of these spins can be studied in terms of a field theory. The non-linear sigma model is particularly useful for understanding low-energy excitations and critical phenomena in ferromagnetic systems, especially in the context of two-dimensional materials.

What are the key differences between the Phi4 model and the non-linear sigma model?

The key difference lies in their treatment of the order parameter and the type of symmetries they address. The Phi4 model focuses on scalar fields and is typically employed to study phase transitions and symmetry breaking in systems with discrete symmetry. In contrast, the non-linear sigma model deals with continuous symmetries and is more suited for studying the low-energy behavior of systems like ferromagnets, where the order parameter can be represented as a vector on a manifold.

Can the Phi4 model be used to study quantum phase transitions in ferromagnets?

Yes, the Phi4 model is often used to investigate quantum phase transitions in ferromagnets. By incorporating quantum fluctuations and analyzing the behavior of the order parameter as a function of temperature and other parameters, researchers can explore how the system transitions between different phases, such as from a disordered to an ordered state, and the role of critical phenomena in these transitions.

How do these models help in understanding real-world ferromagnetic materials?

Both the Phi4 model and the non-linear sigma model provide theoretical frameworks that can be used to gain insights into the behavior of real-world ferromagnetic materials. They allow researchers to explore critical phenomena, phase transitions, and the effects of fluctuations in a controlled manner. By comparing theoretical predictions with experimental data, scientists can refine their understanding of magnetic ordering and the underlying mechanisms in ferromagnets.

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

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

What is the relationship between ferromagnets and the Phi4 model?

How does the non-linear sigma model relate to ferromagnets?

What are the key differences between the Phi4 model and the non-linear sigma model?

Can the Phi4 model be used to study quantum phase transitions in ferromagnets?

How do these models help in understanding real-world ferromagnetic materials?

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