# Mean field theory in the Heisenberg model

• A
• Jeff Chen
In summary, when using mean field approximation in the Heisenberg model, the convention is to assume a uniform magnetization in the z direction and take the magnetization in other directions to be zero. This is due to choosing a coordinate system where the z direction aligns with the unit vector in the direction of magnetization.
Jeff Chen
I have a problem about Heisenberg model.When applying mean field approximation,why does the average of spin flip terms be zero(<S+>=<S-> = 0 )? Thanks

It's just a convention - you take the "mean field" such that the system magnetizes in the z direction, and take the magnetization in the other directions to zero.

More generally, in an arbitrary coordinate system, you assume that the system has a uniform magnetization given by
$$\mathbf{M} = \langle \mathbf{S}_i \rangle = M \hat{e}$$
where ##\hat{e}## is a unit vector in some direction. Now just choose your coordinate system such that the z direction points along ##\hat{e}##, and then you have ##\langle S^+ \rangle = \langle S^- \rangle = 0## in that coordinate system.

## 1. What is mean field theory in the Heisenberg model?

Mean field theory is a theoretical framework used to study the behavior of interacting particles in a system, such as atoms in a material. In the Heisenberg model, it is used to describe the interactions between magnetic moments of particles.

## 2. How does mean field theory work in the Heisenberg model?

In mean field theory, the interactions between particles are approximated by an average field, rather than considering the individual interactions between each particle. This simplifies the calculations and allows for easier analysis of the system.

## 3. What are the assumptions made in mean field theory for the Heisenberg model?

The main assumptions are that the particles are independent and that their interactions can be approximated by a mean field. Additionally, it assumes that the particles are in thermal equilibrium and that the system is in a state of minimum energy.

## 4. What are the limitations of mean field theory in the Heisenberg model?

Mean field theory is a simplified approach and does not take into account fluctuations or correlations between particles. It also does not accurately describe systems at very low temperatures or near phase transitions.

## 5. How is mean field theory in the Heisenberg model used in real-world applications?

Mean field theory is commonly used in condensed matter physics to study the behavior of materials, such as magnets. It has also been applied in other fields, such as statistical mechanics and quantum field theory, to study the behavior of complex systems.

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