# Ising Model for Spins: External Magnetic Field Effect

• aaaa202
In summary, the conversation discusses the use of mean-field theory in the Ising model with an external magnetic field. The model successfully introduces a term that couples the spins to the external field, but it is unclear whether the mean-field approximation can still be used separately for the Ising term. The speaker mentions that the average spin is affected by the external field, but it is not obvious if the deviation from the average is still small in the presence of the field.
aaaa202
I think most of you are familiar with this model (sum runs over nearest neighbours):

H = -J ∑S_iz * S_jz

It demonstrates one of the succeses of meanfield theory as one can succesfully introduce:

S_iz = <S_iz> + S_iz - <S_iz> = <S_iz> + δS_iz

Such that:

S_iz*S_jz ≈ 2S_iz<S_jz> + const

Where I have neglected the second order term. Now my question:
In the presence of an external magnetic field the Hamiltonian gets introduced a second term which couples the spins to the external magnetic field:

H=H_ising + ∑ S_iz * B

In this case can I still use the mean-field approximation separately for the Ising term? My book certainly does it, but I am a bit confused because in my head the average <S_iz> is affected by the external magnetic field's effect on the spin, and in this case it is not for me obvious that just because the deviation from the average is small in the case of no external field, it should be too in the presence of one.

The average $\langle S_j^z \rangle$ is certainly changed by the external magnetic field - consider the extreme case with an antiferromagnetic $J$ and a strong field along some axis. This will change the mean field from zero in the case of no field to its maximal value in the case of a strong field. There are different mean fields for different Hamiltonians, if you will.

## 1. What is the Ising Model for Spins?

The Ising model for spins is a mathematical model used in statistical mechanics to describe the behavior of a large number of interacting magnetic moments or spins in a material. It was first proposed by physicist Ernst Ising in the 1920s and has been extensively studied and applied in various fields, including condensed matter physics, statistical physics, and materials science.

## 2. How does the Ising Model for Spins work?

The Ising model for spins is based on a lattice structure where each lattice point represents a magnetic moment or spin. The spins can either be in an "up" or "down" state, corresponding to a positive or negative magnetization. The model assumes that the interactions between spins are only between nearest neighbors and can be described by a simple energy function. By minimizing this energy function, the model can predict the ground state and other properties of the system.

## 3. What is the significance of an external magnetic field in the Ising Model for Spins?

An external magnetic field is an important factor in the Ising model for spins as it introduces an additional energy term that affects the orientation of the spins. This external field can either be parallel or antiparallel to the spins and can greatly influence the behavior of the system. It can cause the spins to align in the same direction, leading to a ferromagnetic state, or in opposite directions, resulting in an antiferromagnetic state.

## 4. How does the external magnetic field affect the phase transition in the Ising Model for Spins?

The presence of an external magnetic field can significantly alter the phase transition behavior in the Ising model for spins. The critical temperature at which the system transitions from a disordered to an ordered state is affected by the strength of the external field. In the absence of a field, the critical temperature is known as the Curie temperature. However, in the presence of an external field, this critical temperature can shift to higher or lower values depending on the strength and direction of the field.

## 5. What are some real-world applications of the Ising Model for Spins?

The Ising model for spins has been widely used in various fields, including material science, condensed matter physics, and computational neuroscience. It has been applied to study the behavior of magnetic materials, such as ferromagnets and antiferromagnets, and to understand phase transitions in physical systems. In computational neuroscience, the model has been used to simulate neural networks and study the emergence of complex behavior. It has also found applications in data compression and optimization problems.

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