Calculate the spin wave dispersion relation for a ferromagnetic heisenberg model

Your name]In summary, to calculate the spin wave dispersion relation E_k for the ferromagnetic Heisenberg model with j_tot = 1/2 on a 1d square lattice, we follow the steps of writing down the Hamiltonian, expressing the spin operators in terms of creation and annihilation operators, diagonalizing the Hamiltonian, and solving for the eigenvalues in Fourier space. This will give us the desired dispersion relation as a function of the wave vector k.
  • #1
matt_crouch
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1

Homework Statement


Calculate the spin wave dispersion relation Ek for the ferromagnetic Heisenberg model with jtot = 1/2
Assume a 1d square lattice and interactions of strength J between nearest neighbours and zero elsewhere

Homework Equations



H|k> = [E0 +2jtot[itex]\sum J(r)(1-Exp(ik.r) [/itex]] |k>


The Attempt at a Solution



Not really sure where to start :/
 
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  • #2


Thank you for your question. I am happy to assist you in solving this problem.

To calculate the spin wave dispersion relation E_k for the ferromagnetic Heisenberg model with j_tot = 1/2, we can use the following steps:

1. Start by writing down the Hamiltonian for the system. In this case, it will be the Heisenberg model with nearest neighbor interactions:

H = -J ∑ _<i,j> S_i · S_j

where J is the interaction strength and <i,j> indicates summation over nearest neighbor pairs.

2. Next, we need to express the spin operators S_i in terms of creation and annihilation operators. For the j_tot = 1/2 case, we can use the Holstein-Primakoff transformation:

S_i^+ = a_i† √(2S - a_i†a_i)
S_i^- = √(2S - a_i†a_i) a_i
S_i^z = S - a_i†a_i

where a_i† and a_i are the creation and annihilation operators, respectively, and S is the spin quantum number.

3. Now, we can substitute these expressions into the Hamiltonian and use the commutation relations to simplify the terms. This will result in a quadratic Hamiltonian in terms of the creation and annihilation operators.

4. Next, we can diagonalize the Hamiltonian by performing a Fourier transform:

a_i = √(1/N) ∑ _k e^(ik.r_i) a_k
a_i† = √(1/N) ∑ _k e^(-ik.r_i) a_k†

where N is the total number of lattice sites and r_i is the position of the i-th site.

5. Finally, we can solve for the eigenvalues E_k by diagonalizing the Hamiltonian in the Fourier space. This will give us the spin wave dispersion relation E_k as a function of the wave vector k.

I hope this helps you get started on solving the problem. Please let me know if you have any further questions or need clarification on any of the steps.
 

1. What is a spin wave dispersion relation?

A spin wave dispersion relation is a mathematical relationship that describes the energy and momentum of spin waves in a material. It is used to understand the behavior and properties of magnetic systems.

2. How is the spin wave dispersion relation calculated?

The spin wave dispersion relation is calculated using a theoretical model, such as the Heisenberg model, which describes the interactions between magnetic moments in a material. This model takes into account factors such as the strength of magnetic interactions and the crystal structure of the material.

3. What is a ferromagnetic Heisenberg model?

A ferromagnetic Heisenberg model is a theoretical model used to describe the behavior of ferromagnetic materials, such as iron or nickel. It is based on the Heisenberg exchange interaction, which is the dominant interaction between neighboring magnetic moments in a ferromagnet.

4. Why is calculating the spin wave dispersion relation important?

Calculating the spin wave dispersion relation is important because it allows us to understand the magnetic properties of materials and how they may change under different conditions, such as temperature or external magnetic fields. This information is crucial for developing new technologies and materials for applications in data storage, electronics, and spintronics.

5. Can the spin wave dispersion relation be experimentally measured?

Yes, the spin wave dispersion relation can be experimentally measured using techniques such as inelastic neutron scattering, which can directly probe the energy and momentum of spin waves in a material. These experimental results can then be compared to theoretical calculations to validate the model and gain a better understanding of the material's magnetic properties.

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