Heisenberg ferromagnet and spin waves

In summary: S}} \frac{1}{\sqrt{N}} \sum_j \sum_i \exp(ik\cdot r_j) s_j^- s_i^{+} s_i^{-} |0> - \frac{1}{\sqrt{2S}} \frac{1}{\sqrt{N}} \sum_j \sum_i \exp(ik\cdot r_j) s_j^- s_i^{-} s_i^{+} |0> - \frac{g\mu_B H}{\sqrt{2S}} \frac{1}{\sqrt{N}} \
  • #1
JohnGringo
1
0
Hey
Given an anisotropic hamiltonian
[itex]
\mathcal{H} = -\sum_{j,\rho} \left( J_\rho^z s_j^z s_{j+\rho}^z + \frac{J_\rho^{xy}}{2}\left( s_j^+ s_{j+\rho}^- + s_j^- s_{j+\rho}^+ \right)\right) - g\mu_B H\sum_j s_j^z
[/itex]
Here [itex] \rho [/itex] is a vector connecting the neighbouring sites.
How do I show that the state
[itex]
|k> = \frac{1}{\sqrt{2S}}s_{k}^{-}|0>
[/itex]
where
[itex]
s_{k}^-=\frac{1}{\sqrt{N}}\sum_j\exp(ik\cdot r_j)s_j^-
[/itex]
is an eigenstate of the hamiltonian?
So the plan is the use the Fourier transform some how, but I am kind of lost with this. What do I substitute where and why?

Thanks!
 
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  • #2
To show that the state |k> is an eigenstate of the Hamiltonian, we need to calculate the action of the Hamiltonian on the state and show that it is equal to an eigenvalue. We can do this by first writing out the Hamiltonian as a matrix in the basis of the spin operator states, then using the Fourier transform to express the state |k> in terms of these basis states. We then calculate the action of the Hamiltonian on the state |k> in terms of the matrix elements, and finally show that this is equal to an eigenvalue.Let's start by writing out the Hamiltonian in the basis of the spin operator states. For an N-site chain, the Hamiltonian matrix will be a NxN matrix with entries given by:H_{ij} = -J_z^z s_i^z s_j^z -J_{xy}^{xy} (s_i^{+}s_j^{-} + s_i^{-}s_j^{+}) -g\mu_B H\delta_{ij}s_i^zwhere J_z^z and J_{xy}^{xy} are the exchange parameters and H is the applied magnetic field.Now we can use the Fourier transform to express the state |k> in terms of the basis states. Recall that the Fourier transform is defined as:s_k^- = \frac{1}{\sqrt{N}} \sum_j \exp(ik\cdot r_j) s_j^-where r_j is the position vector of site j. Plugging this into the definition of |k>, we have:|k> = \frac{1}{\sqrt{2S}} \frac{1}{\sqrt{N}} \sum_j \exp(ik\cdot r_j) s_j^- |0>Now we can calculate the action of the Hamiltonian on the state |k>. Using the definition of the Hamiltonian matrix above, we have:H|k> = \frac{1}{\sqrt{2S}} \frac{1}{\sqrt{N}} \sum_j \sum_i \exp(ik\cdot r_j
 

1. What is a Heisenberg ferromagnet?

A Heisenberg ferromagnet is a type of magnetic material in which the magnetic moments of individual atoms align in the same direction, resulting in a macroscopic magnetization. This type of magnetism is described by the Heisenberg model, which takes into account the quantum mechanical behavior of electrons in a material.

2. How do spin waves arise in a Heisenberg ferromagnet?

Spin waves are collective excitations of the magnetic moments in a Heisenberg ferromagnet. They arise due to the interactions between neighboring magnetic moments, causing them to oscillate. These oscillations propagate through the material, similar to how sound waves propagate through a medium.

3. What is the significance of spin waves in Heisenberg ferromagnets?

Spin waves play an important role in the behavior and properties of Heisenberg ferromagnets. They can affect the magnetic ordering and stability of the material, as well as its magnetic properties such as susceptibility and magnetization. Spin waves also have practical applications in the field of spintronics, where they can be utilized for information storage and processing.

4. How are spin waves studied and measured in Heisenberg ferromagnets?

Spin waves can be studied and measured using various experimental techniques, such as neutron scattering, Brillouin light scattering, and ferromagnetic resonance. These methods allow researchers to analyze the energy and dispersion of spin waves, as well as their lifetimes and interactions with other excitations in the material.

5. Can spin waves be controlled and manipulated in Heisenberg ferromagnets?

Yes, spin waves can be manipulated and controlled in Heisenberg ferromagnets by applying external magnetic fields or using spin currents. This can influence the propagation and behavior of spin waves, opening up possibilities for potential applications in spintronics and magnonics.

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