Ground State Degeneracy in ferromagnetic Heisenberg model

In summary, the ground state of the Heisenberg FM model is degenerate due to the spin-rotation invariance of the Hamiltonian, with a degeneracy factor of (2LS+1) determined by the total spin and number of lattice sites.
  • #1
sam12
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I am reading the book "Lecture notes on Electron Correlation and Magnetism" by Patrik Fazekas.

It says, "The ground state (of Heisenberg FM model) is not unique. We have just found that the system has the maximum value of the total spin Stot = LS. Sztot = LS state which is maximally polarized in the z-direction. However the Hamiltonian is spin-rotationally invariant , hence turning the total spin in another direction does not change the energy: the ground state must be (2LS+1)-fold degenerate."
where, L is the no. of lattice sites.

I don't understand why the ground state should be (2LS+1)-fold degenerate and not infinitely degenerate. Are we not considering a global rotation symmetry of the system? I understand it in the following way: All the spins in the system have the same quantization axis which is along z-direction. Now, if every spin is rotated by the same angle,it produces a new state but the energy of the system remains unchanged because it depends on the scalar product of spins in the Heisenberg Hamiltonian. Therefore, a global rotation chosen from any of the infinite no. of possible rotations should produce infinitely many possible states and leave the energy of the Hamiltonian invariant (equal to the ground state energy). "

Does the system has a single quantization axis(i.e. same quantization works for each spin)? or Do we need to consider a unique quantization axis for each spin?
What happens to the quantization axis/axes of the system/spins as we consider a different state which has been globally rotated by some angle?
If the quantization axis (z-axis) remains fixed, then the system no longer has the maximum value of Sz in the new state obtained after a global rotation. But, if the quantization axis rotates with the system, then, why are there only, (2LS+1) possible rotations?
In other words, how does L get into the degeneracy?
Thanks
 
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  • #2
for your question and interest in the book "Lecture notes on Electron Correlation and Magnetism" by Patrik Fazekas.

The ground state of the Heisenberg FM model is not unique because of the spin-rotation invariance of the Hamiltonian. This means that the energy of the system is not affected by rotating the total spin in any direction. Therefore, the ground state must be degenerate, meaning there are multiple possible states with the same energy.

To answer your question about the quantization axis, in the Heisenberg model, the quantization axis is the same for all spins in the system. This is because the interactions between spins are the same in all directions, so there is no preferred axis for the spins to align along.

When we consider a different state that has been globally rotated by some angle, the quantization axis remains the same for all spins. However, the orientation of the spins relative to the quantization axis will change. This means that the total spin of the system will also change, but the energy will remain the same.

The reason why there are only (2LS+1) possible rotations is because the total spin of the system is fixed at LS. This means that there are only (2LS+1) possible combinations of individual spin values that can add up to the fixed value of LS. This is where the L factor comes into play in the degeneracy.

I hope this helps clarify the concept of degeneracy in the ground state of the Heisenberg FM model. If you have any further questions, please let me know.
 

1. What is the ground state degeneracy in the ferromagnetic Heisenberg model?

The ground state degeneracy refers to the number of possible energy states that a system can have at its lowest energy level. In the ferromagnetic Heisenberg model, the ground state degeneracy depends on the number of spins in the system and the strength of the magnetic interactions between them.

2. How does ground state degeneracy affect the behavior of the ferromagnetic Heisenberg model?

The ground state degeneracy can affect the behavior of the ferromagnetic Heisenberg model by determining the stability of the system and its susceptibility to external perturbations. A higher ground state degeneracy can lead to a more stable system with a lower susceptibility to perturbations.

3. What factors can influence the ground state degeneracy in the ferromagnetic Heisenberg model?

The ground state degeneracy in the ferromagnetic Heisenberg model can be influenced by the number of spins, the strength of the magnetic interactions, and the geometry of the lattice structure. Other factors such as temperature and external magnetic fields can also play a role.

4. How is ground state degeneracy related to quantum mechanics in the ferromagnetic Heisenberg model?

The ground state degeneracy in the ferromagnetic Heisenberg model is a result of quantum mechanical principles, specifically the Pauli exclusion principle which states that two identical fermions cannot occupy the same quantum state. In this model, the spins of the particles behave as fermions, leading to a degeneracy in the ground state energy level.

5. Can ground state degeneracy be broken in the ferromagnetic Heisenberg model?

While the ground state degeneracy in the ferromagnetic Heisenberg model is a fundamental property of the system, it can be broken by introducing external perturbations such as magnetic fields or by changing the geometry of the lattice structure. This can lead to a shift in the energy levels and a decrease in the ground state degeneracy.

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