Broken symmetry in ferromagnetism

In summary, the conversation discusses the concept of broken symmetry in ferromagnetism. P.W. Anderson argues that there is no broken symmetry because the ground state is an eigenstate of the spin rotation operator, and therefore does not have Goldstone's mode. However, spin waves are considered to be Goldstone's mode in ferromagnetic systems. The concept of a tower of states in spontaneous symmetry breaking is also brought up, with the idea that the ground state in ferromagnetism is degenerate and an eigenvector of the Hamiltonian. The connection is made to symmetry breaking in electroweak theory, but the notion of a tower of states is not seen in this case. The conversation also mentions articles by Rudolf Ha
  • #1
jean194
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Hello,

Today I found this paper and this one where P.W. Anderson says that there is no broken symmetry in ferromagnetism because the ground state is an eigenstate of the spin rotation operator. And so we don't have in this system Goldstone's mode for example.
But I thought spin waves were Goldstone's mode of ferromagnetic systems. So I'm a little confused, especially since in the second article Peierls and Kaplan do not seem to agree with Anderson.
 
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  • #2
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  • #3
Thanks, the pdf is very interesting.
Tell me if I'm wrong but from what I understand, the uniqueness of the ground state and the notion of tower of states is fundamental in spontaneous symmetry breaking. The case of ferromagnetism is special in the sense that the ground state is degenerate and is an eigenvector of the Hamiltonian. So in this sense the symmetry is not broken.

Now I'm trying to make the connection with symmetry breaking in electroweak theory; I can't see where the notion of tower of states appears in the latter case.
 
  • #4
This tower of states is rather a special topic of one of the authors.
The ground state of macroscopic systems is always highly degenerate in the thermodynamic limit and by its very definition is an eigenvector of the hamiltonian.
My favourite article on the subject is by Rudolf Haag:

https://link.springer.com/article/10.1007/BF02731446

In the case of ferromagnetism, it is also an eigenstate of one of the generators of the broken symmetry, which makes it somewhat special. As is explained in the article, this leads, for example, to a reduced number of Goldstone bosons.
 
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  • #5
Thanks I will read that.
 
  • #6
Here is another article (that has an author in common with the article above).
https://arxiv.org/abs/physics/0609177
Footnote 1`1: Notice that ferromagnetism is explicitly not included in this list. The ferromagnet has a large number of possible exact groundstates which are all precisely degenerate, and which all have a finite magnetization. The singling out of one of these eigenstates is in a sense more like classical symmetry breaking than like the quantum symmetry breaking discussed here. The quantum symmetry breaking causes a state which is not an eigenstate of the Hamiltonian to be realized, and thus goes much further than only singling out one particular eigenstate.
 
  • #7
I am not sure whether this statement about "not an eigenstate of the Hamiltonian" is correct. We are talking here about phases, which usually require the idealisation of systems of infinite extent. But in infinite systems, also broken symmetry states are eigenstates of a Hamiltonian.
 

Related to Broken symmetry in ferromagnetism

1. What is broken symmetry in ferromagnetism?

Broken symmetry in ferromagnetism refers to the phenomenon where the magnetic properties of a material are not symmetric or uniform throughout the material. This is due to the alignment of magnetic moments in one direction, creating a net magnetic field.

2. How does broken symmetry occur in ferromagnetic materials?

Broken symmetry in ferromagnetic materials occurs due to the interaction between the spin of electrons and the crystal structure of the material. When these interactions are strong enough, the spins of the electrons align in the same direction, creating a net magnetic field.

3. What are some examples of broken symmetry in ferromagnetism?

Some examples of broken symmetry in ferromagnetism include iron, cobalt, and nickel. These materials exhibit strong ferromagnetic properties due to the alignment of magnetic moments in one direction.

4. What are the potential applications of broken symmetry in ferromagnetism?

Broken symmetry in ferromagnetism has various applications in technology, including in the production of magnetic storage devices, such as hard drives, and in the development of magnetic sensors and actuators.

5. Can broken symmetry in ferromagnetism be controlled?

Yes, broken symmetry in ferromagnetism can be controlled through the application of an external magnetic field or by adjusting the temperature of the material. This allows for the manipulation of the magnetic properties of the material, making it useful for various applications.

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