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Spontaneous symmetry breaking. Ferromagnet. 
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#1
Feb1613, 10:31 AM

P: 316

I'm not sure what people meant about this. Heisenberg hamiltonian is ##O(3)## invariant.
[tex]H=J\sum_{\langle i,j \rangle} \vec{S}_i \cdot \vec{S}_j[/tex] ##\langle \rangle## denotes nearest neighbors. It has ##O(3)## symmetry. If I understand well ground state is infinitely degenerate. But system must choose some ground state. Now where is symmetry breaking here? 


#2
Feb1613, 10:42 AM

Mentor
P: 16,316

The solutions to the ground state do not have O(3) symmetry, do they? (The sum of solutions does, but an individual solution does not)



#3
Feb1613, 10:51 AM

P: 316

I don't understand what do you say. Could you be more explicit? What is spontaneous symmetry breaking?



#4
Feb1613, 11:48 AM

Sci Advisor
P: 8,515

Spontaneous symmetry breaking. Ferromagnet.
Roughly, it means that a ground state does not have the rotational symmetry. Rotating brings you from one ground state to another. But how about the superposition of ground states which is also a ground state? I found Eq 9 and the introduction of http://arxiv.org/abs/physics/0609177v1 helpful.
"The general idea behind spontaneous symmetry breaking is easily formulated: as a collection of quantum mechanical particles grows larger, the object as a whole becomes ever more unstable against small perturbations. In the end even an inﬁnitesimal perturbation is enough to cause the collective system to break the underlying symmetry of the Hamiltonian. The fact that the symmetry breaking can happen spontaneously is then signaled by a set of noncommuting limits: In the complete absence of perturbations even a macroscopic system should conform to the symmetry of the Hamiltonian. If on the other hand the system is allowed to grow to macroscopic size in the presence of even just an inﬁnitesimal perturbation, then it will be able to break the symmetry and end up in a classical state." 


#5
Feb1613, 11:53 AM

P: 320

To answer the original question: The breaking of this "macroscopic" symmetry resulting in a stable permanent magnetization occurs in practice, not in the abstract math model. Real materials have impurities and anisotropies. 


#6
Feb1613, 12:20 PM

P: 316

I'm interesting in symmetry breaking in classical physics. Why I can't say that with rotation of ground state I get same ground state. Also is there symmetry breaking for finite lattice?



#7
Feb1613, 12:25 PM

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#8
Feb1613, 01:59 PM

P: 316

Ok. But what is in that case SPONTANEOUSLY SYMMETRY BREAKING? States are not equivalent. Ok. Let in T=0 all spins are in direction of zaxis. And what now?



#9
Feb1613, 02:34 PM

P: 320

You have to admit that it is you who claimed that there was spontaneous symmetry breaking, so it's somewhat strange to ask us where it is. Example origins of symmetry breaking would be an external magnetic field in the zdirection or impurities with net magnetic moment in the zdirection.



#10
Feb1713, 03:17 AM

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P: 3,593

In the limit of an infinite crystal, e.g. the state where all spins are pointing in zdirection is orthogonal to the state were all spins point in x direction. Furthermore it requires changing the direction of an infinity of spins to move one state into the other. This is not possible in one Hilbert space. Hence the operation which rotates one ground state in the other cannot be implemented in one Hilbert space. So the O(3) symmetry has really ceased to exist on a quantum level.



#11
Feb1713, 03:52 AM

P: 316

Thanks you all for discussion. If I understand it you need infinite energy to flip infinite many spins. What if lattice is finite? For example we have 24x24x24 lattice sites which all pointing up in the z direction. I don't need infinite energy to flip them all. Right?
I have a problem to understand this problem in classical level. What will you say that the spontaneous symmetry breaking is? 


#12
Feb1713, 04:05 AM

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P: 3,593

So it's an idealisation. Maybe more physical is the characterisation in terms of long range order, as this is directly observable. I regard spontaneous symmetry breaking as mainly a quantum mechanical phenomenon. However on scholarpedia, there are two articles, one for symmetry breaking in classical and one for breaking in quantum systems: http://www.scholarpedia.org/article/...ssical_systems http://www.scholarpedia.org/article/...uantum_systems 


#13
Feb1713, 04:07 AM

Sci Advisor
P: 3,593

The point is that you cannot represent this rotation as a quantum mechanical operator. 


#14
Feb1713, 04:15 AM

P: 316

Sorry but what quantum mechanics operator if I talk about classical spins systems?



#15
Feb1713, 05:10 AM

Sci Advisor
P: 3,593

I didn't understand from your OP that you are interested specifically in classical systems.
The point is that, if you were living in an infinite ferromagnet, you would perceive the magnetization axis as singled out and you would not have any means to rotate it. So although your underlying hamiltonian is invariant under rotations, you do not have the means to implement these rotations. That's what is meant with spontaneous symmetry breaking. 


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