The story of anti-ferromagntism

[wdlang]

i just read the obituary of Neel

http://www.nature.com/nature/journal/v409/n6818/full/409302a0.html

it mentions that landau predicted that only ferromagnetism can exist in nature

what is the story?

why did landau have this prediction?

 

[Naty1]

Perhaps whatever is mentioned stemmed from his work on quantum work on diamagnetism:

http://en.wikipedia.org/wiki/Lev_Landau

This statement is not technically correct:

...only ferromagnetism can exist in nature...

Ferromagnetism (including ferrimagnetism)[1] is the strongest type; it is the only type that creates forces strong enough to be felt, and is responsible for the common phenomena of magnetism encountered in everyday life. Other substances respond weakly to magnetic fields with two other types of magnetism, paramagnetism and diamagnetism, but the forces are so weak that they can only be detected by sensitive instruments in a laboratory.

http://en.wikipedia.org/wiki/Ferromagnetism

 

[OhYoungLions]

I think the issue is that allow ferromagnets break symmetry, their broken symmetry state is still an eigenstate of the quantum Heisenberg Hamiltonian. The Neel state on the other hand is not an eigenstate of the Hamiltonian. There was a long debate about whether such a state was actually possible. Not sure how Landau figured into this, but a lot of scientists were of the opinion that quantum fluctuations would "melt" the order, which is the case, of course, in low dimensional systems.

So, both sides were right, really.

 

[M Quack]

I think the issue is that allow ferromagnets break symmetry, their broken symmetry state is still an eigenstate of the quantum Heisenberg Hamiltonian. The Neel state on the other hand is not an eigenstate of the Hamiltonian. There was a long debate about whether such a state was actually possible. Not sure how Landau figured into this, but a lot of scientists were of the opinion that quantum fluctuations would "melt" the order, which is the case, of course, in low dimensional systems.

So, both sides were right, really.

There are a lot of things mixed up here, and the end result is a big mess.

• The Néel state exists and has been observed by many methods in thousands of materials (literally!). Good techniques to study details of antiferromagnetic order are neutron and x-ray scattering
• Unlike ferro- (FM) and ferrimagnets (FIM) , antiferromagnets (AFM) do not have a large macroscopic magnetization. I guess this is what Landau was referring to. In fact, the antiferromagnetic state has a smaller magnetization than the paramagnetic state of the same material. You can clearly observe this by measuring magnetization or susceptibility as function of temperature across the phase transition at the Néel temperature.
• FM, FIM and AFM all break several symmetries - e.g. time reversal symmetry. AFM breaks translation symmetry. FM, FI and often AFM break discrete rotational symmetries. E.g. when a cubic material becomes FM then macroscopic magnetization defines a special direction.
• All magnetic order, including AFM, FM and FIM can be broken by thermal fluctuations when the material is heated above the Néel- or Curie temperature, where a order-disorder phase transition takes place.
• All of these states are perfectly stable and are thus eigenstates of the Hamiltonian
• In some cases the FM, FIM or AFM order parameter can be suppressed by pressure, doping or magnetic fields, even at zero temperature (at least in theory, in practice nobody can ever reach zero temperature). In that case the order gets destroyed by quantum fluctuations, rather than "normal" thermal fluctuations. This is a field of current research, and in particular it is thought to be related to high-temperature superconductivity.
• There has been a long debate about 1D and 2D magnetism. The Mermin-Wagner theorem states that at finite temperature for dimensions <= 2 continuous symmetries cannot be broken.
  http://en.wikipedia.org/wiki/Mermin–Wagner_theorem
  The 2D Ising model, on the other hand, shows that for discrete symmetries this is possible.

The Neel state on the other hand is not an eigenstate of the Hamiltonian.

I believe this is wrong. Can you provide a reference?

There was a long debate about whether such a state was actually possible. Not sure how Landau figured into this, but a lot of scientists were of the opinion that quantum fluctuations would "melt" the order, which is the case, of course, in low dimensional systems.

As a general, sweeping statement this is clearly wrong, unless (maybe) you refer to discussions that took place in the 1930ies. Could you please quote a specific example where this was discussed?

 

[OhYoungLions]

There are a lot of things mixed up here, and the end result is a big mess.

Trust me, there's no mess here (anymore) :P
Keep in mind, the debates I was referring to occurred in the 1930s. Since then we've sorted out a lot of things.

• The Néel state exists and has been observed by many methods in thousands of materials (literally!). Good techniques to study details of antiferromagnetic order are neutron and x-ray scattering

You have to keep in mind that neutron diffraction really only emerged in the late 1940s / early 1950s, and I think was the first measurement to give clear indication of the Neel state. Before then it was not clear that antiferromagnets broke translational / time reversal / whatever symmetry. Neel showed such a broken symmetry state was a good mean field solution, but it wasn't clear whether it was stable. The alternative is a sort of resonating valence bond state, which in some cases gives a lower variational energy.

• Unlike ferro- (FM) and ferrimagnets (FIM) , antiferromagnets (AFM) do not have a large macroscopic magnetization. I guess this is what Landau was referring to. In fact, the antiferromagnetic state has a smaller magnetization than the paramagnetic state of the same material. You can clearly observe this by measuring magnetization or susceptibility as function of temperature across the phase transition at the Néel temperature.

True, but the bulk response such as magnetic susceptibility doesn't tell you anything about the microscopic details of the state in question. So it isn't useful in the debate in question.

• FM, FIM and AFM all break several symmetries - e.g. time reversal symmetry. AFM breaks translation symmetry. FM, FI and often AFM break discrete rotational symmetries. E.g. when a cubic material becomes FM then macroscopic magnetization defines a special direction.
• All magnetic order, including AFM, FM and FIM can be broken by thermal fluctuations when the material is heated above the Néel- or Curie temperature, where a order-disorder phase transition takes place.

Agreed.

• All of these states are perfectly stable and are thus eigenstates of the Hamiltonian

That is strictly not true!

• In some cases the FM, FIM or AFM order parameter can be suppressed by pressure, doping or magnetic fields, even at zero temperature (at least in theory, in practice nobody can ever reach zero temperature). In that case the order gets destroyed by quantum fluctuations, rather than "normal" thermal fluctuations. This is a field of current research, and in particular it is thought to be related to high-temperature superconductivity.

Here I believe you're referring to quantum phase transitions / quantum criticality. These issues are somewhat related to the discussion, but not really. That field is certainly more modern than the 1930s.

• There has been a long debate about 1D and 2D magnetism. The Mermin-Wagner theorem states that at finite temperature for dimensions <= 2 continuous symmetries cannot be broken.
  http://en.wikipedia.org/wiki/Mermin–Wagner_theorem
  The 2D Ising model, on the other hand, shows that for discrete symmetries this is possible.

Sure.

I believe this is wrong. Can you provide a reference?

You don't really need a reference - you can prove it for yourself. On two sites, the eigenstates of the Heisenberg Hamiltonian are the m_s = +1,0,-1 states in the triplet, and the m_s=0 state in the singlet. The Neel state is the broken symmetry state obtained from a linear combination of the m_s=0 triplet and singlet states. Write out the Hamiltonian, and you'll see it's not an eigenstate. Some thought the AFM state should be some generalization of the m_s=0 singlet on the two sites, which is "quantum disordered". This is the solution in 1D, for example.

As a general, sweeping statement this is clearly wrong, unless (maybe) you refer to discussions that took place in the 1930ies. Could you please quote a specific example where this was discussed?

Well, let me be more specific - quantum fluctuations may prevent ordering in low dimensional systems at least at T = 0, when the Hamiltonian has a continuous symmetry. The point here is that the Mermin-Wagner theorem states there is no order at finite temperature in classical systems in 1D or 2D because there are sort of "too many" low energy fluctuations that are thermally driven at finite temperature. But what about at T=0? Shouldn't ordering be possible when thermal fluctuations are absent? The answer, if you're dealing with a quantum system is "not necessarily". The key idea is there is a mapping between quantum systems at T=0 to classical systems at finite temperature, and in one higher dimension. The effective temperature in the classical analogue has to do with the details of the model, such as the value of the spin at each site. The quantum fluctuations present in the quantum model transform to thermal fluctuations in the classical mapping.

Okay, so in 1D, at T=0, for a quantum Heisenberg model, the mapping takes us to a 2D classical model at finite temperature. Mermin-Wagner says there is no long range order. So, even at T=0 there is no order in the 1D quantum model, which has nothing to do with thermal fluctuations, and everything to do with quantum fluctuations. For the 2D quantum model, you map onto a 3D classical model, which may be ordered, or may not, depending on your effective temperature.

I'm pretty sure this is discussed in Assa Auerbach's book.

Anyway, if you don't believe me, here's what Phil Anderson has to say:
source: http://www.aip.org/history/ohilist/23362_1.html

The whole question of why anti-ferromagnetism occurs had always been an interesting one for theoretical physicists. Bethe did the solution of the one dimensional anti-ferromagnetic chain way back in 1931. And the one dimensional anti-ferromagnetic chain with anti magnetic exchange integrals is not anti- ferromagnetic. It doesn't exhibit order—it doesn't make anti-ferromagnetic order. A lot of very famous physicists had played with this problem of is there anti-ferromagnetic order in principle?Ê It was discovered in practice after the second war as soon as neutron defraction became possible. The Oak Ridge group found anti-ferromagnetic order. So now the question was experimentally solved. But why was it possible? Why shouldn't it be possible? It's because to put it in theoretical terms (Landau, incidentally was among the unbelievers.) Landau, although he was one of the discoverers of the phenomenon of anti-ferromagnetism, the co-discoverer with Neel in the 1930s, he justified it by having ferromagnetic layers that then just happened to be opposite to each other. But he didn't really believe that a true three dimensional anti-ferro magnet could happen. Kramers had worked on it a great deal. So many of the great figures had worked on it and thought about it and found it puzzling. The reason is that the ground state of the anti-ferro magnet is clearly not the nominal low temperature state, which is not an eigenstate, (it's not the grand state). The ground state of the ideal anti-ferro magnet can be proved to be a singlet and they all knew the proof goes back to Bethe I believe. But therefore, it can not have a preferred orientation in space. And the question is why does it have a preferred orientation in space? And this is the core of the phenomenon that I named later, much later. I named it broken symmetry.

 

[M Quack]

Trust me, there's no mess here (anymore) :P
Keep in mind, the debates I was referring to occurred in the 1930s. Since then we've sorted out a lot of things.

Maybe you should have pointed that out. The understanding of magnetism as advanced a bit since then.

You have to keep in mind that neutron diffraction really only emerged in the late 1940s / early 1950s, and I think was the first measurement to give clear indication of the Neel state. Before then it was not clear that antiferromagnets broke translational / time reversal / whatever symmetry.

Yes, but today neutron scattering is a standard technique. There are several schools each year where you can learn the basic techniques.

Here I believe you're referring to quantum phase transitions / quantum criticality. These issues are somewhat related to the discussion, but not really. That field is certainly more modern than the 1930s.

Yes and yes. Again, I was not aware that this whole discussion is about the 1930s.

You don't really need a reference - you can prove it for yourself. On two sites, the eigenstates of the Heisenberg Hamiltonian are the m_s = +1,0,-1 states in the triplet, and the m_s=0 state in the singlet. The Neel state is the broken symmetry state obtained from a linear combination of the m_s=0 triplet and singlet states. Write out the Hamiltonian, and you'll see it's not an eigenstate. Some thought the AFM state should be some generalization of the m_s=0 singlet on the two sites, which is "quantum disordered". This is the solution in 1D, for example.

I will have to sit down with a piece of paper to check this.

Well, let me be more specific - quantum fluctuations may prevent ordering in low dimensional systems at least at T = 0, when the Hamiltonian has a continuous symmetry. The point here is that the Mermin-Wagner theorem states there is no order at finite temperature in classical systems in 1D or 2D because there are sort of "too many" low energy fluctuations that are thermally driven at finite temperature.

But what about at T=0? Shouldn't ordering be possible when thermal fluctuations are absent? The answer, if you're dealing with a quantum system is "not necessarily". The key idea is there is a mapping between quantum systems at T=0 to classical systems at finite temperature, and in one higher dimension. The effective temperature in the classical analogue has to do with the details of the model, such as the value of the spin at each site. The quantum fluctuations present in the quantum model transform to thermal fluctuations in the classical mapping.

Okay, so in 1D, at T=0, for a quantum Heisenberg model, the mapping takes us to a 2D classical model at finite temperature. Mermin-Wagner says there is no long range order. So, even at T=0 there is no order in the 1D quantum model, which has nothing to do with thermal fluctuations, and everything to do with quantum fluctuations. For the 2D quantum model, you map onto a 3D classical model, which may be ordered, or may not, depending on your effective temperature.

I'm pretty sure this is discussed in Assa Auerbach's book.

That book looks interesting. If I ever run out of thinks I *have* to read...

Anyway, if you don't believe me, here's what Phil Anderson has to say:
source: http://www.aip.org/history/ohilist/23362_1.html

Back to the 1930s, or 50s or whatever.

From a history of science point of view this is very interesting, and I guess Landau's statement was made in a historic context.