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.
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.
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.
That is strictly not true!!!
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.
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.
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: