What is meant by zero point spin fluctuations? I have hard time understanding the concept of "zero point spin fluctuations".
It appears in different papers on magnetism, e.g. here or here
They seem to draw a similarity between the zero point energy of the quantum harmonic oscillator and spins systems, which is very strange to me.
My attempt at understanding that concept is the following:
first this has nothing to do with QED vacuum or this sort of things, this is pure regular many-body non relativistic QM.
Now following the Kittel, in the part of chapter 12 about magnons, he writes :

The quantization of spin waves proceeds as for
photons and phonons. The energy of a mode $\omega_k$ of frequency k with $n_k$ magnons
is given by
$$\epsilon_k = (n_k + \frac{1}{2}) \hbar \omega_k$$

The equation is given without more details but I can see here the appearance of a zero point energy similar to the harmonic oscillator.
By looking at more "formal" derivations, using the Holstein-Primakoff transformation on the Hubbard model, e.g. here the result is a bit different: there is indeed a zero point energy like term for antiferromagnets, but not for ferromagnets.
But ok, let's say there are system for which the Kittel equation above is valid. My problem is that, I can understand the concept of zero point energy, or zero point "motion" for a quantum harmonic oscillator. One way of looking at it is that the position operator doesn't commute with the hamiltonian, so that eigenstates of $\hat x$ are not eigenstates of the system. The ground state is therefore a linear combinations of the $\hat x$ eigenstates. Another way is to say that even in the ground state the expectation value of $\hat x^2$ is different from zero and will depend on the stiffness $\omega$ of the potential. So there is indeed a connection between the uncertainty principle, or the commutators, and the idea of zero point "motion".
Now for spin systems, say an Heinsenberg model with $\hat H = -J \sum_{<ij>} \hat S_i \cdot \hat S_j$, well the Hamiltonian commutes with the total spin operator, so that I can use eigenstates of the total spin operator to label the energy eigenstates. The commutations relations only give me an uncertainty for the measurement on $x$ and $y$ directions of the total spin, but I can have an arbitrary low uncertainty on the $z$ axis.
This is a very different situation from the harmonic oscillator and I can't make sense of the appearance of any kinds of "spin fluctuations" at zero temperature.
I would appreciate any answer or reference that could help me understand this concept.
 A: If $z$ is the quantization axis, then the projections on $x$ and $y$ axes are undetermined - due to the uncertainty relations, following from non-zero commutators between $S_x, S_y$, and $S_z$.
This is the same kind of uncertainty that we have, e.g., in an oscillator placed in its ground state: the energy is well-defined, but position and momentum are not, since neither commutes with the energy. This is what we call zero-point fluctuations.
In case of an oscillator we could characterize these fluctuations by their probability distribution $|\psi_0(x)|^2$ or by their variance $\langle (x-\langle x\rangle)^2\rangle$. In the same way we could calculate the distribution and the variance, e.g., of the angle that the spin makes with the quantization axis.
A: I think the key is to realize what the difference is between ferromagnets and antiferromagnets.
In the case of the ferromagnet, as you pointed out, the total magnetization does commute with the Hamiltonian, since the Hamiltonian is rotationally symmetric. However, the zero-point fluctuations happen in antiferromagnets, not ferromagnets. In antiferromagnets, the relevant order parameter is not the total magnetization, but the staggered magnetization, which is defined as follows: Assume for simplicity that our antiferromagnet splits into an up sublattice and a down sublattice. Then the staggered magnetization is $\mathbf{M}_\mathrm{AFM} = (\sum_{i \in \uparrow} \mathbf{S}_i)-(\sum_{i \in \downarrow} \mathbf{S}_i)$. This operator does not commute with the Heisenberg Hamiltonian. This is then like the standard harmonic oscillator, where the operator of interest does not commute with the Hamiltonian.
Rephrased, the cartoon picture of a ferromagnet (where all the spins point up) is also the ground state of the FM Heisenberg model, while the cartoon picture of an antiferromagnet (where the spins point up, down, up, down, etc.) is not the ground state of the AFM Heisenberg model (actually it's not even an eigenstate of the Hamiltonian. It's an eigenstate of the staggered magnetization, but the staggered magnetization does not commute with the Hamiltonian!)
An example to clarify the point, and more closely link it to the papers you cited: Assume we have two spins which are coupled antiferromagnetically. Thinking classically, we would expect the ground state to be $|\uparrow \rangle |\downarrow \rangle$. However, the ground state is actually the singlet state given by $\frac{1}{\sqrt{2}}(|\uparrow \rangle |\downarrow \rangle- |\downarrow \rangle|\uparrow \rangle)$. Now imagine we are an experimentalist who wants to measure, say, $S_z$ on spin 1. From our classical intuition, we expect $\langle S^1_z\rangle = S$. However, calculating $\langle S^1_z\rangle$ in the actual ground state of the system yields $\langle S^1_z\rangle = 0$. This "melting" of the spin order is essentially what people would say is due to quantum fluctuations.
Edit to address your specific papers/comments: So far I've been talking about this subject assuming the spins are localized. This is fine in magnetic (Mott) insulators, and I believe it is also fine in the papers you cite, where there is a localized moment which is interacting with the rest of the system. Looking into the ferromagnetic case, they seem to refer specifically to ferromagnetic metals. In this case, as you suggest, the ground state is obviously a complicated many-body state, which can be approximated by a Slater determinant, but not exactly (due to correlation effects.) Essentially, the electron-electron repulsion and the Pauli principle together means the electrons try to avoid each other. I think one can think of this handwavingly as an effective antiferromagnetic exchange interaction (albeit smaller than the ferromagnetic interaction), which then reduces the effective spin moments (in other words, I agree with you.)
