Can we identify configurations of the classical Heisenberg model with pure states of the spin-${1over2}$ quantum Heisenberg model?

Question

My background is mostly probability theory with some elementary quantum mechanics. Consider the following (very informal) "dictionary" between classical and quantum models for a particle in some interval:

Classical particle in $[0,1]$
Quantum particle in $[0,1]$

state space
$[0,1]times mathbb R$ for position/momentum, respectively
$L^2([0,1])$

pure state
delta distribution at $(x,p)in[0,1]timesmathbb R$
projection on some $fin L^2([0,1])$ which has a "sharp peak" at $x$ with some restrictions due to the uncertainty principle

The following "dictionary", however, doesn't seem nowhere near as telling as the above:

Classical Heisenberg model (on one site)
 spin-$S$ quantum Heisenberg model (on one site)

state space
unit sphere in $mathbb R^3$
 $L^2({-S,-S+1,dots,S})$ (???)

pure state
delta distribution on any unit vector of $mathbb R^3$
projection on some $fin L^2(-S,dots,S)$; this can be identified with a probability distribution on some set of cardinality $2S+1$ (???)

As you can see, when I tried to come up with a similar "dictionary" for the classical and quantum versions of the Heisenberg model, I quickly ran into several issues. I don't see at all how to identify pure quantum states with configurations of the classical model. I feel like this lack of understanding also makes the following questions more obscure than they would otherwise need to be:

The spin $Sin{1over2}mathbb N$ is an additional free parameter for the model in the quantum case. If I understand correctly, Lieb proved that the free energy of the quantum Heisenberg models converges to their classical counterparts as $Stoinfty$, but that doesn't seem to shed a lot of light on my question.
In the classical Heisenberg model, the continuous (rotational) symmetry is obvious, whereas it seems kind of surprising at first that a model whose pure states correspond to probability distributions over a finite set can have any continuous symmetry at all and even just looking at the Hamiltonian, ignoring the state space, one needs to know the right unitaries to conjugate with to obtain something like a rotation (see e.g. Lemma 3.3 of these notes)
In particular, Mermin-Wagner seems considerably more complicated (also conceptually, not just due to technicalities as far as I can tell; see e.g. this paper) than for the classical model where it seems to be a mere energy-entropy argument.

Could anyone help shed light on whether there is a more natural identification between the pure states of the classical and the quantum model or whether the naming is merely a historical accident/artefact?

octonion · Answer

The dictionary you want that corresponds with the classical-quantum one you have for a particle on $[0,1]$ is between a classical particle on sphere and a quantum rotor. The eigenfunctions of the quantum rotor correspond to spherical harmonics on a sphere and you can expand sharply peaked functions on a sphere in terms of these.
Despite the name, the classical Heisenberg model has more to do with the quantum side of this dictionary. In the classical Heisenberg model you are integrating over all field configurations in a partition function and this is like a path integral. You can treat the quantum mechanics of particle on an interval the same way by integrating over well defined trajectories in configuration space in a path integral.
There are a few different names for this system: Classical Heisenberg model, $O(N)$ non-linear sigma model, quantum rotor model. These are all related by rather 'trivial' connections. If you put the sigma model on a lattice in a path integral you might call it a classical Heisenberg model or an n-vector model. If you put the sigma model on a spatial lattice but still consider operators you might call it a quantum rotor model. Since the regularization and the interpretation of the Euclidean time direction isn't something essential these are all the same model.
The quantum Heisenberg model is something completely different since as you point out there is a finite Hilbert space at each lattice point. There are some connections you can make between the quantum and classical Heisenberg models in 2D which usually goes by the name "Haldane conjecture" but this is not on the same footing as the dictionary you describe in your question.

Can we identify configurations of the classical Heisenberg model with pure states of the spin-${1over2}$ quantum Heisenberg model?

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	Classical particle in $[0,1]$	Quantum particle in $[0,1]$
state space	$[0,1]times mathbb R$ for position/momentum, respectively	$L^2([0,1])$
pure state	delta distribution at $(x,p)in[0,1]timesmathbb R$	projection on some $fin L^2([0,1])$ which has a "sharp peak" at $x$ with some restrictions due to the uncertainty principle

	Classical Heisenberg model (on one site)	spin-$S$ quantum Heisenberg model (on one site)
state space	unit sphere in $mathbb R^3$	$L^2({-S,-S+1,dots,S})$ (???)
pure state	delta distribution on any unit vector of $mathbb R^3$	projection on some $fin L^2(-S,dots,S)$; this can be identified with a probability distribution on some set of cardinality $2S+1$ (???)