1D transverse-field Ising model - what is the difference between its classical and quantum treatment?

[Duda Jarek]

The 1D transverse field Ising model

is usually solved in quantum way, but we can also solve it classically - parametrize angles of spins and use Boltzmann ensemble of sequences of spin angles:

Pr(σ)∝exp(−H(σ))  for  σ=((cos(αi),sin(αi)))i∈Z

getting Markov process of angles, which can be easily approximated with Maximal Entropy Random Walk, for example getting below

joint distributions for (αi, α_i+1) for various parameters (Section III here ) :

As intuition suggests, there is some thermal wobbling of spin directions: (anti)aligned for dominating J, in x axis for dominating h.

However, in quantum approaches there are only considered spins in four directions, so should we imagine that intermediate angles are obtained by superposition?

Should there be thermal wobbling of spin directions as in densities above?

What are the differences in interpretation and predictions between such looking natural classical treatment and the quantum one?

Edited April 22, 2021 by Duda Jarek