Quantum vs. Classical Ising model and the Bohr van Leeuwen Theorem

Question

The classical Ising model is often described as a simplistic model for ferromagnetism, and the Bohr-Van Leeuwen theorem is understood to preclude classical physics origins of magnetization in matter.

So if one tries to understand the underlying physics from which the Ising model emerges as a fair description of matter, the relevant degrees of freedom are the electronic (or atomic) moments that arise from quantum mechanical origin (spin and angular momentum number). So this means the "classical" Ising model is also quantum.

Therefor my question is, what make the classical Ising model classical? And how come considering a classical model sees to violate the Bohr-Van Leeuwen theorem?

Answer 1

Classical Ising model is about a system with discrete states. So this model is actually quantum. And that was said in the answer proposed by Yvan Velenik in his comment.

But I still want to focus on one nuance. To see the difference between classical and quantum Ising models, just look at their Hamiltonians. Hamiltonian of the classical model $$ \hat{H}_c = -\sum_{j,k} J_{j,k}\hat{\sigma}_j^z\hat{\sigma}_k^z $$ does not contain non-commuting operators while Hamiltonian of the quantum model $$ \hat{H}_q = -\sum_{j,k} J_{j,k}\hat{\sigma}_j^z\hat{\sigma}_k^z - h \sum_j \hat{\sigma}_j^x $$ does. In this respect, one model is classical, and the other is quantum.