Both the Ising and the Heisenberg Models describe spin lattices with interaction on first neighbors. The Hamiltonian in each case is quite similar, despite the fact of treating de spins as Ising variables (1 or -1) or as quantum operators. In the Ising case it looks like
$$H_\textrm{Ising} = -~J \sum_{\langle i\ j\rangle} s^z_{i}\ s^z_{j}$$
where J is the coupling constant ($J>0$ for ferromagnet and $J<0$ for anti-ferromagnet), $\langle i\ j\rangle$ represents sum over first neighbors and $s^z$ is the spin in z direction. On the other hand, the Heisenberg model is
$$H_\textrm{Heisenberg} = -~J \sum_{\langle i\ j\rangle} \hat{S}_{i} \cdot \hat{S}_{j}$$
where the only difference lies in the spins being operators. (In both cases I took away the interaction with an external field for simplicity)
My Question is: What new phenomena does treating the spins as operators brings? I can see that $\hat{S}_{i}\ .\ \hat{S}_{j}$ takes account of the spin in every direction and not just z, but I can't see the physical implication of that.
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