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.
Best Answer
As this is a list-like question, let me list a few things (without much discussion -- feel free to ask specific questions about individual points). Each item mentions what the Heisenberg model (HM) has as opposed to the Ising model (IM).
continuous symmetry vs. discrete symmetry
as a consequence: gapless excitations whenever the symmetry is broken (i.e. in all cases except the 1D antiferromagnet -- in that case, however, there are gapless modes due to the Lieb-Schultz-Mattis theorem)
as a consequence thereof: no spontaneous symmetry breaking in one and two dimensions at finite temperature (as opposed to the 2D HAFM), this is the Mermin-Wagner theorem
non-commuting terms: The eigenstates will typically not have a simple form (as opposed to the IM, which has commuting terms)
the IM Hamiltonian has integer eigenvalues (times $J$), while we cannot easily characterize the eigenvalues of the HM
Some caveats, however:
some of these properties hold because of continuous vs. discrete symmetry, rather than classical vs. quantum
some of the properties hold because of commuting vs. non-commuting rather than discrete vs. continuous symmetry
some of these properties only hold for (infinite) lattices, others already on the level of just a few spins