The free quantum field theories are classified by representation spaces of the irreducible unitary representations of the Poincare group, which serve as the corresponding 1-particle spaces. Their classical counterparts are the symplectic spaces defined on the coadjoint orbits of the Poincare group. In the massive case, this is for spin 0 the cotangent space $R^6$ of one classical spinless particle in $R^3$, and for spin $>0$ the space $R^6\times S^2$ of a classical spinning particle; classically, the spin can take any positive value, not only half-integral ones.

The quantum harmonic oscillator is the special case of a single mode of a bosonic free quantum field theory (typically a mode with fixed momentum, hence energy $E=\hbar\omega$, giving the frequency $\omega$); its $N$th excitation is the corresponding $N$-particle state. The fixed momentum modes themselves are obtained by a spectral analysis of the solutions of the wave equation of the corresponding 1-particle representation, hence of the Klein-Gordon equation for spin 0, the Maxwell equations for spin 1, etc..

For fermions, one doesn't get a harmonic oscillator for each mode but (because of the exclusion principle) a qubit, a quantum 2-state system. For spin 1/2, each fixed momentum modes is obtained by a spectral analysis of the solutions of the Dirac wave equation.

The Ising model is a lattice version of a nonrelativistic spinor field theory; it has at each lattice point a qubit, i.e., the fermionic version of the harmonic oscillator. Spin waves are the lattice analogues of the fixed momentum modes. There are many variants of the Ising model since you can put on each lattice point not just a spin degree of freedom but more complicated objects, and you can also decorate the edges (corresponding to interactions) in more complex ways. Finally, one can replace the lattice by more general graphs or related topological networks.