Quantum exchange symmetry can make the energy of a many-electron system depend on the relative orientation of spins even when no classical magnetic force is invoked. At low energies this behavior is often represented by an effective exchange Hamiltonian.

A common model for localized spins is the Heisenberg form $$H=-J\sum_{\langle i,j\rangle}\mathbf S_i\cdot\mathbf S_j.$$ With this sign convention, $J>0$ favors parallel neighboring spins and can support ferromagnetism, while $J<0$ favors antiparallel alignment and can support antiferromagnetism.

A simpler discrete model is the Ising Hamiltonian $$H=-J\sum_{\langle i,j\rangle}s_is_j-h\sum_i s_i,$$ where $s_i=\pm1$. For $J>0$, neighboring equal spins lower the interaction energy. At low temperature, energy favors large aligned regions; at high temperature, entropy favors disorder. Their competition produces a phase transition in sufficiently high-dimensional systems.

These effective models deliberately discard much microscopic electronic detail while preserving the collective variable that controls magnetic ordering. Their parameters are not universal constants: $J$ emerges from the material's quantum electronic structure.

Spin models provide a bridge between quantum mechanics and statistical physics. They show how a microscopic interaction can produce macroscopic spontaneous order, domains, correlations and critical behavior.