Ising Model

Hamiltonian

The Ising model places spins $s_i=\pm1$ on a lattice with Hamiltonian

$$H=-J\sum_{\langle i,j\rangle}s_i s_j-h\sum_i s_i.$$

For $J>0$, neighboring spins prefer alignment.

Physics

The model captures competition between energy and entropy. In dimensions two and higher it can exhibit a finite-temperature phase transition with spontaneous magnetization.