Phase Transitions & the Ising Model
Spontaneous magnetization, critical exponents, and a live 2D Monte Carlo.
The Ising model places spins $s_i = \pm 1$ on a lattice; energy
$$H = -J \sum_{\langle ij\rangle} s_i s_j - h \sum_i s_i.$$In 1D there's no finite-$T$ phase transition. In 2D, Onsager (1944) computed exactly
$$T_c = \frac{2J}{k_B \ln(1+\sqrt 2)} \approx 2.269\, J/k_B.$$Below $T_c$ the system spontaneously magnetizes (breaks the $\mathbb Z_2$ symmetry). Near $T_c$ observables show power-law behavior with critical exponents:
$$m \sim |t|^\beta, \quad \chi \sim |t|^{-\gamma}, \quad \xi \sim |t|^{-\nu},$$where $t = (T - T_c)/T_c$. 2D Ising values: $\beta = 1/8$, $\gamma = 7/4$, $\nu = 1$. Universality: critical exponents depend only on dimension and symmetry, not the microscopic details — explained by the renormalization group.