Curie–Weiss model

Remarks

The energy depends on $\sigma$ only through the magnetization $m_N = \frac{1}{N} \sum_i \sigma_i$, since $H_N = -N (J m_N^2 / 2 + h m_N)$. This makes the free energy computable exactly as $N \to \infty$. The limit magnetization solves the equation $m = \tanh(\beta (J m + h))$. For $h = 0$ this equation has only the solution $m = 0$ when $\beta J \le 1$, and two further solutions $\pm m^* \ne 0$ when $\beta J > 1$. So the critical point is $\beta_c = 1/J$, and below the critical temperature the magnetization concentrates near $\pm m^*$. It is the model behind mean-field theory, and its critical exponents are the mean-field ones.