Curie–Weiss model

Mean-field Ising ferromagnet with explicit variational free energy and self-consistency equation; exhibits spontaneous magnetization and a second-order phase transition.

Curie–Weiss model

Spins, Hamiltonian, and magnetization

The Curie–Weiss model is a fully connected mean-field analogue of the Ising model . Let $\sigma_i\in\{-1,+1\}$ for $i=1,\dots,N$ and define the empirical magnetization

m_N(\sigma)=\frac{1}{N}\sum_{i=1}^N \sigma_i .

With coupling $J>0$ (ferromagnetic) and external field $h\in\mathbb{R}$ , the Hamiltonian is

H_N(\sigma)=-\frac{J}{2N}\left(\sum_{i=1}^N \sigma_i\right)^2 - h\sum_{i=1}^N \sigma_i = -\frac{J N}{2}\,m_N(\sigma)^2 - h N\,m_N(\sigma).

This is a canonical example of mean-field approximation .

Partition function reduced to a sum over magnetization

In the canonical ensemble , the partition function is

Z_N(\beta,h)=\sum_{\sigma\in\{-1,+1\}^N} e^{-\beta H_N(\sigma)} =\sum_{\sigma} \exp\!\left(\beta N\left(\frac{J}{2}m_N(\sigma)^2+h\,m_N(\sigma)\right)\right).

Grouping configurations by magnetization values $m\in\{-1,-1+2/N,\dots,1\}$ yields

Z_N(\beta,h)=\sum_{m} \binom{N}{\frac{N(1+m)}{2}}\, \exp\!\left(\beta N\left(\frac{J}{2}m^2+h\,m\right)\right).

Variational free energy in the thermodynamic limit

Using Stirling’s approximation,

\frac{1}{N}\ln \binom{N}{\frac{N(1+m)}{2}} \to s(m), \qquad s(m)= -\frac{1+m}{2}\ln\frac{1+m}{2}-\frac{1-m}{2}\ln\frac{1-m}{2},

the pressure/free-energy per spin is obtained by Laplace’s method. Equivalently, the limiting free energy density can be written as

f(\beta,h)= -\frac{1}{\beta}\lim_{N\to\infty}\frac{1}{N}\ln Z_N(\beta,h) =\inf_{m\in[-1,1]} \Phi_{\beta,h}(m),

where the mean-field Landau functional is

\Phi_{\beta,h}(m)= -\frac{J}{2}m^2 - h m - \frac{1}{\beta}\,s(m).

This is a concrete instance of a Landau free energy functional .

Self-consistency equation

Stationary points satisfy $\partial_m \Phi_{\beta,h}(m)=0$ , which gives the Curie–Weiss mean-field equation

m=\tanh\!\bigl(\beta(Jm+h)\bigr).

Solutions describe possible equilibrium magnetizations; global minimizers of $\Phi_{\beta,h}$ determine the thermodynamic phase.

Phase transition and spontaneous magnetization

At zero field $h=0$ :

For $T>T_c$ (equivalently $\beta J<1$ ), the only solution is $m=0$ and it is the unique minimizer of $\Phi_{\beta,0}$ .
For $T<T_c$ (equivalently $\beta J>1$ ), there are two symmetric nonzero minimizers $\pm m_*(T)$ solving $m_*=\tanh(\beta J m_*).$ This is the appearance of spontaneous magnetization with magnetization as the order parameter .

The critical temperature is

T_c=\frac{J}{k_B}.

This provides a clean mean-field example of a phase transition .

Susceptibility and Curie–Weiss law

For $T>T_c$ and small $h$ , linearizing $m=\tanh(\beta(Jm+h))$ gives

(1-\beta J)m \approx \beta h, \qquad\Rightarrow\qquad \chi := \left.\frac{\partial m}{\partial h}\right|_{h=0} \approx \frac{\beta}{1-\beta J} =\frac{1}{k_B(T-T_c)}.

Thus the susceptibility diverges as $T\downarrow T_c$ (from above).

Large-deviation viewpoint

The reduction to a one-dimensional variational problem reflects concentration of $m_N$ and can be formulated as a large-deviation principle for magnetization (see LDP rate function for magnetization ), where $\Phi_{\beta,h}(m)$ plays the role of a rate-function-like effective potential (up to constants).