Ising energy: $E=-h\sum _i{S}_i-J\sum _{⟨ij⟩}{S}_i{S}_j$ Contribution involving spin $j$:

Approximation: replace $S_k$ with mean value $m=⟨S_k⟩$:

where $z$ is the number of nearest neighbors. System reduces to non-interacting spins in effective field $h_{\text{mf}}$.
Single-spin Boltzmann distribution:

Self-consistency: $m$ predicted by $p(S_j)$ must equal $m$ used in $h_{\text{mf}}$:

Critical temperature ($h=0$)

Solve $m=\tanh (\beta Jzm)$ graphically. Three solutions emerge when slope of $\tanh$ at $m=0$ exceeds 1:

• $T>T_c$: only $m=0$ (paramagnetic)
• $T<T_c$: three solutions — $m=0$ unstable, $m=\pm |m|$ stable (ferromagnetic)

Critical behaviour near $T_c$

Let reduced temperature $t=(T-T_c)/T_c$, so $T/T_c=1+t$ and $T_c/T=1/(1+t)$.

Expand $\tanh$ for small $m$: $\tanh x\approx x-x^3/3$. From $m=\tanh (m\cdot T_c/T)$:

So $m=0$ or:

Susceptibility $\chi =\partial m/\partial h{|}_{h=0}$

Expand eq. (8) to first order in $h$, take $\partial /\partial h$:

Substituting $m=0$ ($T>T_c$) and $m^2=3|t|$ ($T<T_c$) for small $t$:

Both diverge as $t\to 0$ — response to applied field blows up at the critical point.

Correlation function and correlation length

Mean field neglects spin-spin correlations, effectively assuming $⟨S_i{S}_j⟩\approx ⟨S_i⟩⟨S_j⟩$. Define:

For large separation $R_{ij}$: $G_{ij}\sim e^{-R_{ij}/\xi }$ where $\xi$ is the correlation length — size of the largest correlated spin clusters.

Mean field predicts: $\xi \sim |t{|}^{-1/2}$ — diverges at $T_c$.

At $T_c$: clusters span the whole system, correlations at all scales matter, mean field breaks down because it ignores all higher-order correlations.

Critical exponents

Near $T_c$, observables follow power laws in $t$: