Mean Field Theory

Approximation technique for systems where $J \neq 0$ — exact solution is intractable in 2D and unsolved in 3D. Core idea: replace the fluctuating neighbor field each spin experiences with its average value.

Derivation

Rewrite the Ising energy by grouping each spin $σ_{i}$ with the field from its $2 d$ neighbors:

E = - \sum_{i = 0}^{N} σ_{i} (H + J \sum_{⟨ j ⟩} σ_{j}) \approx - \sum_{i = 0}^{N} σ_{i} (H + 2 d J ⟨ σ_{j} ⟩^{'})

Approximation: replace $⟨ σ_{j} ⟩^{'}$ (average over nearest neighbors of $i$ ) with $⟨ σ_{i} ⟩ = M$ (average over the whole system):

E \approx - (H + 2 d J M) \sum_{i = 0}^{N} σ_{i}

This is identical to the non-interacting Ising model with an effective field $H_{eff} = H + 2 d J M$ . Using the single-spin result $M = \tanh (\hat{h})$ :

M = \tanh (\frac{H + 2 d J M}{k T})

Solutions

Condition	$H = 0$	$H \neq 0$
$2 d J / k T \leq 1$	one solution: $M = 0$	one solution: $M$ same sign as $H$
$2 d J / k T > 1$	three solutions: $M = 0$ (unstable, high energy) and $M = \pm m^{*}$ (both stable, equal energy)	three solutions: stable one is most magnetized in direction of $H$
$H$ large	one solution regardless of $T$ : $M$ set by $H$	—
At low $T$ , energy dominates → system takes the most strongly magnetized solution. At $H = 0$ below $T_{c}$ : two equally stable phases, spontaneous symmetry breaking.

Critical temperature

The transition from one to three solutions happens when the slope of $\tanh$ at $M = 0$ equals 1:

\frac{d}{d M} \tanh (\frac{2 d J M}{k T}) |_{M = 0} = \frac{2 d J}{k T} = 1 ⟹ T_{c} = \frac{2 d J}{k}

Phase diagram

$T < T_{c}$ , $H = 0$ : ferromagnetic — two phases $M > 0$ and $M < 0$ , discontinuous jump as $H$ crosses 0, $M \neq 0$ even as $H \to 0$
$T > T_{c}$ : paramagnetic — single phase, $M = 0$ when $H = 0$ , $M$ changes continuously with $H$
Critical point at $T_{c}$ : discontinuity vanishes, coexistence curve ends