Ising Model and Mean-Field Theory

The Ising model is the simplest model of interacting spins on a lattice. Each site $i$ has a spin variable $\sigma_i \in \{-1, +1\}$.

$\mathcal{H} = -J\sum_{\langle i,j \rangle}\sigma_i\sigma_j - h\sum_i \sigma_i$

Where $J > 0$ is the ferromagnetic coupling, $\langle i,j \rangle$ denotes nearest-neighbor pairs, and $h$ is the external magnetic field.

Partition function (in 1D with periodic boundary conditions, $N$ spins):

$Z = \sum_{\{\sigma\}} \exp\!\left(\beta J \sum_i \sigma_i \sigma_{i+1} + \beta h \sum_i \sigma_i\right)$

This can be evaluated using the transfer matrix method. Define:

$\mathbf{T} = \begin{pmatrix} e^{\beta J + \beta h} & e^{-\beta J} \\ e^{-\beta J} & e^{\beta J - \beta h} \end{pmatrix}$

Then $Z = \text{Tr}(\mathbf{T}^N) = \lambda_+^N + \lambda_-^N$ where $\lambda_\pm$ are the eigenvalues of $\mathbf{T}$.

In the thermodynamic limit ($N \to \infty$), $Z = \lambda_+^N$ where:

$\lambda_+ = e^{\beta J}\cosh(\beta h) + \sqrt{e^{2\beta J}\sinh^2(\beta h) + e^{-2\beta J}}$

Key result: The 1D Ising model has no phase transition at $T > 0$. The magnetization $m = \langle\sigma\rangle \to 0$ as $h \to 0$ for all finite $T$.

The mean-field (Weiss) approximation replaces each neighboring spin by its thermal average:

$\sigma_i\sigma_j \approx \sigma_i\langle\sigma_j\rangle + \langle\sigma_i\rangle\sigma_j - \langle\sigma_i\rangle\langle\sigma_j\rangle$

The effective Hamiltonian becomes:

$\mathcal{H}_{\text{MF} = -\sum_i \left(zJm + h\right)\sigma_i + \frac{1}{2}N zJ m^2}$

Where $z$ is the coordination number and $m = \langle\sigma\rangle$.

Each spin is independent, so:

$m = \tanh\!\left[\beta(zJm + h)\right]$

This is a self-consistency equation for $m$. For $h = 0$:

$m = \tanh(\beta zJm)$

Expanding for small $m$: $m \approx \beta zJ m - \frac{1}{3}(\beta zJ)^3 m^3$. Nonzero $m$ exists when:

$\beta zJ > 1 \implies T_c^{\text{MF} = \frac{zJ}{k_B}}$

Onsager”s exact solution for the square lattice gives:

$T_c = \frac{2J}{k_B \ln(1 + \sqrt{2})} \approx \frac{2.269J}{k_B}$

The spontaneous magnetization below $T_c$:

$m = \left[1 - \sinh^{-4}(2\beta_c J)\right]^{1/8}, \quad T < T_c$

The specific heat diverges logarithmically at $T_c$:

$C \sim -A\ln|T - T_c|$

Near the critical temperature, physical quantities follow power-law behaviour characterised by critical exponents:

$m \sim (T_c - T)^{\beta}, \quad T < T_c$ $\chi \sim |T - T_c|^{-\gamma}, \quad C \sim |T - T_c|^{-\alpha}$ $\xi \sim |T - T_c|^{-\nu}$

Where $m$ is magnetisation, $\chi$ is susceptibility, $C$ is specific heat, and $\xi$ is the correlation length.

Mean-field values: $\beta = 1/2$, $\gamma = 1$, $\alpha = 0$ (jump), $\nu = 1/2$.

2D Ising exact values: $\beta = 1/8$, $\gamma = 7/4$, $\alpha = 0$ (log), $\nu = 1$.

Mean-field theory is exact above the upper critical dimension ($d > 4$) but gives incorrect exponents for $d < 4$. The exponents depend only on dimensionality and symmetry — not on microscopic details — a property called universality.

Magnetic materials. The Ising model captures the essential physics of ferromagnetic phase transitions. The spontaneous magnetisation below $T_c$ corresponds to permanent magnetisation in ferromagnets like iron and nickel.

Binary alloys. Replacing spin up/down with atom types A/B, the Ising model describes order-disorder transitions in alloys (e.g., brass, CuZn). The coupling $J$ represents the energy preference for unlike neighbours.

Lattice gases. Mapping $\sigma_i = \pm 1$ to occupation numbers $n_i = (1 + \sigma_i)/2$ gives a model of fluid adsorption on surfaces, where the critical point corresponds to the liquid-gas critical point.

Neural networks. The Hopfield model of associative memory is formally equivalent to an Ising model with random couplings, where stored memories correspond to ground states.