The Ising Model | Physics - Wyatt's Notes

The Ising model is the simplest model of a phase transition. On a lattice of $N$ sites, each site $i$ has a spin variable $s_i \in \{+1, -1\}$. The Hamiltonian is

$H = -J\sum_{\langle i,j\rangle} s_i s_j - h\sum_i s_i$

Where $J$ is the coupling constant, $\langle i,j\rangle$ denotes nearest-neighbour pairs, and $h$ is an external magnetic field.

• $J > 0$: ferromagnetic (spins prefer to align).
• $J < 0$: antiferromagnetic (spins prefer to anti-align).

Theorem 6.1. The 1D Ising model with $h = 0$ has no phase transition at any finite temperature.

Proof (Transfer matrix method). Consider a chain of $N$ spins with periodic boundary conditions ($s_{N+1} = s_1$). The partition function is:

$Z = \sum_{\{s_i\}} \prod_{i=1}^{N} e^{\beta J s_i s_{i+1}}$

Define the transfer matrix $\mathbf{T}$ with elements $T_{s_i, s_{i+1}} = e^{\beta J s_i s_{i+1}}$:

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

The partition function is $Z = \mathrm{Tr}(\mathbf{T}^N) = \lambda_+^N + \lambda_-^N$ where $\lambda_\pm$ are the eigenvalues of $\mathbf{T}$:

$\lambda_\pm = e^{\beta J} \pm e^{-\beta J}$

In the thermodynamic limit ($N \to \infty$), $Z \approx \lambda_+^N$ and the free energy per spin is:

$f = -k_BT \ln(e^{\beta J} + e^{-\beta J}) = -k_BT \ln(2\cosh\beta J)$

The magnetisation $m = -\partial f/\partial h|_{h=0} = 0$ for all $T > 0$Confirming no spontaneous magnetisation and hence no phase transition. $\blacksquare$

Theorem 6.2 (Mean field approximation). In mean field theory, each spin feels an effective field due to its neighbours. Replacing $s_j$ by its average $\langle s_j \rangle = m$ in the Hamiltonian:

$H_{\mathrm{MF} = -Jz\, m\sum_i s_i - h\sum_i s_i}$

Where $z$ is the coordination number (number of nearest neighbours). Each spin behaves as if in an effective field $h_{\mathrm{eff} = h + Jz\,m}$.

The self-consistency equation (mean field equation) is:

$m = \tanh\!\left(\frac{\beta(h + Jz\,m)}{k_B}\right) = \tanh\!\left(\frac{h + Jz\,m}{k_BT}\right)$

For $h = 0$: $m = \tanh(Jz\,m / k_BT)$.

Critical temperature. Expanding $\tanh x \approx x - x^3/3$ for small $x$:

$m = \frac{Jz\,m}{k_BT} - \frac{1}{3}\left(\frac{Jz\,m}{k_BT}\right)^3$

For $m \neq 0$Dividing by $m$:

$1 = \frac{Jz}{k_BT_c} - \frac{1}{3}\left(\frac{Jz}{k_BT_c}\right)^3$

At $T = T_c$: $T_c = Jz/k_B$.

Near the critical point, thermodynamic quantities follow power laws:

$m \sim (T_c - T)^{1/\beta}, \quad \chi \sim |T - T_c|^{-\gamma}, \quad C \sim |T - T_c|^{-\alpha}$

Mean field theory predicts:

$\beta = \frac{1}{2}, \quad \gamma = 1, \quad \alpha = 0\ \text{(jump discontinuity)}$

These are the classical critical exponents. They are independent of the spatial dimension $d$ and the lattice structure --- a deficiency of mean field theory. Exact results and renormalisation group calculations give dimension-dependent exponents that agree with experiment.

Problem. For the 2D Ising model on a square lattice ($z = 4$), find $T_c$ in mean field theory and compare with the exact result $k_BT_c / J = 2/\ln(1 + \sqrt{2}) \approx 2.269$.

Problem. Calculate the magnetic susceptibility $\chi = \partial m/\partial h|_{h=0}$ above $T_c$ in mean field theory.

• Confusing the sign convention: $J > 0$ is ferromagnetic, not antiferromagnetic.
• Mean field theory overestimates $T_c$ because it ignores fluctuations. The error grows in lower dimensions.
• The 1D Ising model has no phase transition, but the 2D model does (Onsager, 1944). Do not generalise the 1D result to higher dimensions.
• Critical exponents are universal (depend only on dimension and symmetry), not on lattice details.

flowchart TD
    A[6_The Ising Model] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

The Ising model is the simplest system showing how local interactions create global order. Each spin copies its neighbors’ tendency, and below the critical temperature this copying wins, creating spontaneous magnetization. Mean-field theory averages over neighbors, ignoring fluctuations that become important near the critical point. The 1D model has no phase transition because any domain wall costs only finite energy. In 2D, Onsager’s exact solution shows a phase transition exists. Critical exponents are universal because near the critical point, the correlation length diverges, making microscopic details irrelevant.

• Statistical Mechanics: The Ising model partition function is computed using the same statistical mechanics framework as the canonical ensemble.
• Phase Transitions: The Ising model exhibits a second-order phase transition with universal critical exponents described by Landau theory.
• The Laws of Thermodynamics: The Ising model free energy minimisation at equilibrium follows from the second law of thermodynamics.

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