The Correlation Length
Not every spin flips independently. Not every spin makes its own decision. The Ising model has local interactions — neighbors talk — and that conversation propagates. A spin flip at site i influences its neighbors. Those neighbors influence their neighbors. Information spreads across the lattice like a rumor with infinite patience.
The correlation length ξ measures how far that rumor travels.
Define the correlation function G(r) = ⟨σ_0 σ_r⟩ − ⟨σ_0⟩⟨σ_r⟩. This is the covariance between two spins separated by distance r. If the spins are uncorrelated, G(r) = 0. If they're locked together, G(r) is large. In the ordered phase below T_c, ⟨σ⟩ ≠ 0 and the correlation function doesn't decay to zero — the system is globally ordered. But you can look at the connected correlation function above T_c, where ⟨σ⟩ = 0 and G(r) decays.
Above T_c, the decay is exponential: G(r) ∼ exp(−r/ξ). The parameter ξ is the correlation length. It tells you the characteristic distance over which spins remember each other's state. If ξ = 1, a spin at site i only cares about its nearest neighbors. Forget it. If ξ = 100, a spin reaches across a hundred lattice spacings to find a partner. The system has long-range memory.
The critical point is where ξ diverges.
As T → T_c from above, ξ ∼ (T − T_c)^{−ν}. For the 2D Ising model, ν = 1. The correlation length goes to infinity at the critical point. Spins are correlated across the entire system. The system becomes scale-invariant. Every length scale appears. Fluctuations exist at all sizes — small domains inside larger domains inside even larger domains. A fractal of up and down.
This divergence of ξ is the reason mean-field theory fails below the upper critical dimension. Mean-field theory assumes each spin feels an average field with negligible fluctuations. But when ξ is large, the number of correlated spins N_corr ∼ ξ^d is huge. Fluctuations in the local field scale as 1/√(N_corr). When ξ diverges, the central limit theorem should help — but the correlated spins are not independent. They are a single collective mode. The fluctuations don't average away. They dominate.
The correlation length is not just a number. It's the size of the critical region. Inside a volume of size ξ^d, all spins act together. They're not individual degrees of freedom — they're a single entity fluctuating as one. That's why you need renormalization group theory to handle the critical point. You coarse-grain volumes of size ξ, treat each as a unit, and watch the coupling constants flow. The correlation length sets the scale at which the system looks the same.
Experimentally, ξ is measurable. Neutron scattering measures the structure factor S(q), and near the critical point S(q) has a Lorentzian peak whose width is 1/ξ. X-ray diffraction on ferromagnets, light scattering on fluids near their critical point, concentration fluctuations in binary mixtures — all give you ξ. The correlation length is the most measurable critical quantity, and it connects theory to experiment without any fitting parameters once you know the microscopic lattice spacing.
The correlation length is the distance that one spin's opinion travels through the crowd. Near the critical point, opinions don't just spread — they become the crowd.