The Critical Exponent
Near a continuous phase transition, everything is power laws.
You approach the critical temperature Tc. The correlation length ξ — the typical size of fluctuations — diverges as ξ ∼ |t|^(-ν), where t = (T-Tc)/Tc is the reduced temperature and ν is a critical exponent. The order parameter M (magnetization, density difference) vanishes as M ∼ (-t)^(β) for T < Tc. The susceptibility χ diverges as χ ∼ |t|^(-γ). The specific heat C diverges as C ∼ |t|^(-α). The correlation function at criticality decays as a power law, G(r) ∼ r^(-(d-2+η)), with exponent η. And so on.
These are not fitting parameters. They are universal numbers, determined by the symmetry of the order parameter, the dimensionality of space, and the range of the interactions. They do not depend on the lattice structure, the strength of the coupling, the mass of the particles, or any other microscopic detail. This is one of the most surprising facts in all of physics: the microscopic world disappears, and only the broadest features of the system survive to determine the numbers you measure.
The mean-field theory gives specific values: ν = 1/2, β = 1/2, γ = 1, α = 0 (a jump), η = 0. These are correct above the upper critical dimension (d = 4 for short-range interactions) but wrong below it. The Ising model in two dimensions — solved by Onsager in 1944 — gives β = 1/8, ν = 1, γ = 7/4, α = 0 (a logarithmic divergence), η = 1/4. In three dimensions, no exact solution is known, but numerical methods and the conformal bootstrap have pinned the exponents to extraordinary precision: β ≈ 0.326, ν ≈ 0.630, γ ≈ 1.237, η ≈ 0.036. These numbers are the same for a magnet, for a fluid near its liquid-gas critical point, for a binary alloy near its demixing transition, and for any system in the Ising universality class. They are properties of the group Z₂ and the number 3, not of any particular material.
The scaling relations connect these exponents. Rushbrooke's law: α + 2β + γ = 2. Widom's law: γ = β(δ - 1). Fisher's law: γ = ν(2 - η). These are not independent. They follow from the scaling hypothesis, which says that near the critical point, the free energy is a generalized homogeneous function of its arguments. If you rescale lengths by a factor b, the free energy transforms in a predictable way, with critical exponents determined by the scaling dimensions of the relevant operators.
The renormalization group explains why. Near a fixed point of the RG flow, the only parameters that matter are the relevant ones. Everything else flows to zero. The irrelevant variables — the microscopic details — do not affect the critical behavior. The fixed point is an attractor in theory space, and basins of attraction are the universality classes. The critical exponents are properties of the fixed point, not of the system. This is why different systems share the same numbers. This is also why the critical exponents can be computed from a theory that has nothing to do with the actual system — you can compute Ising exponents from a lattice model and apply them to a fluid, because at the critical point the fluid looks like a lattice model.
The exponents are measured by experiment with increasing precision. The susceptibility of liquid xenon near its critical point, the magnetization of yttrium iron garnet, the specific heat of neon — all agree with the three-dimensional Ising values within experimental error. This is an astonishing agreement between theory and reality, mediated by nothing more abstract than the idea that systems forget their details at large scales.
There is a deeper reason for the power laws. At the critical point, the correlation length diverges, so there is no characteristic length scale. The system is scale-invariant. The only functions consistent with scale invariance are power laws. Logarithms appear when the scaling is marginal. Otherwise, it is power laws, and the critical exponents tell you the shape of the power.