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Kenneth G. Wilson developed the renormalization group as a quantitative method for studying a critical phenomenon. Repeatedly changing the length scale reorganizes a model's parameters, exposing fixed points and the perturbations that grow or shrink near them. With Michael Ellis Fisher, he calculated critical exponents by expanding about four spatial dimensions. This explains why systems with different microscopic details can share critical behavior and gives controlled corrections to simple mean-field predictions.^[ref]

A physicist.^[ref]

A theoretical condensed matter physicist who, with Kenneth Geddes Wilson, calculated critical exponents near four dimensions using the renormalization group. Expanding in the difference between the dimension and four gave controlled corrections to simple critical behavior. The method connects fluctuations on many length scales to quantitative predictions for a phase transition.^[ref]

An O(N) model has a real $N$-component order parameter $\mathbf{\varphi }$ with symmetry under the orthogonal group $O(N)$. A standard Landau-Ginzburg theory free energy isThe gradient term penalizes spatial variation, $r$ tunes the transition, and the positive quartic term bounds the local free energy below. The case $N=1$ has a sign symmetry; for $N>1$, rotations mix the components.

In $2<d<4$, the standard short-range model's continuous critical transition is governed by a Wilson-Fisher fixed point for the corresponding $N$-component order parameter. Near $d=4$, the epsilon expansion treats the interaction perturbatively because its engineering dimension is $4-d$. ^[ref]

The dimension restriction matters. At and above the upper critical dimension $d=4$, the Gaussian description controls the usual critical powers, with logarithmic qualifications at $d=4$. Lower-dimensional continuous-symmetry models can have different transition behavior, including the Berezinskii–Kosterlitz–Thouless transition. Thus the fixed-point claim must be matched to dimension, symmetry and the range of interactions.

A renormalization group compares models at different observation lengths. A Wilsonian coarse-grained statistical Hamiltonian sums over short-distance fluctuations; a renormalization-group transformation rescales coordinates and fields for comparison. Kenneth Geddes Wilson's 1982 lecture treats fluctuations across successive length scales to explain critical phenomena.^[ref]

The experimental motivation is Andrews's carbon dioxide experiment. Dried carbon dioxide occupied a capillary tube above mercury; a screw compressed the transmitting water. A second air-filled tube gauged pressure. Flowing water controlled temperature, and a cathetometer read column positions for volume. The liquid meniscus vanished near the liquid-gas critical point. Residual air and imperfect air-gauge corrections limited accuracy.^[ref] These observations motivate critical-point theory without determining a correlation-length critical exponent. The renormalization group explains how critical measurements can become insensitive to many molecular details.

In 1972, Kenneth Geddes Wilson and Michael Ellis Fisher used an epsilon expansion about $d=4$ to compute nonclassical critical exponents. Their paper distinguishes a Gaussian fixed point from an interacting renormalization-group fixed point, tunes the thermal parameter, and extracts powers from a linearized recursion. Its Ising-like result is $\gamma =1+ϵ/6+O({ϵ}^2)$ for $d=4-ϵ$.^[ref] The worked model below isolates this mechanism with chosen coefficients; it is not a precision calculation for the three-dimensional Ising model.