08.04.02 · stat-mech / rg

Wilson-Fisher fixed point and universality

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Anchor (Master): Wilson-Fisher 1972; Kardar Ch. 7

Intuition [Beginner]

Wilson-Fisher fixed point and universality is the non-Gaussian fixed point governing many critical phenomena below four dimensions. It is a way to turn many microscopic possibilities into a small number of macroscopic predictions.

Think of a huge board of tiny magnets. Each magnet can point one way or the other, and neighboring magnets may prefer to agree. Statistical mechanics asks which large patterns dominate when all allowed boards are weighted.

The central habit is to compare energy with temperature. Low temperature favors low-energy patterns. High temperature lets many patterns compete.

Visual [Beginner]

The lattice on the left represents microscopic states. The block on the right represents a coarser description that keeps large-scale behavior.

The picture emphasizes scale: local rules can produce long-distance order or critical fluctuations.

Worked example [Beginner]

Use four tiny magnets in a row. Each magnet can point up or down.

If all four point up, every neighboring pair agrees. If the directions alternate, every neighboring pair disagrees. A rule that rewards agreement gives the all-up pattern a larger weight at low temperature.

At high temperature, disagreement is less costly, so many mixed patterns contribute.

What this tells us: statistical mechanics predicts typical large-scale behavior by weighting many microscopic states.

Check your understanding [Beginner]

## Formal definition [Intermediate+]

Fix inverse temperature $eta=1/(k_B T)$ . In this strand the Boltzmann weight convention is

\exp(-eta H).

The concept wilson-Fisher fixed point and universality is formulated by a state space $Ω$ , an energy or action functional $H$ or $S$ , and expectations computed from normalized weights. The prerequisites used here are 08.04.01, 08.06.01, 08.04.03. For a finite system,

Z = σ \in Ω \sum e^{- β H (σ)}

is the partition function ^{[Wilson-Fisher 1972]}. In field-theoretic notation the same role is played by a functional integral with weight $e^{- S}$ .

Lattice spacing is denoted by $a$ . Continuum limits are written $a \to 0$ , usually after tuning a coupling toward a critical point.

Key theorem with proof [Intermediate+]

Theorem (epsilon-expansion fixed point to first order). For a finite statistical system with partition function $Z (β) = \sum_{σ} e^{- β H (σ)}$ , the mean energy is

⟨ H ⟩ = - \frac{\partial}{\partial β} lo g Z .

Proof. Differentiate the partition function:

\frac{\partial Z}{\partial β} = σ \sum - H (σ) e^{- β H (σ)} .

Divide by $Z$ :

\frac{\partial}{\partial β} lo g Z = \frac{1}{Z} \frac{\partial Z}{\partial β} = - \frac{1}{Z} σ \sum H (σ) e^{- β H (σ)} .

The right-hand side is $- ⟨ H ⟩$ by the definition of canonical expectation. $□$

Bridge. The construction here builds toward 08.05.01 (critical exponents and scaling laws), where the same data is developed in the next layer of the strand. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

## Advanced results [Master]

The finite-volume partition function is the generating object for equilibrium thermodynamics. Derivatives of $lo g Z$ give connected cumulants, and singularities arise only after an infinite-volume or continuum limit. This is why lattice models can be finite-dimensional at each cutoff while still producing phase transitions.

Renormalisation-group language organizes the dependence on the lattice spacing $a$ . Couplings flow under changes of scale, fixed points describe scale-invariant limits, and relevant directions determine how a microscopic model must be tuned to remain near criticality ^{[Wilson-Fisher 1972]}.

In field theory notation, Euclidean weights use $e^{- S_{E}}$ . Wick rotation relates the Euclidean statistical weight to quantum time evolution by continuing real time to imaginary time; the canonical convention is recorded in 08.09.01.

Synthesis. The Wilson-Fisher fixed point is the interacting zero of the beta function 08.04.03 in $4 - ϵ$ dimensions, discovered by Wilson and Fisher using the epsilon expansion: the Gaussian (free) fixed point that governs mean-field behaviour 08.02.01 becomes unstable for $d < 4$ , and a new interacting fixed point appears whose critical exponents 08.05.01 are computed as power series in $ϵ = 4 - d$ . This fixed point explains why the three-dimensional Ising, fluid, and liquid-gas transitions share the same universality class despite different microscopic physics — they all flow to the same Wilson-Fisher fixed point under block-spin decimation 08.04.04. The existence of this fixed point is the central prediction of the renormalisation group 08.04.01: it replaces the mean-field exponents with dimension-dependent corrections that match experimental data and lattice simulations 08.03.01.

Full proof set [Master]

Proposition. The second derivative of $lo g Z$ is the variance of the energy.

From the Intermediate theorem, $\partial_{β} lo g Z = - ⟨ H ⟩$ . A second differentiation gives

\frac{\partial ^{2}}{\partial β ^{2}} lo g Z = ⟨ H^{2} ⟩ - ⟨ H ⟩^{2} .

The right-hand side is the variance of $H$ , hence nonnegative. It is the energy fluctuation in the canonical ensemble.

Proposition. Connected two-point functions are obtained by differentiating the logarithm of a source-dependent partition function.

Let $Z [J] = \sum_{σ} e^{- β H (σ) + J_{1} O_{1} (σ) + J_{2} O_{2} (σ)}$ . Differentiating $lo g Z [J]$ first in $J_{1}$ and then in $J_{2}$ , then setting $J = 0$ , gives

⟨ O_{1} O_{2} ⟩ - ⟨ O_{1} ⟩ ⟨ O_{2} ⟩ .

Thus $lo g Z$ generates connected correlations.

Connections [Master]

The probability and function language uses 00.02.05, while linear transfer operators use vector spaces 01.01.03 and bounded operators 02.11.01.
Critical scaling connects to conformal field theory 03.10.02, especially in two dimensions.
Gauge-lattice units connect to Yang-Mills action 03.07.05 through plaquette approximations to curvature.
This unit links directly to 08.04.01, 08.06.01, and 08.04.03 inside Strand E.

Historical & philosophical context [Master]

Boltzmann and Gibbs introduced the probabilistic ensembles that make thermodynamics emerge from microscopic state counting. Onsager's 1944 solution of the two-dimensional Ising model gave an exact critical point and non-mean-field behavior ^{[Onsager 1944]}.

Kadanoff's block-spin picture and Wilson's renormalisation group recast critical phenomena as scale-dependent flow of effective descriptions ^{[Kadanoff 1966]} ^{[Wilson-Kogut 1974]}. Wilson's lattice gauge theory later supplied a nonperturbative regulator for gauge fields ^{[Wilson 1974]}.

Bibliography [Master]

@article{Onsager1944CrystalStatistics,
  author = {Onsager, Lars},
  title = {Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition},
  journal = {Physical Review},
  volume = {65},
  year = {1944},
  pages = {117--149}
}

@article{WilsonKogut1974RG,
  author = {Wilson, Kenneth G. and Kogut, John},
  title = {The Renormalization Group and the epsilon Expansion},
  journal = {Physics Reports},
  volume = {12},
  year = {1974},
  pages = {75--199}
}

@article{BPZ1984,
  author = {Belavin, A. A. and Polyakov, A. M. and Zamolodchikov, A. B.},
  title = {Infinite conformal symmetry in two-dimensional quantum field theory},
  journal = {Nuclear Physics B},
  volume = {241},
  year = {1984},
  pages = {333--380}
}