Critical exponents and scaling laws

Intuition [Beginner]

Critical exponents and scaling laws is numbers measuring power-law behavior near a phase transition. 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 critical exponents and scaling laws is formulated by a state space $\Omega$, an energy or action functional $H$ or $S$, and expectations computed from normalized weights. The prerequisites used here are 08.02.01, 08.04.02, 08.01.04. For a finite system,

$$Z=\sum _{\sigma \in \Omega }{e}^{-\beta H(\sigma )}$$

is the partition function ^{[Stanley §3]}. 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 (Rushbrooke scaling relation). For a finite statistical system with partition function $Z(\beta )={\sum }_{\sigma }{e}^{-\beta H(\sigma )}$, the mean energy is

$$⟨H⟩=-\frac{\partial }{\partial \beta }\log Z.$$

Proof. Differentiate the partition function:

$$\frac{\partial Z}{\partial \beta }=\sum _{\sigma }-H(\sigma )e^{-\beta H(\sigma )}.$$

Divide by $Z$:

$$\frac{\partial }{\partial \beta }\log Z=\frac{1}{Z}\frac{\partial Z}{\partial \beta }=-\frac{1}{Z}\sum _{\sigma }H(\sigma )e^{-\beta H(\sigma )}.$$

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

Bridge. The construction here builds toward 08.06.02 (conformal symmetry at criticality), 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 $\log Z$ 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 ^{[Stanley §3]}.

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. Critical exponents ($\alpha ,\beta ,\gamma ,\delta ,\nu ,\eta$) quantify the power-law singularities of thermodynamic quantities at a continuous phase transition: the specific heat diverges as $|T-T_c{|}^{-\alpha }$, the order parameter vanishes as $(T_c-T{)}^{\beta }$, the susceptibility diverges as $|T-T_c{|}^{-\gamma }$, and the correlation length diverges as $|T-T_c{|}^{-\nu }$. Scaling relations (Rushbrooke, Widom, Fisher, Josephson) reduce these six exponents to two independent ones, and the renormalisation group 08.04.01 explains why: the exponents are determined by the eigenvalues of the linearised RG map at the critical fixed point 08.04.02. Mean-field theory 08.02.01 predicts one set of values (e.g., $\beta =1/2$), the Onsager solution 08.03.01 gives another ($\beta =1/8$ in 2D), and universality is the statement that the exponents depend on dimension and symmetry alone, not on microscopic details.

Full proof set [Master]

Proposition. The second derivative of $\log Z$ is the variance of the energy.

From the Intermediate theorem, ${\partial }_{\beta }\log Z=-⟨H⟩$. A second differentiation gives

$$\frac{{\partial }^2}{\partial {\beta }^2}\log Z=⟨H^2⟩-⟨H{⟩}^2.$$

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 $Z[J]={\sum }_{\sigma }{e}^{-\beta H(\sigma )+J_1{O}_1(\sigma )+J_2{O}_2(\sigma )}$. Differentiating $\log Z[J]$ first in $J_1$ and then in $J_2$, then setting $J=0$, gives

$$⟨O_1{O}_2⟩-⟨O_1⟩⟨O_2⟩.$$

Thus $\log Z$ 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 08.02.01, 08.04.02, and 08.01.04 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}
}