Scaling and the Renormalization-Group Idea

At a critical point the correlation length is infinite, so there is no characteristic scale between the lattice spacing and the size of the system. Fluctuations of every size are present, and a photograph of the spin configuration looks statistically the same whether taken at the scale of ten spins or a thousand. The renormalization group turns that self-similarity into a method. Coarsen the description by grouping neighboring spins into blocks and replacing each block by a single effective spin; the coarsened system is another spin model, with different couplings. Iterating the coarsening generates a trajectory in the space of couplings, and the critical behavior is read off from the structure of that trajectory near its fixed points.¹

Self-similarity and block spins

Away from $T_c$ the correlation length $\xi$ is finite, and coarse-graining past $\xi$ reaches a scale where the blocks are uncorrelated: the system looks disordered (high $T$) or uniformly ordered (low $T$). At $T_c$ the correlation length is infinite, and no amount of coarsening reaches an uncorrelated scale — the coarsened system is statistically identical to the original. Criticality is the scale-invariant point of the coarse-graining.

Kadanoff's block-spin construction realizes the coarsening concretely. Partition the lattice into blocks of $b^d$ sites ($b$ blocks on a side, in $d$ dimensions). Replace each block by one block spin, whose value is set by a rule such as the majority sign of the spins it contains. The block spins live on a new lattice with spacing $b$ times larger, and if their interactions can again be written in Ising form, the coarsening has produced a new Ising model with a rescaled coupling.

The renormalization-group transformation

Coarse-graining defines a map on the space of couplings. Let $\vec{K}=(K_1,K_2,\dots )$ collect all the couplings of the model — the nearest-neighbor coupling, and whatever further-neighbor or multi-spin couplings the coarsening generates. One block-spin step sends

the renormalization-group (RG) transformation at scale factor $b$. The transformation preserves the partition function, so it preserves the free energy up to the additive constant produced by integrating out the short-scale spins. Two facts make it powerful.

• The number of spins drops by $b^d$, so the singular part of the free energy density obeys $f_s(\vec{K})=b^{-d}{f}_s({\vec{K}}^{\prime })$.
• The lattice spacing grows by $b$, so the correlation length measured in lattice units shrinks, $\xi ({\vec{K}}^{\prime })=\xi (\vec{K})/b$.

The correlation-length relation already fixes where critical behavior can live. A fixed point ${\vec{K}}^{\ast }=R_b({\vec{K}}^{\ast })$ satisfies $\xi =\xi /b$, so its correlation length is either $0$ or $\infty$. Zero-correlation-length fixed points are trivial sinks (perfect order or perfect disorder). A fixed point with $\xi =\infty$ is a critical fixed point, and the physics of the transition is controlled by it.

Flows, fixed points, and stability

Iterating $R_b$ generates a trajectory through coupling space. Trajectories are pushed toward stable fixed points and away from unstable ones, and the pattern of attraction organizes the phase diagram.

The stability of a fixed point is decided by linearizing the transformation around it. Writing $\vec{K}={\vec{K}}^{\ast }+\delta \vec{K}$, one step acts as $\delta {\vec{K}}^{\prime }=M\delta \vec{K}$ with $M$ the derivative matrix of $R_b$. Diagonalizing $M$ gives eigenvalues written as powers of the scale factor,

with $y_i$ the RG eigenvalues. Each eigendirection is classified by the sign of $y_i$.

• Relevant ($y_i>0$): the deviation grows under coarsening, driving the system away from the fixed point. Relevant couplings must be tuned to zero to reach criticality.
• Irrelevant ($y_i<0$): the deviation shrinks and the trajectory returns to the fixed point. Irrelevant couplings do not affect the critical behavior.
• Marginal ($y_i=0$): the linear analysis is inconclusive and higher orders decide.

For the Ising universality class the critical fixed point has exactly two relevant directions, identified with the reduced temperature $t$ and the field $h$. Every other coupling — further-neighbor interactions, lattice anisotropies, the detailed shape of the potential — is irrelevant, so it flows away and leaves no trace in the exponents. This is the origin of universality: microscopically different systems that flow to the same fixed point share its two relevant eigenvalues and therefore its exponents.

Exponents from the eigenvalues

The scaling relations of the previous lesson follow from the two relevant eigenvalues $y_t$ and $y_h$. Under one RG step the relevant couplings scale as $t\to b^{y_t}t$ and $h\to b^{y_h}h$, while the correlation length divides by $b$. The correlation-length relation $\xi (t)=b\xi (b^{y_t}t)$ is solved by a power law,

The free-energy density relation $f_s(t,h)=b^{-d}{f}_s(b^{y_t}t,b^{y_h}h)$ is likewise solved by a homogeneous function, exactly the Widom scaling form assumed in the previous lesson, now derived. Matching powers gives every exponent in terms of $y_t$, $y_h$, and $d$:

Two numbers fix all six exponents, which is why only two are independent, and the scaling relations are algebraic identities among these expressions. Hyperscaling, $\nu d=2-\alpha$, is immediate from the first two: $\nu d=d/y_t=2-\alpha$. It rests on the free energy scaling as ${\xi }^{-d}$, which holds only below the upper critical dimension; above it a dangerous irrelevant coupling spoils the naive scaling and hyperscaling fails, matching the Landau-theory finding.

The one-dimensional Ising decimation

The one-dimensional Ising chain carries the entire scheme through in closed form. Write the zero-field partition function with coupling $K=\beta J$,

Coarse-grain by decimation: sum over every other spin, leaving a chain with half as many spins and twice the spacing ($b=2$). Summing out a spin $s_2$ between its neighbors $s_1$ and $s_3$,

This depends on $s_1,s_3$ only through their product, so it can be rewritten as a new nearest-neighbor weight $e^{g+K^{\prime }{s}_1{s}_3}$. Matching the two cases $s_1{s}_3=\pm 1$ gives the recursion

with an additive constant $g=\frac{1}{2}\ln \left(4\cosh 2K\right)$ that shifts the free energy but not the couplings. The recursion is the RG transformation for the chain, exact and complete.

The recursion has two fixed points. At $K^{\ast }=0$ (infinite temperature), $\cosh 0=1$ gives $K^{\prime }=0$; expanding for small $K$, $K^{\prime }\approx K^2$, so nearby couplings shrink and $K=0$ is stable. At $K^{\ast }=\infty$ (zero temperature), $\cosh 2K\approx e^{2K}/2$ gives $K^{\prime }\approx K-\frac{1}{2}\ln 2<K$, so the coupling decreases and $K=\infty$ is unstable. There is no fixed point at finite $K$. Every chain at positive temperature flows to the disordered fixed point $K=0$, which reproduces the exact result of the second lesson: the one-dimensional Ising model has no transition except at $T=0$. The only critical fixed point sits at zero temperature, where the correlation length is infinite and the model is scale-invariant.

The same machinery applied in two or more dimensions generates a nontrivial critical fixed point at finite $K$, whose relevant eigenvalue $y_t$ yields the correlation-length exponent $\nu$ and, through the scaling formulas, the rest of the exponents. Carrying that computation out requires approximate RG schemes — the Migdal-Kadanoff bond-moving rules, or Wilson's momentum-shell integration and the $ϵ$-expansion around four dimensions — which lie beyond this module. The conceptual content is complete here: coarse-graining flows in coupling space, a critical fixed point controls the transition, its relevant eigenvalues set the exponents, and its basin of attraction is the universality class.

Summary

• At $T_c$ the correlation length is infinite, so the system is scale-invariant; the renormalization group exploits this by coarse-graining (Kadanoff block spins) and tracking how the couplings change.
• The RG transformation ${\vec{K}}^{\prime }=R_b(\vec{K})$ preserves the free energy and shrinks the correlation length by $b$; its fixed points have $\xi =0$ (trivial sinks) or $\xi =\infty$ (critical fixed points that control transitions).
• Linearizing near a critical fixed point gives eigenvalues $b^{y_i}$: relevant ($y_i>0$) couplings grow and must be tuned to reach criticality, irrelevant ($y_i<0$) couplings shrink and drop out — the source of universality.
• The two relevant eigenvalues $y_t,y_h$ fix all exponents ($\nu =1/y_t$, $2-\alpha =d/y_t$, and so on), deriving the Widom scaling form and the scaling relations, with hyperscaling holding only below the upper critical dimension.
• The 1D Ising decimation $K^{\prime }=\frac{1}{2}\ln \cosh 2K$ has only the stable disordered fixed point $K=0$ and the unstable $K=\infty$; all finite couplings flow to disorder, reproducing the absence of a finite-temperature transition.