--- title: Scaling and the Renormalization-Group Idea module: Phase Transitions moduleNumber: 11 lessonNumber: 5 order: 1105 summary: > At a critical point fluctuations exist on every length scale, so the system looks the same after coarse-graining. The renormalization group makes this self-similarity a computation: group spins into blocks, integrate out the short scales, and track how the couplings change. The transformation has fixed points, and the flow near a critical fixed point separates relevant couplings that grow from irrelevant ones that shrink, which is why only dimension and symmetry survive to set the exponents. The one-dimensional Ising decimation carries the whole scheme through in closed form and reproduces the absence of a finite-temperature transition. topics: [Phase Transitions] sources: - book: Kardar (Statistical Physics of Fields) ref: "Ch. 4 — The Renormalization Group; §4.1–4.3, and Ch. 5 §5.1–5.2" - book: Pathria & Beale ref: "Ch. 14 — Phase Transitions: The Renormalization Group Approach; §14.1–14.4" - book: Wilson ref: "Nobel lecture, The Renormalization Group and Critical Phenomena (1982)" draft: false --- 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.[^wilson] [^wilson]: Wilson's Nobel lecture, , gives the physical picture; Kardar (Fields), §4.1–4.3, and Pathria & Beale, §14.1–14.4, develop the formalism. ## 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. $$ % caption: Kadanoff block-spin coarse-graining: each $3\times 3$ block of the fine lattice is replaced by one block spin (majority rule), producing a lattice with three times the spacing and a new effective coupling. \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % fine 6x6 lattice with 3x3 block boundaries \foreach \x in {0,...,5} \foreach \y in {0,...,5} { \filldraw[acc] (\x*0.42,\y*0.42) circle (1.3pt); } \draw[black,thick] (-0.21,-0.21) rectangle (2.31,2.31); \draw[black,thick] (0.63,-0.21)--(0.63,2.31); \draw[black,thick] (1.47,-0.21)--(1.47,2.31); \draw[black,thick] (-0.21,0.63)--(2.31,0.63); \draw[black,thick] (-0.21,1.47)--(2.31,1.47); \node[black,below] at (1.05,-0.35) {blocks of $3\times 3$}; \draw[->,black,very thick] (2.7,1.05)--(3.9,1.05); \node[black,above] at (3.3,1.05) {coarsen}; % coarse 2x2 lattice \foreach \x in {0,1} \foreach \y in {0,1} { \filldraw[acc] (4.5+\x*0.9,0.15+\y*0.9) circle (2.4pt); } \node[black,below] at (4.95,-0.35) {block spins}; \end{tikzpicture} $$ ## 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 $$ \vec K \;\longmapsto\; \vec K' = R_b(\vec K), $$ 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 $f_s(\vec K) = b^{-d} f_s(\vec K')$. - The lattice spacing grows by $b$, so the correlation length measured in lattice units shrinks, $\xi(\vec K') = \xi(\vec K)/b$. 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. $$ % caption: Renormalization-group trajectories in coupling space; the critical fixed point $C$ has one unstable (relevant) direction along the critical surface's normal and attracts along it, while the order sink $O$ and disorder sink $D$ collect the two phases. \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} % critical surface as a curve \draw[acc,thick] (0.6,0.4) .. controls (2.4,1.6) and (3.6,2.4) .. (5.6,3.4); \node[acc,above left] at (1.2,0.9) {critical surface}; \filldraw[black] (3.0,1.95) circle (2.6pt); \node[black,below right] at (3.05,1.9) {$C$}; % flows along the surface into C \draw[->,black,thick] (1.1,0.72) -- (1.9,1.3); \draw[->,black,thick] (5.0,3.05) -- (4.2,2.55); % relevant direction off the surface, away from C \filldraw[acc] (5.0,0.7) circle (2.4pt); \node[acc,below] at (5.0,0.6) {$D$}; \filldraw[acc] (1.2,3.3) circle (2.4pt); \node[acc,above] at (1.2,3.4) {$O$}; \draw[->,black,thick] (3.0,1.95) .. controls (3.8,1.6) and (4.4,1.1) .. (4.9,0.8); \draw[->,black,thick] (3.0,1.95) .. controls (2.4,2.4) and (1.8,2.9) .. (1.3,3.2); \node[black,right] at (4.1,1.35) {relevant}; \end{tikzpicture} $$ 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 $\delta \vec K' = M\,\delta\vec K$ with $M$ the derivative matrix of $R_b$. Diagonalizing $M$ gives eigenvalues written as powers of the scale factor, $$ \lambda_i = b^{y_i}, $$ 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. $$ % caption: Eigendirections at a critical fixed point; the relevant axis (eigenvalue exponent $y>0$) carries trajectories outward and must be tuned to zero to reach criticality, while irrelevant directions ($y<0$) draw trajectories back in along the critical surface. \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \filldraw[black] (0,0) circle (2.6pt); \node[black,below left] at (-0.1,-0.1) {critical point}; % irrelevant axis (horizontal), arrows inward \draw[acc,thick] (-3.2,0)--(3.2,0); \draw[->,acc,thick] (-3.0,0.0)--(-1.6,0.0); \draw[->,acc,thick] (3.0,0.0)--(1.6,0.0); \node[acc,below right] at (2.2,-0.1) {irrelevant}; \node[acc,above left] at (-2.0,0.1) {$y<0$}; % relevant axis (vertical), arrows outward \draw[black,thick] (0,-2.6)--(0,2.6); \draw[->,black,thick] (0,0.6)--(0,2.3); \draw[->,black,thick] (0,-0.6)--(0,-2.3); \node[black,right] at (0.1,2.1) {relevant}; \node[black,right] at (0.1,1.3) {$y>0$}; \end{tikzpicture} $$ 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, $$ \xi(t) \sim |t|^{-\nu}, \qquad \nu = \frac{1}{y_t}. $$ 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 $d$: $$ 2 - \alpha = \frac{d}{y_t}, \quad \beta = \frac{d - y_h}{y_t}, \quad \gamma = \frac{2 y_h - d}{y_t}, \quad \delta = \frac{y_h}{d - y_h}. $$ 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 $K = \beta J$, $$ Z = \sum_{\{s_i\}} \prod_i e^{K s_i s_{i+1}} . $$ 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 $s_3$, $$ \sum_{s_2 = \pm 1} e^{K s_2 (s_1 + s_3)} = 2\cosh\!\big(K(s_1 + s_3)\big). $$ This depends on $s_1, s_3$ only through their product, so it can be rewritten as a new nearest-neighbor weight $e^{g + K' s_1 s_3}$. Matching the two cases $s_1 s_3 = \pm 1$ gives the recursion $$ K' = \tfrac{1}{2}\ln\cosh(2K), $$ with an additive constant $g = \tfrac12\ln\!\big(4\cosh 2K\big)$ that shifts the free energy but not the couplings. The recursion is the RG transformation for the chain, exact and complete. $$ % caption: The one-dimensional Ising decimation recursion, with the renormalized coupling $K_1 = \tfrac12\ln\cosh 2K$ on the vertical axis, lies below the diagonal for all $K>0$, so every finite coupling flows toward $K=0$: no finite-temperature critical fixed point exists, and there is no transition. \begin{tikzpicture}[>=Latex,font=\footnotesize,scale=1.0] \definecolor{acc}{HTML}{4A6FA5} \draw[->,black] (0,0)--(3.4,0) node[right,black]{$K$}; \draw[->,black] (0,0)--(0,3.4) node[above,black]{$K_1$}; \draw[black] (0,0)--(3.2,3.2); \node[black,above left] at (3.1,3.1) {$K_1=K$}; \draw[acc,very thick] plot[domain=0:3.0,samples=60] (\x,{0.5*ln((exp(2*\x)+exp(-2*\x))/2)}); \node[acc,below right] at (2.5,{0.5*ln((exp(5.0)+exp(-5.0))/2)}) {$K_1=\frac{1}{2}\ln\cosh 2K$}; \filldraw[black] (0,0) circle (2.4pt); \node[black] at (1.15,2.75) {stable sink at $K=0$}; \draw[->,black,thick] (2.4,-0.35) -- (0.5,-0.35); \node[black,below] at (1.4,-0.4) {trajectory toward disorder}; \end{tikzpicture} $$ The recursion has two fixed points. At $K^\ast = 0$ (infinite temperature), $\cosh 0 = 1$ gives $K' = 0$; expanding for small $K$, $K' \approx K^2$, so nearby couplings shrink and $K = 0$ is **stable**. At $K^\ast = \infty$ (zero temperature), $\cosh 2K \approx e^{2K}/2$ gives $K' \approx K - \tfrac12\ln 2 < K$, 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 $\epsilon$-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 $\vec K' = R_b(\vec K)$ 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 $K' = \tfrac12\ln\cosh 2K$ 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.