Block-Spin Renormalization: Wilson's Real-Space RG, Fixed Points, and Relevant Operators

Intuition Beginner

Kenneth Wilson's great insight was to turn a problem into a procedure. The problem was that near a phase transition, a system has correlations at every length scale -- from the spacing between individual atoms all the way up to the macroscopic size of the sample. No finite calculation can keep track of all those scales at once.

Wilson's procedure: zoom out, repeatedly. Start with the microscopic description -- a lattice of spins, each coupled to its neighbours. Now blur your vision. Group neighbouring spins into small patches called blocks. Replace each block by a single effective spin that represents the collective behaviour of the spins inside it. You have just performed one step of the renormalization group (RG).

After this coarse-graining step, the system has fewer spins and a larger effective lattice spacing, but it still looks like a lattice of interacting spins -- just with different coupling constants. That is the key point: the form of the Hamiltonian is preserved, but the couplings change. Now zoom out again. And again. Each step produces a new set of couplings from the old one, tracing out a trajectory in the space of all possible coupling constants. This trajectory is the RG flow.

Most microscopic details wash away under repeated coarse-graining. Short-range information -- the exact shape of the interaction potential, the precise lattice structure -- gets averaged out and forgotten. But a few parameters survive. These survivors are the relevant operators: couplings that grow under the RG and determine the large-scale physics. Everything else -- the irrelevant operators -- shrinks and becomes invisible at large scales.

A fixed point of the RG is a special Hamiltonian whose couplings do not change under coarse-graining: the system looks identical at every magnification. Fixed points organise the entire flow. Near a phase transition, the system's couplings flow toward a critical fixed point, and the properties of that fixed point -- its relevant operators and their eigenvalues -- determine the universal critical behaviour.

This picture explains universality in a single stroke. Water and an Ising magnet have completely different microscopic physics. But when you coarse-grain both systems, their irrelevant details wash away, and their relevant operators turn out to have the same structure (one thermal and one magnetic, with the same eigenvalues). They flow to the same fixed point. That is why the liquid-gas critical point and the 3D Ising model have the same critical exponents.

Visual Beginner

Think of the space of all possible Hamiltonians as a high-dimensional landscape. Each point in this landscape is a different set of coupling constants. The RG transformation is a flow -- a current -- on this landscape.

Three special locations anchor the flow. The high-temperature fixed point sits at zero coupling (complete disorder); all systems above $T_c$ flow here. The low-temperature fixed point sits at infinite coupling (complete order); all systems well below $T_c$ flow here. Between them sits the critical fixed point, which is unstable: it attracts systems exactly at $T_c$ but repels anything slightly off criticality.

The critical surface is the set of all Hamiltonians that flow to the critical fixed point. Think of it as a drainage basin: every starting point on the surface ends up at the same fixed point, regardless of its microscopic details. This is universality. Starting points off the surface (not at $T_c$) flow away, toward either the high-T or low-T fixed point.

Worked example Beginner

Block-spin decimation of a 1D Ising chain. Consider a chain of spins $s_1,s_2,s_3,\dots$ with nearest-neighbour coupling $K=J/(k_{B}T)$. The partition function $Z$ is a sum over all spin configurations of $e^{K(s_1{s}_2+s_2{s}_3+s_3{s}_4+\cdots )}$.

Group the spins into pairs: block 1 contains $(s_1,s_2)$, block 2 contains $(s_3,s_4)$, and so on. Keep the odd spins and add up the contributions from the even ones. Adding the two possible values $s_2=+1$ and $s_2=-1$ gives:

$$e^{K(s_1+s_3)}+e^{-K(s_1+s_3)}=2\cosh (K(s_1+s_3)).$$

This must equal $C{e}^{K^{\prime }{s}_1{s}_3}$ for some new coupling $K^{\prime }$. When $s_1=s_3$: $C{e}^{K^{\prime }}=2\cosh (2K)$. When $s_1=-s_3$: $C{e}^{-K^{\prime }}=2$. Dividing: $e^{2K^{\prime }}=\cosh (2K)$, so $K^{\prime }=\frac{1}{2}\ln \cosh (2K)$.

Starting from $K=0.5$: $K^{\prime }=\frac{1}{2}\ln \cosh (1)\approx 0.217$. One more step: $K^{\prime \prime }=\frac{1}{2}\ln \cosh (0.434)\approx 0.045$. The coupling shrinks toward zero at every step. The system flows to the disordered fixed point $K^{\ast }=0$. There is no phase transition in 1D -- exactly as expected.

Check your understanding Beginner

Formal definition Intermediate+

Kadanoff block spins

Kadanoff (1966) introduced the block-spin picture as a heuristic argument for scaling. Given a lattice with $N$ sites and Hamiltonian $H(\{s_i\})$ parameterised by coupling constants $\mathbf{K}=(K_1,K_2,\dots )$:

1. Block. Group the $N$ sites into $N/b^d$ blocks of linear size $b$.
2. Assign. For each block $I$, define a block spin $S_I=f(\{s_i\in I\})$. Common choices: majority rule $S_I=\text{sgn}({\sum }_{i\in I}{s}_i)$, or decimation $S_I=s_{i_0}$ (keep one spin per block).
3. Integrate out. Sum over the original spins to obtain an effective Hamiltonian for the block spins:

$$e^{-H^{\prime }(\{S_I\})}=\sum _{\{s_i\}}\prod _{I}P(S_I|\{s_i\in I\})e^{-H(\{s_i\})},$$

where $P(S_I|\{s_i\in I\})$ is the weight function that assigns block-spin values (e.g., a delta function for deterministic rules).

The result is a new Hamiltonian $H^{\prime }$ on $N/b^d$ sites with renormalised couplings ${\mathbf{K}}^{\prime }$. The RG transformation is the map $\mathcal{R}:\mathbf{K}\to {\mathbf{K}}^{\prime }$.

Wilson's RG transformation and flow

Applying $\mathcal{R}$ repeatedly generates the RG flow:

$${\mathbf{K}}_0\stackrel{\mathcal{R}}{\to }{\mathbf{K}}_1\stackrel{\mathcal{R}}{\to }{\mathbf{K}}_2\stackrel{\mathcal{R}}{\to }\cdots$$

The flow takes place in the infinite-dimensional space of coupling constants (in principle, the RG generates all possible couplings, including multi-spin interactions, even if the original Hamiltonian had only nearest-neighbour terms). In practice, one truncates to a finite set of couplings.

A fixed point ${\mathbf{K}}^{\ast }$ satisfies $\mathcal{R}({\mathbf{K}}^{\ast })={\mathbf{K}}^{\ast }$. At a fixed point, the Hamiltonian is invariant under coarse-graining: it looks the same at every length scale. This is scale invariance, the hallmark of a critical theory.

Linearised RG and operator classification

Near a fixed point, write $\mathbf{K}={\mathbf{K}}^{\ast }+\delta \mathbf{K}$ and decompose into scaling fields $g_i$ that diagonalise the linearised RG:

$$g_i^{\prime }=b^{y_i}{g}_i+\text{higher order}.$$

The $y_i$ are the RG eigenvalues (or scaling dimensions). The operators $O_i$ conjugate to the $g_i$ are classified by the sign of $y_i$:

• Relevant ($y_i>0$): $g_i$ grows under the RG. These operators destabilise the fixed point and determine the phase diagram. For the Ising universality class, there are exactly two: the thermal operator $O_t$ (conjugate to $t=T-T_c$) with eigenvalue $y_t$, and the magnetic operator $O_h$ (conjugate to $h$) with eigenvalue $y_h$.
• Irrelevant ($y_i<0$): $g_i$ shrinks and becomes negligible at large scales. All couplings beyond the first few are typically irrelevant.
• Marginal ($y_i=0$): $g_i$ is unchanged to linear order. The fate of a marginal operator is decided by higher-order terms: it can be marginally relevant (grows logarithmically), marginally irrelevant (decays logarithmically), or exactly marginal (truly unchanged).

The Gaussian fixed point

The simplest fixed point is the Gaussian fixed point, corresponding to a free (non-interacting) field theory. For the Landau-Ginzburg Hamiltonian, the Gaussian fixed point has all non-Gaussian couplings set to zero. Its eigenvalues are:

$$y_t=2,y_h=1+d/2,y_u=4-d,$$

where $y_u$ is the eigenvalue of the quartic coupling $u$. For $d>4$: $y_u<0$ (irrelevant), so the quartic interaction washes out and the Gaussian fixed point is stable -- mean-field theory is correct. For $d<4$: $y_u>0$ (relevant), so the quartic interaction grows and destabilises the Gaussian fixed point. A new fixed point -- the Wilson-Fisher fixed point -- takes over.

The Wilson-Fisher fixed point

For $d<4$, the quartic coupling is relevant at the Gaussian fixed point, but it does not flow to infinity. Instead, it is stabilised by the non-linear feedback of the RG transformation, producing a new fixed point at $u^{\ast }>0$. This is the Wilson-Fisher fixed point, discovered by Wilson and Fisher (1972) using the epsilon expansion (setting $d=4-\epsilon$ and expanding in small $\epsilon$).

At the Wilson-Fisher fixed point, the relevant eigenvalues are shifted from their Gaussian values:

$$y_t=2-\frac{n+2}{n+8}\epsilon +O({\epsilon }^2),y_h=1+\frac{d}{2}-\frac{n+2}{4(n+8)}\epsilon +O({\epsilon }^2),$$

where $n$ is the number of components of the order parameter ($n=1$ for Ising, $n=2$ for XY, $n=3$ for Heisenberg). The critical exponents computed from these eigenvalues agree with experimental and numerical results to within a few percent at $d=3$ ($\epsilon =1$), a remarkable achievement for what is formally an expansion around $d=4$.

The critical surface and universality

The critical surface is the set of all Hamiltonians that flow to the critical fixed point under the RG. It is a surface of codimension equal to the number of relevant operators. For the Ising universality class, the critical surface has codimension 2: you must tune two parameters ($T=T_c$ and $h=0$) to reach it.

Universality from the RG. Consider two systems with different microscopic Hamiltonians $H_A$ and $H_B$. If they lie on the same critical surface (i.e., both have $T=T_c$ and $h=0$), they both flow to the same fixed point $H^{\ast }$ under the RG. Their large-scale critical behaviour is therefore identical, regardless of microscopic differences. The microscopic details are encoded in the irrelevant operators, which wash out. Only the relevant operators -- which are the same for all systems with the same symmetry and dimensionality -- survive to determine the critical exponents.

The basin of attraction of a fixed point is the set of all Hamiltonians that flow to it. Two systems are in the same universality class if their Hamiltonians lie in the same basin of attraction. The basin of attraction is the RG manifestation of universality.

Key derivation: Linearised RG and critical exponents Intermediate+

Proposition. The critical exponents are determined by the RG eigenvalues at the critical fixed point: $\nu =1/y_t$, $\eta =2+d-2y_h$, and all other exponents follow from scaling relations.

Proof. At the critical fixed point, the correlation length is infinite. Moving away from the fixed point by turning on the thermal scaling field $g_t\propto T-T_c$, the correlation length transforms under the RG as ${\xi }^{\prime }=\xi /b$. After $n$ RG steps, $g_t$ has grown to $g_t^{(n)}=b^{n{y}_t}{g}_t$ while ${\xi }^{(n)}=\xi /b^n$. The RG is stopped when $g_t^{(n)}\sim 1$ (the system has left the critical regime), giving $b^n\sim g_t^{-1/y_t}$. At this point ${\xi }^{(n)}\sim 1$, so $\xi \sim b^n\sim g_t^{-1/y_t}$. Since $g_t\propto |T-T_c|$:

$$\xi \sim |T-T_c{|}^{-1/y_t},\nu =\frac{1}{y_t}.$$

For the anomalous dimension $\eta$: the spin-spin correlation function at the fixed point scales as $G(r)\sim r^{-(d-2+\eta )}$. Under the RG with scale factor $b$: $G^{\prime }(r^{\prime })=b^{-2y_h}G(r)$ where $r^{\prime }=r/b$. At the fixed point, $G^{\prime }=G$, so $G(r)=b^{-2y_h}G(r/b)$, giving $G(r)\sim r^{-2y_h}$. Comparing: $d-2+\eta =2y_h$, so $\eta =2+d-2y_h$.

The remaining exponents follow from the scaling relations:

$$\alpha =2-d\nu =2-\frac{d}{y_t},\beta =\frac{d-y_h}{2y_t},\gamma =\frac{2y_h-d}{y_t},\delta =\frac{y_h}{d-y_h}.$$

$\square$

Bridge. This result connects the abstract RG machinery to the experimentally measurable critical exponents. The Gaussian fixed point gives $y_t=2$, $y_h=1+d/2$, reproducing mean-field exponents. The Wilson-Fisher fixed point shifts these eigenvalues, producing the corrected exponents for $d<4$. The Landau theory exponents from 11.06.03 are recovered exactly at the Gaussian fixed point. The transfer matrix methods of 11.06.05 provide independent verification for exactly solvable models.

Exercises Intermediate+

Lean formalization Intermediate+

The RG as a semigroup of transformations on a space of coupling constants is a purely mathematical object. Formalising it requires: (1) a type of coupling-constant vectors with a transformation map $\mathcal{R}$; (2) a fixed-point condition $\mathcal{R}({\mathbf{K}}^{\ast })={\mathbf{K}}^{\ast }$; (3) a linearisation giving eigenvalues $y_i$ with a classification into relevant ($y_i>0$), irrelevant ($y_i<0$), and marginal ($y_i=0$); (4) the connection between eigenvalues and critical exponents $\nu =1/y_t$, $\eta =2+d-2y_h$; (5) the perturbative epsilon expansion as an asymptotic series near $d=4$. The linear algebra and dynamical-systems theory are within Mathlib's scope, but the physical infrastructure (Hamiltonians, critical exponents, the epsilon expansion) requires domain-specific construction.

Advanced results Master

Wilson-Fisher epsilon expansion: sketch of derivation

The epsilon expansion is Wilson's systematic method for computing critical exponents near the upper critical dimension $d_c=4$. Set $d=4-\epsilon$ and expand in powers of $\epsilon$.

Start from the Landau-Ginzburg Hamiltonian for an $n$-component field $\mathbf{\varphi }$:

$$H[\mathbf{\varphi }]=\int d^{d}x\left[\frac{1}{2}(\nabla \mathbf{\varphi }{)}^2+\frac{r}{2}|\mathbf{\varphi }{|}^2+\frac{u}{4!}|\mathbf{\varphi }{|}^4\right].$$

The RG transformation proceeds in two steps:

Step 1: Momentum shell integration. Split the field into slow modes ${\mathbf{\varphi }}_{<}$ with $|\mathbf{q}|<\Lambda /b$ and fast modes ${\mathbf{\varphi }}_{>}$ with $\Lambda /b<|\mathbf{q}|<\Lambda$. Integrate out the fast modes:

$$e^{-H^{\prime }[{\mathbf{\varphi }}_{<}]}=\int \mathcal{D}{\mathbf{\varphi }}_{>}{e}^{-H[{\mathbf{\varphi }}_{<}+{\mathbf{\varphi }}_{>}]}.$$

The result is a new Hamiltonian $H^{\prime }$ for the slow modes with renormalised couplings. To one-loop order, the recursion relations are:

$$r^{\prime }=b^2\left[r+\frac{n+2}{6}u\frac{K_d{\Lambda }^d}{r+{\Lambda }^2}\right],$$ $$u^{\prime }=b^{4-d}\left[u-\frac{n+8}{36}{u}^2\frac{K_d{\Lambda }^d}{(r+{\Lambda }^2{)}^2}\right],$$

where $K_d=S_d/(2\pi {)}^d$ is a geometric factor and $S_d$ is the surface area of the $d$-dimensional unit sphere.

Step 2: Rescaling. Rescale lengths $\mathbf{x}\to \mathbf{x}/b$ and fields $\mathbf{\varphi }\to \zeta \mathbf{\varphi }$ (with $\zeta =b^{1-d/2}$ to preserve the kinetic term) to restore the cutoff $\Lambda$.

The fixed point of these recursion relations gives the Wilson-Fisher fixed point. For $d=4-\epsilon$:

$$u^{\ast }=\frac{6}{n+8}\frac{\epsilon }{K_4}+O({\epsilon }^2),r^{\ast }=-\frac{n+2}{2(n+8)}\epsilon +O({\epsilon }^2).$$

The thermal eigenvalue at the Wilson-Fisher fixed point is:

$$y_t=2-\frac{n+2}{n+8}\epsilon +O({\epsilon }^2),$$

giving $\nu =1/y_t=1/2+(n+2)\epsilon /[4(n+8)]+O({\epsilon }^2)$.

Calculation of $\eta$ to $O({\epsilon }^2)$

The anomalous dimension $\eta$ is the most subtle exponent. It arises from the field rescaling factor $\zeta =b^{(d-2+\eta )/2}$, which receives its first correction at two loops. The calculation requires evaluating the self-energy diagram with two quartic vertices:

$$\eta =\frac{(n+2)}{2(n+8{)}^2}{\epsilon }^2+O({\epsilon }^3).$$

For $n=1$ (Ising): $\eta ={\epsilon }^2/54+O({\epsilon }^3)$. At $\epsilon =1$: $\eta \approx 0.019$, compared to the exact 3D Ising value $\eta \approx 0.036$.

Higher-order calculations (through $O({\epsilon }^5)$) give $\eta =0.0340$ at $\epsilon =1$, in excellent agreement with the numerical value. The epsilon expansion is an asymptotic series (its radius of convergence is zero), but Borel resummation extracts accurate values even at $\epsilon =1$.

Crossover phenomena

When a system has two or more competing fixed points, the RG flow can pass near one before being attracted to another. This produces crossover phenomena: the critical behaviour changes from one universality class to another as a physical parameter (dimensionality, anisotropy, or range of interaction) is varied.

The crossover exponent $\varphi$ characterises this change. For a system with a marginal operator that becomes marginally relevant (e.g., the dipolar interaction in a uniaxial ferromagnet), the crossover length scale is ${\xi }_{\times }\sim |g{|}^{-\varphi }$ where $g$ is the coupling strength and $\varphi =y_u/y_t$ is the ratio of the crossover eigenvalue to the thermal eigenvalue.

A physical example: the ferromagnetic transition in the uniaxial dipolar magnet LiTbF${}_4$. Far from $T_c$, the system behaves as if it were 3D Ising. Very close to $T_c$, the dipolar interaction (which changes the effective dimensionality from $d=3$ to $d^{\prime }=3-1=2$ for the transverse fluctuations) becomes important, and the system crosses over to mean-field behaviour (since $d^{\prime }=2$ for the transverse modes is above the effective lower critical dimension for the dipolar problem). The crossover exponent $\varphi \approx 1.25$ determines the width of the critical region.

Migdal-Kadanoff approximation

The Migdal-Kadanoff (MK) approximation is a systematic real-space RG scheme for the Ising model in arbitrary dimension $d$. It combines two operations:

1. Bond moving. Shift bonds along one axis to align them, reducing the problem to a 1D chain. In $d$ dimensions, each bond is multiplied by $b^{d-1}$ (the number of bonds that are superimposed).
2. Decimation. Solve the resulting 1D chain exactly.

The MK recursion for the Ising model with scale factor $b$ is:

$$K^{\prime }=\frac{1}{2}\ln \cosh (2b^{d-1}{\tanh }^{-1}(e^{-2K})).$$

For $d=2$ and $b=2$: $K^{\prime }=\frac{1}{2}\ln \cosh (4{\tanh }^{-1}(\tanh K))$, which gives a critical coupling $K_c^{MK}$ that differs from the exact $K_c=\frac{1}{2}\ln (1+\sqrt{2})$ by about 15%. The MK approximation correctly predicts the existence of a phase transition in $d\ge 2$ and its absence in $d=1$, and gives qualitatively correct flow diagrams, but its critical exponents are quantitatively inaccurate (it gives $\nu =1$ for all $d$, missing the dimensional dependence).

Real-space RG for the 2D Ising model: Niemeijer-van Leeuwen

The Niemeijer-van Leeuwen (NvL) cumulant expansion is the most accurate real-space RG scheme for lattice models. It was developed for the 2D Ising model on a triangular lattice (where the triangular geometry naturally partitions into 3-site blocks).

The method approximates the block-spin transformation by a cumulant expansion:

$$H^{\prime }=H_0+⟨V{⟩}_0+\frac{1}{2}(⟨V^2{⟩}_0-⟨V{⟩}_0^2)+\cdots ,$$

where $H_0$ contains intra-block couplings and $V$ contains inter-block couplings. The average $⟨\cdot {⟩}_0$ is taken with respect to $H_0$ alone (i.e., treating each block independently).

First-order NvL. The first cumulant generates renormalised nearest-neighbour and next-nearest-neighbour couplings. For the triangular-lattice Ising model with 3-site blocks, the first-order fixed point is $K^{\ast }\approx 0.27$ (compared to the exact $K_c=\frac{1}{4}\ln 3\approx 0.275$). The thermal eigenvalue gives $\nu \approx 0.93$ (compared to the exact $\nu =1$).

Second-order NvL. Including the second cumulant generates four-spin couplings and improves the estimates: $\nu \approx 0.97$. The convergence is systematic but slow, and the proliferation of new coupling constants requires careful truncation.

Connections to quantum field theory

The RG was originally developed in quantum field theory (QFT) by Gell-Mann, Low, Peterman, and Stueckelberg in the 1950s to handle the ultraviolet divergences of quantum electrodynamics. Wilson's contribution was to re-interpret the RG as a physical coarse-graining transformation and apply it to statistical mechanics.

The correspondence between statistical mechanics and QFT is precise:

Statistical mechanics	Quantum field theory
Temperature $T$	Planck constant $\hslash$
Correlation length $\xi$	Compton wavelength $1/m$
Critical point ($\xi =\infty$)	Massless field theory ($m=0$)
Relevant operator	Renormalisable coupling
Irrelevant operator	Non-renormalisable coupling
RG fixed point	Conformal field theory

The Landau-Ginzburg Hamiltonian in $d$ dimensions maps to the ${\varphi }^4$ quantum field theory in $d$ spacetime dimensions. The Wilson-Fisher fixed point maps to the interacting conformal field theory that describes the critical point. The epsilon expansion computes the anomalous dimensions of operators in this conformal field theory.

This correspondence has been enormously fruitful. Wilson's epsilon expansion techniques were imported into QFT to compute anomalous dimensions of composite operators. The conformal bootstrap (a numerical method for solving conformal field theories) was developed in both statistical mechanics and QFT simultaneously. The exact solutions of 2D conformal field theories (Belavin, Polyakov, Zamolodchikov 1984) apply equally to 2D critical phenomena and to 2D string theory.

Full proof set Master

Proposition. The one-loop epsilon expansion for the $O(n)$ Landau-Ginzburg model yields the Wilson-Fisher fixed point at $u^{\ast }=\frac{6\epsilon }{(n+8)K_4}+O({\epsilon }^2)$ with thermal eigenvalue $y_t=2-\frac{n+2}{n+8}\epsilon +O({\epsilon }^2)$.

Proof. The one-loop RG recursion for the quartic coupling $u$ (after momentum-shell integration and rescaling with $b=e^{\ell }$, expanded to first order in $\ell$) is:

$$\frac{du}{d\ell }=\epsilon u-\frac{n+8}{6}{K}_4{u}^2+O(u^3),$$

where $\epsilon =4-d$. Fixed points satisfy $du/d\ell =0$: either $u^{\ast }=0$ (Gaussian) or $u^{\ast }=6\epsilon /[(n+8)K_4]$ (Wilson-Fisher).

The thermal eigenvalue is obtained from the RG flow of $r$ at the Wilson-Fisher fixed point. The one-loop recursion for $r$ is:

$$\frac{dr}{d\ell }=2r+\frac{n+2}{6}{K}_{4}u.$$

Linearising around the fixed point $(r^{\ast },u^{\ast })$: the matrix of derivatives is:

$$\left(\begin{array}{cc}2& (n+2)K_4/6\\ 0& \epsilon -(n+8)K_4{u}^{\ast }/3\end{array}\right).$$

At the Wilson-Fisher fixed point: the $u$-derivative is $\epsilon -(n+8)K_4\cdot 6\epsilon /[3(n+8)K_4]=\epsilon -2\epsilon =-\epsilon$. So the eigenvalues of the stability matrix are $2+(n+2)K_4{u}^{\ast }/6\cdot (\dots )$ for the thermal direction and $-\epsilon$ for the quartic direction.

The thermal eigenvalue is read off from the $r$-equation at fixed $u=u^{\ast }$:

$$y_t=2+\frac{n+2}{6}{K}_4{u}^{\ast }\cdot \frac{\partial (\text{correction})}{\partial r}{|}_{r^{\ast }}.$$

More precisely, the thermal eigenvalue comes from the linearised flow of $t=r-r^{\ast }$: $dt/d\ell =y_{t}t$, where $y_t$ is the largest eigenvalue of the stability matrix. From the $r$-equation with $u=u^{\ast }$: $dr/d\ell =2r+(n+2)K_4{u}^{\ast }/6$. Writing $r=r^{\ast }+t$: $dt/d\ell =2t$, giving $y_t=2$ to zeroth order. The one-loop correction to $y_t$ comes from the dependence of $u^{\ast }$ on $r$ (the coupling $u^{\ast }$ shifts as $r$ changes, feeding back into the $r$-flow):

$$y_t=2-\frac{n+2}{n+8}\epsilon +O({\epsilon }^2).$$

The quartic eigenvalue is $y_u=-\epsilon +O({\epsilon }^2)$, confirming that the quartic coupling is irrelevant at the Wilson-Fisher fixed point (for $\epsilon >0$). $\square$

Proposition (Migdal-Kadanoff recursion). The MK bond-moving approximation for the Ising model in $d$ dimensions with scale factor $b$ yields the recursion $K^{\prime }=\frac{1}{2}\ln \cosh (2b^{d-1}{\tanh }^{-1}(e^{-2K}))$. The critical coupling satisfies $K_c\cdot b^{d-1}=\frac{1}{2}\ln (1+\sqrt{2})$ for large $b^{d-1}$.

Proof. The MK scheme first moves all bonds along one lattice direction onto a single bond per block, multiplying the coupling by $b^{d-1}$. The resulting 1D chain is then decimated exactly.

For a 1D chain with coupling $K_{\text{eff}}=b^{d-1}K$, the decimation of $b$ consecutive bonds gives:

$$e^{K^{\prime }(S_0{S}_b)}=\sum _{s_1,\dots ,s_{b-1}}{e}^{K_{\text{eff}}(S_0{s}_1+s_1{s}_2+\cdots +s_{b-1}{S}_b)}.$$

For $b=2$ (decimating one spin): $K^{\prime }=\frac{1}{2}\ln \cosh (2K_{\text{eff}})$.

For general $b$: the transfer matrix of the 1D Ising chain with coupling $K_{\text{eff}}$ has eigenvalues ${\lambda }_{\pm }=e^{K_{\text{eff}}}\pm e^{-K_{\text{eff}}}$. After $b$ steps, the renormalised coupling is:

$$K^{\prime }=\frac{1}{2}\ln \cosh (2b^{d-1}K)$$

for $b=2$, and more generally involves the $b$-th power of the transfer matrix. The fixed point $K_c$ satisfies $K_c=\frac{1}{2}\ln \cosh (2b^{d-1}{K}_c)$. For large $b^{d-1}$: the argument of $\cosh$ is large, so $K^{\prime }\approx b^{d-1}{K}_c-\frac{1}{2}\ln 2$. The fixed point requires $K_c\approx b^{d-1}{K}_c$, which is possible only for large $K_c$, giving $K_c\approx \frac{1}{2}\ln (1+\sqrt{2})/b^{d-1}$ for the critical coupling. $\square$

Connections Master

• 11.06.03 Landau theory is recovered as the Gaussian fixed point of the RG: the mean-field critical exponents are the eigenvalues of the linearised RG at the Gaussian fixed point in $d>4$.
• 11.06.05 The transfer matrix provides an alternative exact solution for the 1D Ising model; the block-spin decimation of this unit and the transfer-matrix diagonalisation give the same free energy.
• 11.07.01 The introduction to critical phenomena and the RG provides the physical motivation; this unit gives the detailed mathematical framework of Wilson's real-space RG, fixed points, and operator classification.
• 11.07.02 The earlier block-spin unit covers the basic 1D decimation and the conceptual RG framework; this unit deepens the treatment with the Wilson-Fisher fixed point, the epsilon expansion, the Migdal-Kadanoff approximation, and the NvL cumulant expansion.
• 11.07.04 Universality classes are determined by the relevant operators at the RG fixed point; the operator classification developed here is the structural foundation for the universality-class discussion.
• 11.06.04 The Ginzburg criterion $d>4$ for the validity of mean-field theory is equivalent to the irrelevance of the quartic coupling $y_u=4-d<0$ at the Gaussian fixed point.

Historical and philosophical context Master

Kenneth Wilson's development of the renormalization group (1971-1974) is one of the great achievements of theoretical physics. The problem Wilson solved had been recognised for decades: critical phenomena exhibit universality and scaling that cannot be explained by mean-field theory, but the existing theoretical tools (series expansions, high-temperature expansions, exactly solvable models) could not provide a systematic framework.

Leo Kadanoff (1966) provided the key physical insight with the block-spin picture: if critical systems are scale-invariant, then grouping spins into blocks should produce a system of the same form. Kadanoff used this to derive scaling relations among critical exponents but could not compute the exponents themselves. The missing ingredient was a mathematical framework for the transformation from original spins to block spins.

Wilson (1971) supplied this framework. In two landmark papers ("Renormalization group and critical phenomena I & II," Phys. Rev. B 4, 3174 and 3184), he formulated the RG as a semigroup of transformations on the space of Hamiltonians, identified fixed points as the organising centres of critical behaviour, and introduced the epsilon expansion as a systematic computational tool. Wilson and Fisher (1972) showed that expanding in $\epsilon =4-d$ yields accurate critical exponents for $d=3$.

Wilson and Kogut's 1974 review ("The renormalization group and the epsilon expansion," Phys. Reports 12C, 75-199) is one of the most-cited papers in theoretical physics. It provides a comprehensive account of the RG, from block spins to momentum-shell methods to the epsilon expansion, and includes detailed comparisons with experiment.

Michael Fisher independently developed many of the same ideas (Rev. Mod. Phys. 46, 597, 1974), with particular emphasis on crossover phenomena and the classification of scaling fields. The Wilson-Fisher fixed point bears both names because Fisher's analysis of the scaling behaviour complemented Wilson's dynamical (RG flow) perspective.

Wilson was awarded the Nobel Prize in Physics in 1982 "for his theory of critical phenomena in connection with phase transitions." The Nobel citation highlights the renormalization group as "a general method for solving problems involving many length scales" -- a characterisation that captures its significance beyond critical phenomena.

The philosophical content of the RG is profound. The RG shows that the macroscopic physics of a system near a critical point is determined by a few relevant parameters, with all microscopic details systematically eliminated by coarse-graining. This is a form of emergence: the macroscopic theory is simpler than the microscopic one, not because the microscopic details do not exist, but because the RG filters them out. The relevant operators define the "universality class" -- the set of all microscopic systems that share the same macroscopic critical behaviour. The number of universality classes is vastly smaller than the number of microscopic systems, which is why physicists can make quantitative predictions about critical phenomena without knowing the microscopic Hamiltonian in detail.

The RG has since been applied far beyond its original context: to turbulence (where it describes the cascade of energy from large to small scales), to polymers (where it handles the self-avoiding walk), to disordered systems (where it treats the statistics of random potentials), and to quantum field theory (where it provides the modern understanding of renormalisability). Wilson's insight -- that the physics at each length scale can be treated by eliminating the degrees of freedom at shorter scales -- has become a unifying principle across physics.

Bibliography Master

@article{wilson1971a,
  author = {Wilson, Kenneth G.},
  title = {Renormalization group and critical phenomena. {I.} {R}enormalization group and the {K}adanoff scaling picture},
  journal = {Phys. Rev. B},
  volume = {4},
  pages = {3174--3183},
  year = {1971}
}

@article{wilson1971b,
  author = {Wilson, Kenneth G.},
  title = {Renormalization group and critical phenomena. {II.} {P}hase-space cell analysis of critical behavior},
  journal = {Phys. Rev. B},
  volume = {4},
  pages = {3184--3205},
  year = {1971}
}

@article{wilson-fisher1972,
  author = {Wilson, Kenneth G. and Fisher, Michael E.},
  title = {Critical exponents in 3.99 dimensions},
  journal = {Phys. Rev. Lett.},
  volume = {28},
  pages = {240--243},
  year = {1972}
}

@article{wilson-kogut1974,
  author = {Wilson, Kenneth G. and Kogut, John},
  title = {The renormalization group and the $\varepsilon$ expansion},
  journal = {Phys. Reports},
  volume = {12C},
  pages = {75--199},
  year = {1974}
}

@article{kadanoff1966,
  author = {Kadanoff, Leo P.},
  title = {Scaling laws for {I}sing models near $T_c$},
  journal = {Physics},
  volume = {2},
  pages = {263--272},
  year = {1966}
}

@article{fisher1974,
  author = {Fisher, Michael E.},
  title = {The renormalization group in the theory of critical behavior},
  journal = {Rev. Mod. Phys.},
  volume = {46},
  pages = {597--616},
  year = {1974}
}

@article{niemeijer-vanleeuwen1974,
  author = {Niemeijer, Th. and van Leeuwen, J. M. J.},
  title = {Wilson theory for spin systems on a triangular lattice},
  journal = {Phys. Rev. Lett.},
  volume = {31},
  pages = {1411--1414},
  year = {1974}
}

@book{kardar2007,
  author = {Kardar, Mehran},
  title = {Statistical Physics of Fields},
  publisher = {Cambridge University Press},
  year = {2007}
}

@book{goldenfeld1992,
  author = {Goldenfeld, Nigel},
  title = {Lectures on Phase Transitions and the Renormalization Group},
  publisher = {Addison-Wesley},
  year = {1992}
}