Block-Spin (Real-Space) Renormalization Group

Stand far enough back from a magnet poised at its critical point and you cannot tell how far back you are standing. Blur it, shrink it, and the blurred-shrunk picture is statistically identical to the original — the hallmark of scale invariance. Leo Kadanoff's block-spin idea turns that visual intuition into an actual calculation. Tile the lattice into blocks of $b^d$ spins, replace each block by one coarse-grained "block spin" using a rule such as majority vote, then rescale all lengths by $b$ so the block lattice has the original spacing again. What you get is a new spin system with renormalized couplings — and the map from old couplings to new is the central object of the whole theory. This is the real-space face of the renormalization group, the direct physical-lattice complement to the momentum-shell method that works in Fourier space. Its appeal is transparency: you can point at the spins being averaged, and the procedure works even in $d=2$ and $d=3$ where there is no small parameter like $\epsilon=4-d$ to organize an expansion. Its cost is that an exact block transformation does not stay within a finite set of couplings — summing out the interior of each block couples blocks that were not directly coupled before, generating further-neighbour, three-spin, and plaquette interactions. Practical schemes truncate this proliferation, trading exactness for tractability. The reward for that trade is concrete. Kadanoff's construction shows why the RG eigenvalues must be powers of the block size, gives an explicit recipe (the Niemeijer–van Leeuwen cumulant expansion) that produces a nontrivial fixed point and a critical exponent $\nu\approx1$ for the 2D Ising model directly from a flow equation, and makes the abstract slogan "coarse-grain, rescale, renormalize" into something you can compute.

Key takeaways

Kadanoff's block-spin RG partitions a lattice into blocks of $b^d$ spins, replaces each by one coarse-grained spin (majority or projection rule), and rescales lengths by $b$ to restore the original spacing.
Summing over the internal block spins generates a renormalized Hamiltonian with new couplings, defining a recursion map $K'=R_b(K)$ in coupling space.
The semigroup property $R_{b_1 b_2}=R_{b_1}\circ R_{b_2}$ forces the linearized RG eigenvalues to be powers of the block size, $\lambda_i=b^{y_i}$ with $b$ -independent exponents $y_i$ .
An exact block transformation generates infinitely many couplings; real schemes (cumulant, Migdal–Kadanoff, Niemeijer–van Leeuwen) truncate — uncontrolled but often accurate.
The Niemeijer–van Leeuwen triangular-block cumulant scheme gives a nontrivial fixed point $K^*\approx0.336$ and $\nu\approx0.99$ for the 2D Ising model, from a flow equation rather than the exact solution.
Key idea: group $b^d$ spins into one block spin, sum out the internal spins to get a renormalized Hamiltonian, and rescale lengths by $b$ ; this defines a recursion $K'=R_b(K)$ whose fixed points are critical points and whose linearized eigenvalues $b^{y_i}$ carry the universal information.

Where this lives today

Block-spin RG remains the method of choice wherever there is no small parameter to organize an expansion. In $d=2$ and $d=3$ , where the $\epsilon=4-d$ expansion is unreliable, real-space schemes and their numerical descendant — the Monte Carlo Renormalization Group of Swendsen — deliver high-precision exponents by blocking Monte Carlo configurations repeatedly and estimating the linearized matrix $T_{\alpha\beta}$ from correlations between successive blocking levels. Because it works directly on the lattice without Fourier modes, real-space RG adapts naturally to quenched disorder: the Ma–Dasgupta–Hu strong-disorder RG decimates the strongest bond or field at each step and flows to infinite-randomness fixed points that govern random quantum magnets. And the same block-decimation logic, with the classical block spin replaced by an optimally truncated quantum Hilbert space, is the conceptual ancestor of the density-matrix renormalization group (DMRG) — the dominant tool for one-dimensional quantum many-body physics today. Kadanoff's picture of blurring and shrinking a magnet turned out to be a template for half a century of computation.

The physical picture

Imagine a near-critical magnet photographed and then optically blurred and shrunk so that each $b\times b$ patch becomes a single pixel. At the critical point the blurred-and-shrunk picture is statistically indistinguishable from the original — that is scale invariance, made literal. The block-spin transformation is this blur-and-shrink operation, applied not to a photograph but to the probability distribution over configurations. Self-similarity — the picture looking the same after blurring — is exactly the fixed-point condition on the couplings. This makes concrete the coarse-grain–and–rescale program of the renormalization group idea. Every step removes wiggles shorter than $b$ lattice spacings; the correlation length, re-measured in the new (larger) lattice units, shrinks by $b$ . Tuning temperature to criticality means tuning so that no matter how many times you blur, the system never resolves into "ordered" or "disordered" — it sits forever at the fixed point.

Figure

A 3×3 block on the square lattice: nine spins collapse to one block spin by majority rule, and lengths rescale by b＝3

Why does the recursion generate new couplings? When you sum over the spins inside a block, the resulting effective interaction between two block spins depends on the configurations of all intermediate spins — which couples blocks that were never directly bonded. This "spreading" of interactions is the price of working in real space, and the entire art of real-space RG is choosing a truncation that captures the dominant new couplings while discarding the rest.

Figure

Under coarse-graining the correlation length contracts by b each step; only ξ＝∞ (criticality) is a fixed point

The exponents $y_i$ are properties of the fixed point, independent of the block size $b$ used to reach it. The reason is a clean consistency argument: coarse-graining by $b_1$ then by $b_2$ must equal coarse-graining by $b_1 b_2$ in one shot. The linearized maps therefore compose, their eigenvalues multiply, $\lambda_i(b_1 b_2)=\lambda_i(b_1)\lambda_i(b_2)$ , and the only continuous multiplicative function is a power law $\lambda_i(b)=b^{y_i}$ . This composition rule is the precise statement that the RG is a renormalization "group" — really a semigroup, since coarse-graining throws away information and cannot be inverted.

The core equations

The block transformation is defined by a projection operator $T$ that assigns block spins to each fine configuration, weighted so that probability — and therefore the partition function — is conserved.

e^{-\beta H'[\{S'\}]} = \sum_{\{S\}} T(\{S'\},\{S\})\, e^{-\beta H[\{S\}]},\qquad \sum_{\{S'\}} T(\{S'\},\{S\})=1 \qquad (1)

where: $H$ — original Hamiltonian on the fine lattice; $H'$ — renormalized Hamiltonian for the block spins; $S_i$ — fine spins; $S'_I$ — block spin of block $I$ ; $T\ge0$ — coarse-graining projector. A majority rule is $T=\prod_I\delta_{S'_I,\,\mathrm{sgn}(\sum_{i\in I}S_i)}$ . The normalization $\sum_{\{S'\}}T=1$ guarantees $Z'=Z$ : summing $(1)$ over $\{S'\}$ collapses $T$ to unity and returns the original partition function exactly. The Hamiltonian itself is best viewed as a point in coupling space: an infinite vector of dimensionless couplings, one for each possible local operator.

-\beta H[\{S\}] = \sum_{\alpha} K_\alpha\, S_\alpha[\{S\}] \qquad (2)

where: $K_\alpha$ — dimensionless coupling for operator $\alpha$ ; $S_\alpha$ — a local lattice operator such as $\sum_{\langle ij\rangle}S_iS_j$ (nearest-neighbour), $\sum_i S_i$ (field), or multi-spin terms. The infinite vector $K=(K_1,K_2,\dots)$ specifies the Hamiltonian; the RG acts on this vector. Combining the two gives the central object — the RG recursion, a map from the old coupling vector to the new one.

K' = R_b(K) \qquad (3)

where: $R_b$ — the renormalization map for block size $b$ ; $b$ — the linear rescaling factor (block edge in lattice units). A critical point is a fixed point of this map,

K^* = R_b(K^*), \qquad (4)

because at criticality $\xi=\infty$ and the coarse-grained system must look identical to the original. Near a fixed point the map linearizes, and its eigenvalues are the universal content.

\frac{\partial K'_\alpha}{\partial K_\beta}\bigg|_{K^*}\equiv T_{\alpha\beta},\qquad \sum_\beta T_{\alpha\beta}\,e^{(i)}_\beta=\lambda_i\,e^{(i)}_\alpha,\qquad \lambda_i=b^{\,y_i} \qquad (5)

where: $T_{\alpha\beta}$ — the linearized RG matrix at the fixed point; $\lambda_i$ — eigenvalue along eigen-operator $e^{(i)}$ (a scaling direction); $y_i=\ln\lambda_i/\ln b$ — the scaling exponent, relevant if $y_i>0$ , irrelevant if $<0$ , marginal if $0$ . The full analysis of these eigenvalues belongs to linearization about a fixed point and their classification to relevant, irrelevant, and marginal operators; the extraction of critical exponents from the two relevant $y_t,y_h$ is owned by critical exponents from the RG. Block-spin RG supplies the machine $R_b$ whose eigenvalues those articles interpret. The correlation length transforms simply, which is what makes fixed points critical.

\xi(K)=b\,\xi(K') \qquad (6)

where: $\xi$ — correlation length in lattice units. At a fixed point $\xi=b\,\xi$ , so $\xi=0$ or $\infty$ ; the nontrivial fixed points, with $\xi=\infty$ , are the critical points.

The core derivations

Derivation AWhy the RG eigenvalues are powers of the block size

The single most important structural fact about real-space RG is that the linearized eigenvalues take the form $\lambda_i=b^{y_i}$ with $b$ -independent exponents $y_i$ — this is what makes the exponents intrinsic and universal rather than artifacts of the block size you happened to choose. The goal here is to prove that form from the composition (semigroup) structure of coarse-graining alone, without any specific model. The strategy: argue that two successive block transformations equal a single larger one, deduce that the linearized eigenvalues must multiply, and note that the only continuous function satisfying a multiplicative law is a power. Watch the direction of the logic — it flows from a geometric fact about lengths to an algebraic constraint on eigenvalues.

Step 1. Composition of coarse-grainings. Blocking with edge $b_1$ and then with edge $b_2$ removes structure below $b_1$ , then below $b_1 b_2$ — the same result as one block transformation with edge $b_1 b_2$ :

R_{b_1 b_2}=R_{b_1}\circ R_{b_2}.

Why this is a semigroup and not a group: coarse-graining discards short-distance information irreversibly, so there is no inverse — only forward composition.

Step 2. Linearize each map at the common fixed point $K^*$ . The linearized operators inherit the composition:

T(b_1 b_2)=T(b_1)\,T(b_2).

Step 3. Along a shared eigen-direction $e^{(i)}$ the eigenvalues therefore multiply:

\lambda_i(b_1 b_2)=\lambda_i(b_1)\,\lambda_i(b_2).

Step 4. The only continuous solution of $\lambda(b_1 b_2)=\lambda(b_1)\lambda(b_2)$ is a power law:

\lambda_i(b)=b^{\,y_i},\qquad y_i=\frac{\ln\lambda_i}{\ln b},

with $y_i$ manifestly independent of $b$ — the same exponent emerges whether you block by 2, 3, or 3 then 2. Verified: `verify_block-spin-real-space-rg.py` confirms $\lambda(b_1 b_2)=\lambda(b_1)\lambda(b_2)$ for $\lambda=b^y$ and that $y=\ln\lambda/\ln b$ is $b$ -independent (check 6).

Go deeper · Derivation B · 1 minThe Niemeijer–van Leeuwen cumulant RG for the 2D triangular Ising model

Derivation BThe Niemeijer–van Leeuwen cumulant RG for the 2D triangular Ising model

This is the payoff of the real-space method: a closed, computable RG that produces a genuine nontrivial fixed point and a critical exponent for the 2D Ising model, from a flow equation rather than from Onsager's exact solution. The construction partitions the triangular lattice into three-site blocks, assigns each a majority-rule block spin (unambiguous for three Ising spins), treats the three intra-block bonds exactly, and treats the inter-block bonds perturbatively through a first-order cumulant expansion. The strategy is: split $H=H_0+V$ into intra- and inter-block parts, average $e^{-\beta V}$ over intra-block fluctuations consistent with the fixed block spins, and read off the renormalized coupling. What to watch: the block-spin's average magnetization is the bridge that converts intra-block alignment into inter-block coupling.

Step 1. Block partition and splitting. Group the triangular lattice into three-spin triangular blocks; each carries $S'_I=\mathrm{sgn}(S_1+S_2+S_3)$ . Write $-\beta H=-\beta H_0-\beta V$ , with $H_0$ the three intra-block bonds (exact) and $V$ the inter-block bonds (perturbative).

Step 2. First-order cumulant expansion:

-\beta H' = \ln\langle e^{-\beta V}\rangle_0 \approx \langle -\beta V\rangle_0 + \tfrac12\big(\langle V^2\rangle_0-\langle V\rangle_0^2\big)+\cdots,

where $\langle\cdot\rangle_0$ averages over intra-block configurations at fixed block spins, weighted by $e^{-\beta H_0}$ .

Step 3. Effective block magnetization. A boundary spin of a block with $S'_I=+$ has average value

\langle S_i\rangle_0^{(+)}=\frac{e^{3K}+e^{-K}}{e^{3K}+3e^{-K}},

where $e^{3K}$ weights the fully aligned internal state and the $e^{-K}$ terms the two-up-one-down states consistent with $S'_I=+$ .

Step 4. First-order recursion. Each pair of neighbouring blocks shares two inter-block bonds (coordination factor 2), and the coupling passes through the squared block magnetization:

K' = 2K\left(\frac{e^{3K}+e^{-K}}{e^{3K}+3e^{-K}}\right)^{2}.

Step 5. Fixed point and exponent. Solving $K'=K$ for the nonzero root gives $K^*\approx0.3356$ . Linearizing with the triangular rescaling $b=\sqrt3$ ,

y_t=\frac{\ln(dK'/dK|_{K^*})}{\ln\sqrt3}\approx0.882\ \Rightarrow\ \nu=\frac{1}{y_t}\approx1.13.

Carrying the cumulant expansion to second order improves this to $\nu\approx0.99$ , close to the exact $\nu=1$ . Verified: `verify_block-spin-real-space-rg.py` finds the nonzero fixed point $K^*=0.3356$ by bracketing $R(K)=K$ , and computes $y_t=0.882$ , $\nu=1.134$ at first order (checks 9–10), matching Niemeijer–van Leeuwen (1973–74).

Limiting cases

Real-space schemes interpolate between exactly solvable cases and physically motivated approximations. The exponent formulas quoted in the last column are owned and derived by critical exponents from the RG; here they are used only to interpret the fixed points this method locates.

Case	Result	Meaning
1D, $b=2$ decimation	$K'=\tfrac12\ln\cosh 2K$	Exact for the 1D Ising chain; the only fixed points are $K^=0$ (disorder) and $K^=\infty$ (order). No finite- $T$ critical point, so $T_c=0$ — see decimation.
Large block $b\to\infty$	$\lambda_i=b^{y_i}$ , $y_i$ fixed	Exponents are $b$ -independent if the scheme is exact; residual $b$ -dependence in a truncated scheme measures its error.
Majority rule vs decimation	$S'_I=\mathrm{sgn}(\sum_{i\in I}S_i)$ vs $S'_I=S_{i_0}$	Different projectors give different non-universal flows and fixed-point locations, but the same universal exponents.
Migdal–Kadanoff bond-moving	$K'=\tfrac12\ln\cosh(2b^{\,d-1}K)$ (schematic)	Approximate; exact on hierarchical lattices; produces a genuine finite- $T$ fixed point in $d\ge2$ with reasonable but not exact exponents.

Worked examples

Example 1 — Majority-rule outcomes for a small block, counted exactly. We make the coarse-graining rule concrete by enumerating it on a $2\times2$ block of four Ising spins, with the tie broken toward $+$ .

Step 1. There are $2^4=16$ fine configurations. Under majority rule $S'=\mathrm{sgn}(\sum_{i}S_i)$ , a configuration maps to $+$ if more spins are up, to $-$ if more are down, with ties (two up, two down) broken to $+$ .

Step 2. Count the strict-plus configurations: all four up ( $\binom{4}{4}=1$ ) and exactly three up ( $\binom{4}{3}=4$ ), giving $5$ . Count the ties (two up, two down): $\binom{4}{2}=6$ . Count strict-minus: three down ( $4$ ) and four down ( $1$ ), giving $5$ .

Step 3. With the tie-break to $+$ : configurations mapping to $S'=+$ number $5+6=11$ ; to $S'=-$ number $5$ . Check: $11+5=16$ . 11 configurations → $S'=+$ , 5 → $S'=-$ .

Step 4. The lesson: the projector is deterministic (each fine configuration maps to exactly one block spin, so $\sum_{S'}T=1$ holds), but it is not symmetric under $S'\to-S'$ once a tie-break is fixed — a reminder that the coarse-graining rule is a genuine choice that can bias the flow, which is why symmetric weighted rules are often preferred.

Example 2 — Reading the exact Onsager exponents off the eigenvalues. The universal payoff of any block scheme is that the exact eigenvalues, once known, deliver every critical exponent. For the 2D Ising model the exact RG eigenvalues are $y_t=1$ and $y_h=15/8$ ; we feed them into the exponent formulas (owned by critical exponents from the RG) to recover Onsager's numbers.

Step 1. Correlation-length exponent: $\nu=1/y_t=1/1=1$ .

Step 2. Order-parameter exponent: $\beta=(d-y_h)/y_t=(2-15/8)/1=1/8=0.125$ .

Step 3. Susceptibility exponent: $\gamma=(2y_h-d)/y_t=(15/4-2)/1=7/4=1.75$ .

Step 4. Critical-isotherm exponent: $\delta=y_h/(d-y_h)=(15/8)/(1/8)=15$ . These are exactly the Onsager values for the 2D Ising model — a genuine, if truncated, block scheme aims to reproduce the eigenvalues $y_t=1$ , $y_h=15/8$ that make them so. $\nu=1,\ \beta=1/8,\ \gamma=7/4,\ \delta=15$ .

Origins · 1 mincoarse-graining and how this was discovered

coarse-graining

the central operation of the renormalization group · Coarse-graining is the act of systematically discarding short-distance detail to obtain an effective description at a larger scale. In Kadanoff's block-spin realization it is the averaging of

b^d

spins into one; in Wilson's momentum-shell version it is the integration over high-wavevector modes. The operation is deliberately irreversible — information is thrown away — which is why the renormalization "group" is really a semigroup. Coarse-graining is the reason universality exists: two microscopically different systems, coarse-grained enough, look the same, because the differences lived at the short scales that were averaged out.

History

1944

— Onsager solves the 2D Ising model exactly, fixing the target values ν＝1, y_h＝15/8

1966

— Kadanoff introduces block-spin scaling for Ising models near T_c (Physics Physique Fizika 2, 263)

1971

— Wilson casts coarse-graining as an explicit flow of couplings, giving the block idea its general framework (Physical Review B 4)

1973

— Niemeijer and van Leeuwen compute the triangular-block cumulant RG, obtaining a nontrivial fixed point and ν from a flow equation

1982

— Wilson receives the Nobel Prize in Physics for the theory of critical phenomena in connection with phase transitions

Going further

References. Kadanoff, Scaling laws for Ising models near $T_c$ , Physics Physique Fizika 2, 263 (1966) · Niemeijer & van Leeuwen, Phys. Rev. Lett. 31, 1411 (1973); Physica 71, 17 (1974) · Kardar, Statistical Physics of Fields, Ch. 4–5 · Pathria & Beale, Statistical Mechanics, 3rd ed., Ch. 14 · Sethna, Statistical Mechanics: Entropy, Order Parameters and Complexity, Ch. 12 · Wilson & Kogut, Phys. Rep. 12, 75 (1974) · Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Ch. 9.

Provenance. Recursion structures and exponents are standard. A verifier should: (1) reproduce the 1D decimation $K'=\tfrac12\ln\cosh 2K$ by the cosh identity; (2) check the Niemeijer–van Leeuwen first-order recursion and $K^*\approx0.336$ , $\nu\approx1.13$ at first order improving to $\approx0.99$ at second order; (3) verify that $d=2$ , $y_t=1$ , $y_h=15/8$ give Onsager's $\nu=1,\beta=1/8,\gamma=7/4,\delta=15$ via the exponent formulas owned by the Critical Exponents from the RG article. All numbers are textbook-standard.

Builds on: The Renormalization Group Idea: Coarse-Graining and Rescaling · Decimation and the One-Dimensional Ising RG · RG Flow in Coupling Space. Leads to: Fixed Points of the RG · Linearization About a Fixed Point and Eigenvalue Spectrum · Critical Exponents from the RG · Relevant, Irrelevant, and Marginal Operators.

Is block-spin RG exact or approximate?

The definition is exact — the block transformation with a normalized projector conserves the partition function identically. What is approximate is any attempt to keep the flow within a finite set of couplings. An exact transformation generates infinitely many interactions (further-neighbour, three-spin, plaquette), so practical schemes truncate to a manageable few. The 1D chain with $b=2$ decimation is the rare case where the exact transformation happens to close on a single coupling. In $d\ge2$ you always truncate, which introduces uncontrolled but often small errors in the exponents.

How is this different from momentum-shell RG?

They implement the same coarse-grain–rescale–renormalize cycle in different variables. Block-spin RG works in real space: you group nearby spins and average them. Momentum-shell RG works in Fourier space: you integrate out field modes with large wavevector in a thin shell near the cutoff. Real-space RG is intuitive and works in any dimension without a small parameter, but truncation is uncontrolled; momentum-shell RG gives a systematic $\epsilon=4-d$ expansion but needs a field-theory description. Both define the same flow $K'=R_b(K)$ , the same fixed points, and the same eigenvalue exponents — the physics is independent of which implementation you choose, which is itself a statement of universality.

Why does the majority rule need a tie-break, and does it matter?

For an odd number of spins per block (like three on the triangular lattice) the majority is always well-defined, so no tie-break is needed. For an even number (like four in a $2\times2$ square block) you can have two up and two down, and the rule must decide. A fixed tie-break (say to $+$ ) makes the projector deterministic but breaks the $S'\to-S'$ symmetry of the coarse-graining, which can bias the non-universal flow. It does not change the universal exponents if the scheme is otherwise sound, but it is one reason people often prefer weighted or randomized rules that preserve the symmetry. Odd block sizes sidestep the issue entirely.

What does it mean that the RG is a "semigroup" and not a group?

A group requires every element to have an inverse. Coarse-graining does not: when you average $b^d$ spins into one, you discard the information about how those spins were individually arranged, and you cannot recover it. So you can compose coarse-grainings (block by $b_1$ , then by $b_2$ , equalling a block by $b_1 b_2$ ) but you cannot "un-block." That forward-only composition is a semigroup. The name "renormalization group" is historical and slightly inaccurate; the semigroup structure is nonetheless exactly what forces the eigenvalues to be powers of $b$ .

If different projection rules give different fixed points, which one is "correct"?

All well-chosen rules are correct in the only sense that matters: they give the same universal exponents. The location of the fixed point in coupling space, and the flow trajectories, are non-universal — they depend on the rule, just as the critical temperature depends on microscopic details. What must agree across rules are the eigenvalues $y_i$ at the fixed point, because those are physical. If two rules disagree on the exponents, at least one has been truncated too severely to capture the fixed point accurately. This is why cross-checking a real-space scheme against exact results (like the 2D Ising eigenvalues) is essential.

How does the Niemeijer–van Leeuwen scheme get a nontrivial fixed point where 1D decimation does not?

Dimensionality. In 1D, decimation gives $K'<K$ always, so the flow runs monotonically to disorder — no fixed point at finite coupling. On the 2D triangular lattice, the inter-block coupling is enhanced by the shared bonds and the squared block magnetization, so at low temperature $K'$ can exceed $K$ (order strengthens under coarse-graining) while at high temperature it decreases. Between these regimes sits an unstable fixed point $K^*\approx0.336$ — the critical point. That is the whole content of " $d=2$ orders but $d=1$ does not," seen through the RG: a nontrivial fixed point exists only when the geometry lets the coarse-grained coupling grow.