Near Tc the only length scale besides the lattice spacing is the correlation length ξ, which diverges at the critical point. Coarse-graining the lattice and rescaling it reduces ξ by a factor b, so away from Tc the system flows toward one of two trivial fixed points, while at Tc it stays self-similar. Turning this picture into equations justifies Widom’s scaling ansatz, the scaling form of the correlation function, and the universality of critical exponents.
Based on lecture notes by Sang Hoon Lee for Chapter 2, following K. Christensen and N. R. Moloney, Complexity and Criticality (2005). Units: kB = 1 and J = 1. References
Three square lattices are simulated live at reduced temperatures t < 0, t = 0 and t > 0. Each column applies the block-spin transformation Rb twice, top to bottom, and redraws every stage at the same size. The lattices update once every 0.1 seconds, with local single-spin moves (Metropolis sweeps), so domains evolve smoothly from one picture to the next. After a new start the lattices first equilibrate for 3 000 sweeps at full speed, which takes about 15 seconds; averaging begins after that. Near Tc single-spin updates relax slowly, so the t = 0 column keeps coarsening for a while.
Since spins are correlated up to ξ, regions smaller than ξ might behave like a single block spin. The real-space renormalization procedure is to (1) divide the lattice into blocks of linear size b, each with bd spins, (2) replace each block by a single block spin sI according to a coarse-graining rule, and (3) rescale all lengths by b to restore the original lattice spacing. The rule is not unique: the majority rule sI = sign(Σi∈I si), with ties broken at random, or decimation, where sI takes the value of one chosen spin in the block. Either way, fluctuations on scales below b are averaged out.
Every length shrinks by b, including the correlation length, ξ′ = ξ/b. At the critical point ξ = ∞ is unchanged; away from it the system moves away from t = 0. To first order t′ = λt(b)t, and composing two transformations, λt(b2)λt(b1) = λt(b1b2) with λt(1) = 1, has the unique solution of a power law:
Summing first over the configurations consistent with each block-spin configuration includes all 2N microstates, and defines the renormalized energy E′ through e−βE′{sI} = Σ e−βE{si}. The partition function is invariant, Z(t, h, N) = Z(t′, h′, N′) with N′ = N/bd, so the free energy per spin obeys f(t, h) = b−df(bytt, byhh). Choosing b = |t|−1/yt ∝ ξ gives exactly Widom’s form fs = |t|2−α𝓕±(h/|t|Δ) with 2 − α = νd and Δ = yh/yt. The same argument for g(r, t, h) gives d − 2 + η = 2β yt. Two RG eigenvalues fix every exponent:
Take b = 2 and let each block spin be the odd spin (decimation). Summing over each even spin s2 between s1 and s3 gives 2 cosh[K1(s1 + s3)] = exp(K0′ + K1′s1s3), which only depends on s1s3. Matching the cases s1 = ±s3:
The partition function keeps its form with half the spins and a renormalized coupling. Since K1′ < K1 for every K1 > 0, the flow runs to the stable fixed point K1* = 0 (T = ∞, non-interacting spins, ξ = 0) away from the unstable one at K1* = ∞ (T = 0, fully aligned, ξ = 0). No transition at finite T, exactly as the transfer matrix showed. Because tanh K1′ = tanh² K1, the correlation length ξ = −1/ln tanh K halves exactly at each step, and the offsets K0′ add up to the free energy.
On the square lattice, sum out every second spin (b = √2, followed by a 45° rotation). The decimated spin s5 couples to its four neighbors through 2 cosh[K1(s1 + s2 + s3 + s4)], which can only be rewritten exactly by generating new couplings: a nearest-neighbor coupling K1′, a next-nearest-neighbor coupling K2′ (s1 and s3 now interact through s5), and a four-spin coupling K3′. Solving the eight conditions gives
Each further step generates couplings over longer distances, so the flow lives in an infinite-dimensional coupling space and must be truncated. Odd couplings such as sisjsk never appear because E must be symmetric under si → −si at H = 0. Compare two truncations:
A generalized Ising model in zero field is a point K = (K1, K2, K3, …) in the space of all couplings that respect the up–down symmetry: −βE = K0N + K1Σnnsisj + K2Σnnnsisj + K3Σ□sisjsksl + …. The original model lies on the K1-axis, at K = 0 for T = ∞ and at infinity for T = 0. The transformation K′ = Rb(K) is a recursion that generates new couplings even if they started at zero. It obeys Rb2Rb1 = Rb1b2 but has no inverse, so it forms a semigroup rather than a group.
Since ξ(K′) = ξ(K)/b, a fixed point Rb(K*) = K* must have ξ(K*) = 0 (the trivial weak- and strong-coupling fixed points) or ξ(K*) = ∞ (the non-trivial fixed point). Each fixed point has a basin of attraction; that of the non-trivial one, where ξ = ∞, is the critical surface. Near K*, δK′ = M(b)δK with Mij = ∂Ki′/∂Kj, and in its eigenbasis the scaling fields obey ui′ = λiui with λi = byi. A field is relevant if yi > 0 (driven away from K*), irrelevant if yi < 0 (it dies out), and marginal if yi = 0. The irrelevant directions span the critical surface; the Ising model has only two relevant fields, t and h.
Below is the classic two-coupling approximation of the square-lattice decimation, keeping only the lowest order in K: K′ = 2K2 + L and L′ = K2, where K is the nearest-neighbor and L the next-nearest-neighbor coupling. Click anywhere to start a trajectory.
Trajectories that start on the critical line flow into the fixed point along the irrelevant direction. Any small offset along the relevant direction grows by λ1 each step and is expelled toward one of the trivial fixed points: high temperature (toward the origin) or low temperature (toward infinity).
Apart from the constant factor eN′K0′, Z is invariant, so the singular part of the free energy obeys fs(K) = b−dfs(K′). Written in scaling fields after n steps, fs(t, h, u3, …) = b−ndfs(bnytt, bnyhh, bny3u3, …). The irrelevant fields vanish as n → ∞, and choosing bnyt = |t|−1 gives fs = |t|d/yt𝓕±(h/|t|yh/yt): Widom’s ansatz, with 2 − α = d/yt = dν and Δ = yh/yt. Any model whose couplings flow to the same fixed point shares its yt and yh, and therefore its exponents. That is universality.
Key reference. K. Christensen and N. R. Moloney, Complexity and Criticality, Imperial College Press, London (2005), Chapter 2.