Part 6 of 6. The Ising Hamiltonian is unchanged when every spin flips, and so is the Monte Carlo rule that samples it. Below the critical temperature the state picks a sign anyway.
Parts 4 and 5 ran deterministic rules on the grid; this part adds thermal noise. Put a spin σ = ±1 on each site and reward neighbours for agreeing, and you have the Ising model (Lenz 1920, Ising 1924). The earlier parts asked which rotations a lattice keeps. The symmetry here is internal, flipping every spin, and the lattice keeps it exactly. The question is how a state can lose a symmetry its rules never break.
Here J > 0 rewards aligned neighbours and h is an external field. The order parameter is the magnetisation per site:
Flipping every spin leaves each product σiσj unchanged, so at h = 0
The symmetry group is ℤ₂ = {identity, global flip}. A field breaks it explicitly.
An 80×80 grid has 26400 configurations, far too many to sum. The Metropolis algorithm (1953) instead builds a Markov chain whose stationary distribution is the Boltzmann weight: pick a site, compute the cost of flipping it,
and accept with probability
ΔE takes one of five values, so acceptance is a table lookup. One sweep is N proposals.
Figure 1. Metropolis on an 80×80 periodic lattice, pre-run 150 sweeps from random spins at Tc. Dashed lines on the trace are Onsager's ±M0(T). At 4.0 M jitters around zero, at 1.5 it sits on a plateau, at Tc it swings. In Wolff mode a cluster larger than half the grid is swapped for its complement, which the flip symmetry makes equivalent and which stops the screen flashing.
A symmetric Hamiltonian does not guarantee a symmetric simulation; an update rule could favour one sign through how it spends random numbers. The test: run a second copy that starts as the exact flip of the first, and feed both the same random numbers. At h = 0 flipping σi costs the same in both, since σi and si both change sign, so every decision matches and the overlap
stays at exactly −1. A field adds 2σih to ΔE, with opposite signs in the two copies.
Figure 2. Two pairs of 48×48 lattices at T = 2.3, each pair sharing one random stream. The grids and the violet line are the h = 0 pair; the pink line is a pair run in a small field, which breaks the mirror within a few sweeps and ends with the two copies identical.
The Wolff update used below passes the same test, because it builds clusters from whether neighbours agree, which a global flip leaves unchanged.
At h = 0 a finite lattice has ⟨M⟩ = 0 exactly, since each configuration and its flip carry the same weight. A broken phase needs the limits in this order:
Swap the limits and ⟨M⟩ = 0 again. On a finite grid the question becomes how long the chain keeps its sign. To cross over, a periodic grid must pass through a strip bounded by two walls that each span the box, so the barrier grows with L.
Figure 3. Histograms of M on two grids started all up: 200,000 Metropolis sweeps at L = 10, 40,000 at L = 32. A reversal is a crossing from beyond +M0/2 to beyond −M0/2 or back. The small grid splits its time evenly between the signs; the large one has not left its starting sign.
Going from T = 2.2 to 2.0 stretches the small grid's wait by about an order of magnitude, because the walls cost more as the grid orders.
Onsager's 1944 solution puts the square-lattice critical point at
and the spontaneous magnetisation (announced by Onsager in 1949, proved by Yang in 1952) is
It vanishes as (Tc − T)1/8, with infinite slope. A finite grid rounds this off: its correlation length stops at the box size.
Figure 4. Average |M| from 500 Wolff sweeps (after 100 of warm-up) at every 0.1 from 1.0 to 3.5 and at Tc, on 8×8 and 32×32 grids, against Onsager's curve in green. Both overshoot near and past Tc, the 8×8 grid far more.
Most lattice models have no exact solution, and a rounded curve does not say where the transition is. The Binder cumulant does:
It tends to 2/3 in the ordered phase and to 0 in the disordered one. At the critical point the shape of the distribution of M stops depending on size, so curves for different L cross there.
Figure 5. U for L = 8, 16, 32 at eleven temperatures, each from 2,000 Wolff sweeps. The readout interpolates each pair's crossing; the dashed green line is Onsager's Tc. Crossings move by a few hundredths between seeds, most for the smallest pair.
That number is what the lattice sets. The triangular lattice has Tc = 4/ln 3 ≈ 3.64 but the same exponent 1/8, because exponents belong to a universality class fixed by the ℤ₂ symmetry, short-range couplings and two dimensions. Layered magnets such as K2CoF4, whose planes barely interact and whose spins prefer one axis, measure β close to 1/8. Binary alloys like β-brass and the liquid-gas critical point are Ising too, but in the three-dimensional class, with β ≈ 0.326. Weak bond disorder is marginal in 2D by the Harris criterion: the exponents survive with logarithmic corrections.
Metropolis turns a domain over one spin at a time, and near Tc domains fill the box. The cost shows in the autocorrelation ρ(t) of |M| and its integrated time τ = ½ + Σt≥1 ρ(t), roughly the sweeps per independent sample. The Wolff update (1989) grows a cluster by adding each agreeing neighbour with probability 1 − e−2J/kT, the value that makes it satisfy detailed balance, and flips it whole.
Figure 6. ρ(t) of |M| at Tc: Metropolis runs of 20,000, 30,000 and 80,000 sweeps at L = 8, 16, 32, and 4,000 Wolff sweeps at L = 32 (a Wolff sweep is enough clusters to flip N spins). τ is summed until the lag passes 6τ. Equal steps on the log axis mean a power law, τ ∝ Lz; published estimates give z ≈ 2.17 for Metropolis.
Each doubling of L multiplies the Metropolis time by four or five, while Wolff stays under one sweep. That is why Figures 4 and 5 use Wolff.
Two relatives appear in the Emergence series: quench this ferromagnet to zero temperature and its domains coarsen (Coarsening and Consensus); draw each coupling's sign at random and you get the spin glass.