Ising model (2D & 3D)

Experiment

Compare hot and cold starts, change temperature T, coupling J or field H, and watch domains evolve. Positive J favours equal neighbours; negative J favours opposite neighbours. A positive field favours +1, a negative field −1. The triangular antiferromagnet is frustrated: all three bonds around a triangle cannot be satisfied simultaneously.

Model and lattice

E = −J Σ⟨ij⟩ sᵢsⱼ − H Σᵢ sᵢ, with sᵢ = ±1 and each bond counted once. T, J and H share a fixed energy unit, with kB = 1; T is not divided by the adjustable J. Boundaries are periodic. Square: four axial neighbours. Triangular: six neighbours along three equal-length directions; displayed in an oblique cell. Cubic: six axial neighbours. Cubic + diagonals: six axial plus eight body-diagonal neighbours, all coupled by the same J. This last graph is an extended simple-cubic model.

Updates and reproducibility

One sweep makes N random single-spin proposals with replacement. ΔE = 2sᵢ(J Σⱼsⱼ + H). Accept if ΔE ≤ 0, otherwise with probability exp(−ΔE/T); at T = 0 reject positive ΔE and accept neutral moves. Sweeps count Monte Carlo updates, not physical seconds. A fixed random seed reproduces the initial state and updates. Restart restores the initial experiment; Reset clears the result while keeping inputs. Playback uses sweeps per second independently of display frame rate, slowing under load. Hidden tools and tabs suspend it without catching up; reduced motion starts paused.

Measurements

m = Σsᵢ/N is signed; |m| is its magnitude. Time means of m and |m| are distinct. On even bipartite lattices with J < 0, staggered magnetization weights each spin by (−1)^(x+y+z). Each accepted flip updates total magnetization and energy; rejected proposals still occupy Monte Carlo time. Samples begin after the chosen warm-up and are taken at the selected sweep interval. Changing T, J, H or sampling settings clears means and starts warm-up again without replacing the spins. Restart sampling does the same. Acceptance is counted since that measurement reset. Samples can remain correlated; no confidence interval or automatic equilibrium test is implied.

Critical references and limits

For H = 0 and J > 0 on an infinite lattice, square T꜀ = 2J/ln(1+√2), triangular T꜀ = 4J/ln3. The cubic estimate is T꜀ ≈ 4.5115J. No critical preset is supplied for the 14-neighbour graph. Finite lattices have rounded transitions; the square magnetization does not jump discontinuously from zero to one. Single-spin updates can equilibrate very slowly near criticality, at low temperature or under frustration. These are educational finite-lattice experiments, not critical-exponent estimates.

Sources

Potoyan: Ising model and Metropolis sampling; Hinczewski & Berker: triangular critical temperature; Butera & Comi: cubic critical coupling.