Ising Explorer — 2D Ferromagnet Phase Transition

Lattice 128×128

Convention: H = −J Σ⟨ij⟩ sᵢsⱼ − H Σ sᵢ
β = 1/T · spins ∈ {+1, −1}

Click the lattice to play / pause · Space also toggles

▶ Start simulation

Temperature T 2.269

External field H 0.000

Size L 128

Algorithm Metropolis

Wolff cluster updates do not satisfy detailed balance for H≠0. Auto-falling back to Metropolis while |H| > 0.

Presets:

|M| / N

—

⟨|Σsᵢ|⟩ / N

E / N

—

−J⟨s·s⟩ − H⟨s⟩

χ (per spin)

—

β(⟨M²⟩−⟨|M|⟩²)/N

C (per spin)

—

β²(⟨E²⟩−⟨E⟩²)/N

Guided experiments

Find Tc by scanning

Anneal 5 → 0.1 and watch χ, C peak near 2.27.

Hysteresis at T<Tc

T=1.8, sweep H from −0.5 → +0.5 → −0.5.

Critical slowing

Compare Metropolis vs Wolff at T=Tc.

Finite-size

Watch χ-peak sharpen as L: 50→200.

What am I looking at?

Each pixel is a spin sᵢ ∈ {+1, −1} (orange = up, teal = down). We evolve toward the Boltzmann distribution P(state) ∝ exp(−E/T) with E = −J Σ⟨ij⟩ sᵢsⱼ − H Σᵢ sᵢ using periodic boundaries. Below Tc the system spontaneously magnetizes — a global sᵢ ↔ −sᵢ symmetry breaks. Above Tc thermal noise wins and ⟨M⟩ → 0.

Metropolis vs Wolff

Metropolis proposes a single spin flip and accepts with min(1, exp(−ΔE/T)). Near Tc the correlation length diverges and single-spin dynamics get stuck (critical slowing down, τ ∼ L^z).

Wolff grows a cluster of aligned spins by adding neighbors with probability p = 1 − exp(−2βJ) and flips the whole cluster. This decorrelates configurations near Tc dramatically faster, but only respects detailed balance for H=0 — so this app disables it when |H|>0.

Onsager's exact result (infinite L, H=0)

For the infinite 2D square-lattice Ising model at zero field, Tc = 2 / ln(1+√2) ≈ 2.26919, and the magnetization below Tc is |m(T)| = (1 − sinh(2β)⁻⁴)^1/8. The dashed curve on the M–T chart is this exact result — finite L rounds it near Tc.

Observables T=2.269, H=0.000

|M|/N vs time (sweeps)

|M|/N

E/N vs time (sweeps)

E/N

|M|(T) — measurements vs Onsager

samples Onsager current