Ising Model Visualization

Phase Transitions and Critical Phenomena in Statistical Mechanics - Interactive Monte Carlo Simulation

Wilhelm Lenz (1920) · Ernst Ising (1925) · Lars Onsager (1944)

Temperature (T) 2.27
T/Tc 1.00
Energy E -1.50
Magnetization |M| 0.85
Monte Carlo Steps 0

Simulation Controls

Low 2.27 High
Critical Temperature Tc ≈ 2.269
-2.0 0.0 +2.0
Antiferro -1.0 1.0 Ferro +1.0
20 50 100
1 20 100

Energy Evolution

Magnetization Evolution

Ising Model Theory

The Ising model is one of the most iconic models in statistical mechanics, describing the interaction behavior of spins on a lattice.

Hamiltonian

H = -J Σ<ij> sᵢsⱼ - h Σᵢ sᵢ
  • sᵢ = ±1 - Spin direction (up/down)
  • J - Coupling constant (J>0 ferromagnetic, J<0 antiferromagnetic)
  • h - External magnetic field strength
  • Σ<ij> - Sum over nearest neighbor spins

Milestones

  • 1920 - Wilhelm Lenz proposes the model
  • 1925 - Ernst Ising solves 1D case (no phase transition)
  • 1944 - Lars Onsager exactly solves 2D case (discovers phase transition)

Phase Transition Phenomena

Critical Temperature

Tc = 2/ln(1+√2) ≈ 2.269

Near the critical temperature, the system undergoes a transition from ordered to disordered state.

Three Regimes

Low Temperature T < Tc

Ferromagnetic ordered phase. Spontaneous symmetry breaking, most spins point in the same direction, magnetization |M| > 0.

Critical Point T ≈ Tc

Critical fluctuations. Large-scale clusters appear, critical slowing down, magnetic susceptibility diverges.

High Temperature T > Tc

Paramagnetic disordered phase. Spins randomly oriented, average magnetization M = 0.

Metropolis-Hastings Algorithm

Using Monte Carlo methods to simulate the thermodynamic behavior of the system.

Algorithm Steps

  1. Randomly select a spin sᵢ
  2. Calculate energy change ΔE if flipped
  3. If ΔE ≤ 0, accept the flip
  4. If ΔE > 0, accept with probability exp(-ΔE/kT)
  5. Repeat N×N times for one Monte Carlo step

Acceptance Probability

P(accept) = min(1, e^(-ΔE/kT))
Note: Near the critical point, the system exhibits 'critical slowing down' phenomenon, with significantly slower convergence. Wolff algorithm (cluster flipping) can be used to accelerate.

Observation Guide

Low-T Ferromagnetism (T < 2.0)

Set T ≈ 1.5, observe large regions of same color. This is ferromagnetic ordered state with spontaneous symmetry breaking.

Critical Fluctuations (T ≈ 2.27)

Set T = 2.27, observe formation and death of large-scale clusters. This is the most interesting region!

High-T Paramagnetism (T > 3.0)

Set T ≈ 4.0, observe random spin flips. This is disordered paramagnetic state.

External Field Effect

Adjust external field h, observe bias in spin direction. h > 0 favors up, h < 0 favors down.

Antiferromagnetic Phase (J < 0)

Set J = -1.0, stripe-like antiferromagnetic ordered state forms at low temperature.

Interactive Formula Understanding

H

Total Energy

The Hamiltonian of the system, representing total energy. System tends toward lowest energy state.

-J Σ sᵢsⱼ

Interaction Term

Interaction energy of nearest neighbor spins. J > 0: same direction has lower energy (ferromagnetic); J < 0: opposite direction has lower energy (antiferromagnetic).

-h Σ sᵢ

External Field Term

Energy of external field acting on spins. h > 0: up spins have lower energy; h < 0: down spins have lower energy.