🧲 Ising Model
Explore ferromagnetic phase transitions with the 2D Ising model and Metropolis Monte Carlo algorithm. Watch spontaneous magnetisation emerge from pure thermal noise as you cool through the critical temperature Tc � 2.269.
The Ising Model: From Spins to Phase Transitions
The Ising model is one of the most important models in theoretical physics � a minimal lattice model that yet exhibits one of the richest phenomena in nature: a second-order phase transition. Every site i on a 2D square lattice carries a binary spin si ? {+1, -1} (representing a magnetic moment pointing up or down). The energy of any spin configuration {s} is:
where J = 1 is the ferromagnetic coupling (aligned spins lower the energy), the first sum is over nearest-neighbour pairs, and h is an external magnetic field. At temperature T, the probability of configuration {s} is the Boltzmann weight e-H/k_BT / Z.
🌡 Two Phases
The system has two distinct thermodynamic phases separated by the critical temperature Tc:
- T < Tc (ordered phase): Thermal fluctuations are too weak to randomise spins. Below Tc, a spontaneous magnetisation M ? 0 appears � all spins prefer to align even with no external field. This is the ferromagnetic phase.
- T > Tc (disordered phase): Thermal noise dominates. Spins fluctuate independently, domains are short-lived, and the time-averaged magnetisation M = 0. This is the paramagnetic phase.
- T = Tc (critical point): Domain clusters exist at every length scale simultaneously � the system is scale-invariant. The correlation length ? ? 8.
📐 Onsager's Exact Solution
In 1944, Lars Onsager derived the exact free energy of the 2D Ising model on the square lattice � one of the most celebrated calculations in mathematical physics. The critical temperature is:
The spontaneous magnetisation was also solved exactly by C. N. Yang (1952):
The exponent 1/8 is a critical exponent � a universal quantity shared by all models in the same universality class.
⚙️ Metropolis Monte Carlo
Because the number of spin configurations (2N�) is astronomically large, we cannot sum the partition function directly. Instead, the Metropolis algorithm (Metropolis et al. 1953) generates a sequence of configurations that samples the Boltzmann distribution:
- Pick a spin at random.
- Compute ?E = 2si(J Snn sj + h).
- If ?E = 0, accept the flip (energetically favourable).
- If ?E > 0, accept with probability exp(-?E/kBT).
This satisfies detailed balance, ensuring the steady-state distribution is the correct Boltzmann distribution. One sweep = N� attempted flips. This simulation uses an optimised inner loop with pre-computed acceptance probabilities for speed.
📊 Critical Exponents & Universality
Near Tc, physical observables diverge or vanish as power laws. The 2D Ising exponents are exact and universal (shared by all 2D Z2-symmetric systems):
| Observable | Power law | Exponent (2D) |
|---|---|---|
| Magnetisation M | (Tc-T)� | � = 1/8 = 0.125 |
| Susceptibility ? | |T-Tc|-? | ? = 7/4 = 1.75 |
| Correlation length ? | |T-Tc|-? | ? = 1 |
| Specific heat C | -ln|T-Tc| | a = 0 (log div.) |
| Correlation fn g(r) | r-(d-2+?) at Tc | ? = 1/4 |
These exponents satisfy the scaling relations 2� + ? = 2 - a and d? = 2 - a (hyperscaling). The universality class concept � that microscopic details are irrelevant near Tc � is one of the deepest unifying principles in physics.
Domain Walls and Symmetry Breaking
Below Tc, the two ground states (+all spins up, -all spins down) are degenerate. The system must choose one � spontaneous symmetry breaking. At finite temperature, domains of opposite magnetisation are separated by domain walls: 1D interfaces whose free energy cost grows with their length. Above Tc, domain walls proliferate and destroy long-range order.
Critical slowing-down: Near Tc, the relaxation time t diverges as t ? ?z where z � 2.17 for the 2D Ising model with local Metropolis dynamics. This is why the simulation appears to freeze up at T � Tc � the system takes exponentially longer to equilibrate, a fundamental computational challenge in Monte Carlo simulation addressed by cluster algorithms (Wolff, Swendsen�Wang) which achieve z � 0.
Real-World Applications
🧲 Ferromagnetism
Iron, cobalt, and nickel become ferromagnetic below their Curie temperature (Fe: 1043 K, Co: 1388 K, Ni: 627 K). The Ising model captures the essential physics: exchange interaction between atomic spins, thermal fluctuations, and the phase transition. Permanent magnets, hard drives, and MRI machines all rely on ferromagnetic order.
🧬 Protein Folding & Biology
Ising-type models extended to describe two-state systems appear throughout biology: neuron firing (on/off), protein conformational states, genetic switches (gene on/off), and population genetics (allele frequency). The universality of the physics means insight transfers across radically different systems.
📈 Economics & Social Systems
Economists use Ising-like models for opinion dynamics (binary opinions), financial market herding (buy/sell), and social contagion. The Schelling segregation model is formally equivalent to a lattice-gas version of Ising. Phase transitions in these models correspond to tipping points � sudden shifts in consensus or market regimes.
🖥 Quantum Computing
The quantum transverse-field Ising model is the basis of D-Wave quantum annealers. Many NP-hard combinatorial optimisation problems can be mapped to finding the ground state of an Ising Hamiltonian: graph colouring, max-cut, travelling salesman, and protein structure prediction. The Ising model is literally the hardware model of quantum advantage.
Phase Diagram and Hysteresis
The 2D Ising model phase diagram in (T, h) space has a critical point at (Tc, 0). For h = 0 and T < Tc, the two phases coexist. For h ? 0, one phase is always preferred and the transition disappears (no singularity). The line of first-order transitions along h = 0, T < Tc terminates at the critical point.
Hysteresis: If you start in the h > 0 phase below Tc, then sweep h ? negative, the system does not immediately reverse its magnetisation � it stays in a metastable state until the field is strong enough to nucleate droplets of the reversed phase. This is the microscopic origin of magnetic hysteresis, which is exploited in magnetic data storage: the bistability between +M and -M represents a classical bit.
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📖 Recommended Reading
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Curriculum Relevance
| Level | Topic | Relevance |
|---|---|---|
| A-Level Physics | Magnetism, thermal physics | Ferromagnetism, phase transitions, Curie temperature |
| IB / AP Physics | Statistical thermodynamics | Boltzmann factor, partition function, entropy |
| Undergraduate Physics | Statistical mechanics | Mean-field theory, critical exponents, renormalisation group |
| Postgraduate | Condensed matter physics | Universality classes, Monte Carlo methods, lattice field theory |