Phase Transitions

1. Ehrenfest classification

Paul Ehrenfest (1933) classified phase transitions by the lowest derivative of the Gibbs free energy G(T, P) that is discontinuous at the transition point:

Modern terminology reserves "phase transition" for first-order transitions and "critical point" or "continuous transition" for second-order ones, though both are colloquially called "phase transitions". Third-order transitions (where G″ is continuous but G‴ is not) are rare — the Heatley superconductor transition and certain liquid-crystal transitions.

2. Order parameter and symmetry breaking

Lev Landau (1937) introduced the concept of an order parameter ψ to quantify the degree of order in a phase. The order parameter is:

• Zero in the disordered (high-temperature, high-symmetry) phase
• Non-zero in the ordered (low-temperature, lower-symmetry) phase
• Continuous at second-order transitions; discontinuous at first-order transitions

Spontaneous symmetry breaking occurs when the ordered phase has lower symmetry than the Hamiltonian (the microscopic energy function). A ferromagnet's Hamiltonian is symmetric under spin flip (H is the same whether all spins are up or down), but below T_c the system chooses one direction of magnetisation — breaking the symmetry spontaneously.

Goldstone's theorem states that when a continuous symmetry (like rotational invariance) is spontaneously broken, massless bosonic excitations called Goldstone modes appear. In magnets these are spin waves (magnons); in superfluids they are phonons.

3. The Ising model

The Ising model is the simplest statistical-mechanical model that exhibits a phase transition. Proposed by Wilhelm Lenz (1920) and named after his student Ernst Ising (who solved the 1D case in 1925), it consists of binary spins σᵢ ∈ {−1, +1} on a lattice with nearest-neighbour interactions:

Exact solutions

1D Ising (Ising 1925): No phase transition at T > 0. The correlation length ξ ~ exp(2J / k_B T) diverges only as T → 0.

2D Ising (Onsager 1944): One of the landmark results of 20th-century physics — an exact free energy for the square lattice with h = 0. The critical temperature is:

3D Ising: No exact solution known. Critical temperature for the simple cubic lattice: k_B T_c / J ≈ 4.5116. Critical exponents computed by renormalisation group and numerical methods: β ≈ 0.326, ν ≈ 0.630, η ≈ 0.036.

4. Mean-field theory and Landau theory

Mean-field theory replaces the actual fluctuating neighbours of a spin with their average value ⟨σ⟩ = M:

Landau free energy expansion

Near the critical point where ψ (order parameter) is small, the free energy can be expanded in powers of ψ by symmetry:

The sign of the coefficient b determines the order of the transition. This is the core of the Landau paradigm: the global physics near a transition follows from the symmetry of the order parameter, not the microscopic details. This universality is why completely different systems can share the same critical exponents.

5. Critical exponents and universality

Near a second-order critical point, all thermodynamic quantities diverge or vanish as power laws in the reduced temperature t = (T − T_c) / T_c:

Universality is the remarkable fact that the critical exponents depend only on: (1) the spatial dimension d and (2) the symmetry of the order parameter (its number of components n). They do not depend on the microscopic details — the lattice type, bond strength, or interaction range. This is why iron and nickel share the same critical exponents despite different electronic structures: both are 3D Ising-class magnets.

6. Renormalisation group — a brief sketch

The renormalisation group (RG), developed by Wilson and Fisher (1972), explains why universality holds and provides a systematic method for computing critical exponents. The core idea is a coarse-graining transformation that zooms out from the microscopic scale to observe the effective long-wavelength theory.

The RG transformation

1. Block spin: Group nearby spins into blocks; replace each block with a single effective spin by majority rule or averaging.
2. Rescale: Restore the original lattice spacing by multiplying all lengths by the blocking factor b.
3. Renormalise: Rescale spin amplitudes to restore normalisation.

Each RG step produces a new effective Hamiltonian with renormalised coupling constants. Iterating the transformation traces a flow in the space of coupling constants. The key objects are:

The deep result is that at the critical point, the system is scale-invariant — it looks the same at every length scale. This is precisely why power laws appear: power laws are the only scale-invariant functions. All microscopic details are irrelevant operators under the RG flow, leaving only the universality class (determined by d and n) to determine the critical exponents.

7. Examples across physics

Ferromagnetic transitions

Below the Curie temperature T_c, ferromagnets develop spontaneous magnetisation even without an external field. The spins align parallel due to quantum exchange interaction. For iron, T_c = 1043 K. At T_c, the correlation length ξ → ∞ — spin fluctuations occur at all length scales, producing the characteristic critical opalescence (visible as large fluctuations if the sample is ferro-fluidically transparent).

Superconductivity

The superconducting transition is a second-order phase transition from normal metal to superconductor at T_c. The order parameter is the macroscopic quantum wavefunction ψ = |ψ|e^{iφ} of Cooper pairs. The broken symmetry is U(1) gauge symmetry — the phase φ is spontaneously chosen. The GL free energy (Ginzburg-Landau) is the Landau expansion in |ψ|²:

Liquid-gas critical point

The liquid-gas transition is first-order below T_c (the liquid and gas coexist at the vapour pressure curve). At the critical point (T_c, P_c), the first-order transition line ends — the distinction between liquid and gas disappears. At the critical point, the order parameter (density difference) vanishes continuously with 3D Ising exponents (β ≈ 0.326), not the mean-field value β = 1/2.

Percolation and social networks

Percolation is a purely geometric phase transition: bonds on a lattice are occupied with probability p. Below a threshold p_c, all connected clusters are finite. Above p_c, a giant connected cluster spanning the entire lattice appears — this is the percolation transition. The cluster size S ~ |p − p_c|^β diverges at p_c with 2D Ising universality class exponents. Percolation theory models forest fire propagation, epidemic spread, and network resilience.

8. JavaScript Monte Carlo simulation

The Metropolis-Hastings algorithm simulates the 2D Ising model by proposing single-spin flips and accepting or rejecting them based on the Boltzmann factor:

// 2D Ising Model — Metropolis Monte Carlo
// canvas: draws the spin lattice; logs magnetisation and energy

const N    = 80;    // lattice size (N × N spins)
const J    = 1.0;   // coupling constant (ferromagnetic)
const kB   = 1.0;   // Boltzmann constant (natural units)
let   T    = 2.5;   // temperature (change to sweep through Tc ≈ 2.269)

// Initialise lattice randomly
let spins = Array.from({ length: N }, () =>
  Array.from({ length: N }, () => (Math.random() < 0.5 ? 1 : -1))
);

// Periodic boundary conditions
const idx = i => ((i % N) + N) % N;

// Energy change for flipping spin (i,j)
function deltaE(i, j) {
  const s    = spins[i][j];
  const sumN = spins[idx(i-1)][j] + spins[idx(i+1)][j]
             + spins[i][idx(j-1)] + spins[i][idx(j+1)];
  return 2 * J * s * sumN;
}

// One Monte Carlo sweep = N² spin flip attempts
function mcSweep() {
  for (let k = 0; k < N * N; k++) {
    const i  = Math.floor(Math.random() * N);
    const j  = Math.floor(Math.random() * N);
    const dE = deltaE(i, j);
    if (dE <= 0 || Math.random() < Math.exp(-dE / (kB * T))) {
      spins[i][j] = -spins[i][j];   // accept flip
    }
  }
}

// Compute magnetisation (order parameter)
function magnetisation() {
  let sum = 0;
  for (let i = 0; i < N; i++)
    for (let j = 0; j < N; j++)
      sum += spins[i][j];
  return Math.abs(sum) / (N * N);
}

// Compute total energy per spin
function energy() {
  let E = 0;
  for (let i = 0; i < N; i++)
    for (let j = 0; j < N; j++)
      E -= J * spins[i][j] * (spins[i][idx(j+1)] + spins[idx(i+1)][j]);
  return E / (N * N);
}

// Render lattice to canvas
function render(canvas) {
  const ctx  = canvas.getContext('2d');
  const cell = canvas.width / N;
  for (let i = 0; i < N; i++) {
    for (let j = 0; j < N; j++) {
      ctx.fillStyle = spins[i][j] === 1 ? '#6366f1' : '#1e1e2e';
      ctx.fillRect(j * cell, i * cell, cell, cell);
    }
  }
}

// Animation loop
function runSimulation(canvas, steps = 100) {
  let step = 0;
  function frame() {
    mcSweep();
    if (++step % 5 === 0) render(canvas);
    if (step < steps) requestAnimationFrame(frame);
  }
  requestAnimationFrame(frame);
}

// Usage: runSimulation(document.getElementById('ising-canvas'));
        

Measuring the phase transition

// Sweep temperature to measure M(T) and identify T_c
async function tempSweep(canvas, Tmin = 1.5, Tmax = 3.5, nT = 20) {
  const results = [];
  const dT = (Tmax - Tmin) / (nT - 1);

  for (let ti = 0; ti < nT; ti++) {
    T = Tmin + ti * dT;

    // Equilibrate
    for (let s = 0; s < 500; s++) mcSweep();

    // Measure (average over 200 sweeps)
    let mSum = 0, eSum = 0;
    for (let s = 0; s < 200; s++) {
      mcSweep();
      mSum += magnetisation();
      eSum += energy();
    }
    results.push({ T, M: mSum / 200, E: eSum / 200 });
    render(canvas);

    // Yield to browser
    await new Promise(r => setTimeout(r, 0));
  }

  return results;
  // results: array of { T, M, E }
  // Plot M vs T to see the second-order phase transition at T_c ≈ 2.269
}

// Binder cumulant (finite-size scaling)
// U_L = 1 − <M⁴> / (3 <M²>²)
// At T_c, U_L is the same for all system sizes L → locate T_c without extrapolation
function binderCumulant(spins) {
  function moment(n) {
    let m = 0;
    for (let i = 0; i < N; i++)
      for (let j = 0; j < N; j++)
        m += spins[i][j];
    m /= N * N;
    return Math.pow(Math.abs(m), n);
  }
  // Average <M²> and <M⁴> over many sweeps for accuracy...
}