Landau Theory of Phase Transitions

Definition

Landau theory describes a continuous phase transition by expanding the free energy as a power series in an order parameter. The order parameter, written m, is something that is zero in the symmetric (disordered) phase and grows to a nonzero value in the ordered phase — magnetization in a ferromagnet, density difference in a fluid, the pairing amplitude in a superconductor.

Near the transition you don't need the microscopic Hamiltonian. You only need symmetry. For a system symmetric under m → −m (a magnet with no applied field), odd powers are forbidden, leaving:

F(m, T) = F0 + a(T)·m² + b·m⁴

Landau's stroke was to let the quadratic coefficient carry the temperature dependence in the simplest analytic way that can change sign:

a(T) = a0·(T − Tc),   with a0 > 0,  b > 0

So the full Landau free energy is:

F(m, T) = F0 + a0·(T − Tc)·m² + b·m⁴

Everything — the transition, the symmetry breaking, the critical scaling — falls out of how this one curve reshapes as T sweeps through Tc.

How it works — the shape of the curve

The free energy is a function of m; the equilibrium state sits at the value of m that minimizes it. The competition is between two terms:

• The quadratic term a(T)·m² controls the curvature at the origin. If a > 0 it pulls the curve up on both sides — m = 0 is a valley. If a < 0 it pushes the curve down near the origin — m = 0 becomes a hilltop.
• The quartic term b·m⁴ with b > 0 always turns the curve back up at large |m|, keeping the free energy bounded below. It is what catches the order parameter and fixes where the new minima settle.

Above Tc (T > Tc, so a > 0): the curve is a single bowl with its bottom at m = 0. The disordered phase is the only stable state.

At Tc (a = 0): the curve is flat to fourth order at the origin — pure m⁴. Infinitesimally soft. This flatness is what makes fluctuations diverge at the critical point.

Below Tc (T < Tc, so a < 0): the origin becomes a local maximum and two symmetric minima appear at ±m. The curve is now a double well — the celebrated "Mexican hat" in cross-section. The system rolls into one well or the other. The act of choosing is spontaneous symmetry breaking: the equations are symmetric in ±m, but the realized state is not.

Worked example — finding m and the jump in heat capacity

Minimize the free energy. Take the derivative and set it to zero:

dF/dm = 2·a0·(T − Tc)·m + 4·b·m³ = 0
      = 2m·[ a0·(T − Tc) + 2·b·m² ] = 0

Two families of solutions:

• m = 0 — always a stationary point.
• m² = −a0·(T − Tc) / (2b) = a0·(Tc − T) / (2b) — real only when T < Tc.

So below Tc the equilibrium order parameter is:

m(T) = ± √[ a0·(Tc − T) / (2b) ]   ∝   (Tc − T)^(1/2)

This is the mean-field result: the order parameter grows as the square root of how far you are below Tc. The critical exponent is β = 1/2.

Put numbers on it. Suppose a0 = 1, b = 1 (in convenient units), Tc = 100. At T = 99, m = √(1·1/2) = √0.5 ≈ 0.707. Cool further to T = 96: m = √(4/2) = √2 ≈ 1.414. Quadrupling the distance below Tc (1 → 4) only doubles m — that is the square-root law in action. Right at T = 100, m = 0 exactly, and it rises with vertical (infinite) slope, dm/dT → −∞, the signature of a continuous transition.

Substitute m² back to get the equilibrium free energy below Tc:

F_min = F0 − a0²·(Tc − T)² / (4b)

It bends below the m = 0 branch quadratically. Differentiating twice with respect to T gives the heat capacity, which is continuous in value but discontinuous in its derivative — there is a finite jump in specific heat at Tc:

ΔC = a0² · Tc / (2b)   (mean-field specific-heat jump at Tc)

No latent heat is released — entropy is continuous — which is exactly what makes this a second-order (continuous) transition rather than first-order. The classic experimental fingerprint is the lambda-shaped heat-capacity curve near Tc.

Variants and regimes

Variant	Free-energy form	Transition order	Physical example
Standard symmetric	F0 + a(T−Tc)m² + bm⁴, b>0	Second order	Ising ferromagnet, no field
With external field h	… + bm⁴ − h·m	Crossover (no sharp Tc)	Magnet in applied field
Cubic term allowed	… + c·m³ + …	First order	Liquid crystal isotropic→nematic
Tricritical (b → 0)	F0 + a m² + d m⁶, d>0	Tricritical point	He-3/He-4 mixtures
Negative quartic	… − |b|m⁴ + d m⁶	First order (jump in m)	Some structural transitions
Ginzburg–Landau (spatial)	∫[ (∇ψ)²/2 + a|ψ|² + b|ψ|⁴ ] dV	Second order	Superconductors, superfluids
Complex order parameter	a|ψ|² + b|ψ|⁴ (U(1) symmetric)	Second order, continuous symmetry	Superfluid ⁴He, Higgs field

The Ginzburg–Landau version adds a gradient term (∇m)² that penalizes spatial variation of the order parameter. It introduces a length scale — the correlation length ξ ∝ |T − Tc|^(−1/2) in mean field — which diverges at Tc. That divergence is why critical points look the same when zoomed at any scale, and it is the doorway to the renormalization group.

Common pitfalls and misconceptions

• Thinking a(T) = a0(T − Tc) is derived. It isn't — it's the simplest analytic guess that changes sign at Tc. Landau theory is phenomenological. The coefficients a0, b, and Tc are inputs you fit or compute from a microscopic model (e.g. mean-field Ising).
• Forgetting why odd powers vanish. The m and m³ terms drop only because of the m → −m symmetry. If your order parameter doesn't have that symmetry (density, nematic order), a cubic term is allowed and the transition is generically first order.
• Believing β = 1/2 is universal. It is universal within mean field, but real 2D and 3D systems with short-range interactions have different exponents (β ≈ 0.125 in 2D Ising, ≈ 0.326 in 3D Ising). Mean field becomes exact only above the upper critical dimension d = 4.
• Ignoring fluctuations. Landau theory pins the system at the free-energy minimum and ignores thermal fluctuations of m. Near Tc those fluctuations grow without bound. The Ginzburg criterion tells you the temperature window where mean field is untrustworthy.
• Confusing "the symmetric state is unstable" with "it disappears." Below Tc, m = 0 is still a stationary point — it's just a maximum now (or a metastable point in first-order cases). Supercooling and hysteresis come precisely from systems lingering in such metastable states.
• Treating b < 0 as a typo. A negative quartic coefficient is physically meaningful — it signals a first-order transition that must be stabilized by a positive m⁶ term. The point where b passes through zero is a tricritical point.

JavaScript — minimizing the Landau free energy

// Landau free energy F(m, T) = F0 + a0*(T - Tc)*m^2 + b*m^4
function landauF(m, T, { F0 = 0, a0 = 1, b = 1, Tc = 100 } = {}) {
  return F0 + a0 * (T - Tc) * m * m + b * m * m * m * m;
}

// Analytic equilibrium order parameter (the minimizing |m|)
function orderParameter(T, { a0 = 1, b = 1, Tc = 100 } = {}) {
  if (T >= Tc) return 0;                    // disordered phase
  return Math.sqrt(a0 * (Tc - T) / (2 * b)); // m ~ (Tc - T)^(1/2)
}

console.log(orderParameter(100)); // 0       — exactly at Tc
console.log(orderParameter(99));  // 0.7071  — sqrt(0.5)
console.log(orderParameter(96));  // 1.4142  — sqrt(2): 4x deeper, only 2x bigger
console.log(orderParameter(101)); // 0       — above Tc, fully disordered

// Verify the square-root law numerically by brute-force minimization
function minimizeNumerically(T, opts = {}) {
  let best = 0, bestF = Infinity;
  for (let m = -3; m <= 3; m += 0.0005) {
    const f = landauF(m, T, opts);
    if (f < bestF) { bestF = f; best = m; }
  }
  return Math.abs(best);
}

for (const T of [101, 100, 99, 96, 84]) {
  console.log(`T=${T}: analytic=${orderParameter(T).toFixed(4)}` +
              ` numeric=${minimizeNumerically(T).toFixed(4)}`);
}
// Analytic and numeric agree — the double well's minima sit at ±sqrt(a0(Tc-T)/2b)

// Specific-heat jump at Tc: ΔC = a0^2 * Tc / (2b)
function specificHeatJump({ a0 = 1, b = 1, Tc = 100 } = {}) {
  return (a0 * a0 * Tc) / (2 * b);
}
console.log(specificHeatJump()); // 50 — finite discontinuity, no latent heat

Applications

• Ferromagnetism. The order parameter is magnetization. Above the Curie temperature spins point every which way (m = 0); below it they align (m ≠ 0). Iron's Curie point is 1043 K — cool past it and the metal spontaneously magnetizes.
• Superconductivity (Ginzburg–Landau, 1950). The order parameter is a complex pairing wavefunction ψ. The theory predicts the penetration depth, coherence length, and the type-I/type-II distinction — and won Ginzburg a share of the 2003 Nobel Prize. It later grounded the microscopic BCS theory.
• Superfluidity. Liquid ⁴He below 2.17 K develops a complex order parameter with a continuous U(1) symmetry, flowing without viscosity.
• Liquid crystals. The isotropic-to-nematic transition uses a tensor order parameter; symmetry permits a cubic term, so it's first order — exactly what Landau theory predicts and experiment confirms.
• The Higgs mechanism. The Higgs potential μ²|φ|² + λ|φ|⁴ is a Landau free energy for a quantum field. When μ² goes negative, the vacuum spontaneously breaks electroweak symmetry and elementary particles acquire mass — the same Mexican-hat geometry as a ferromagnet.
• Cosmology. Symmetry-breaking phase transitions in the early universe (GUT scale, electroweak scale) are modeled with effective Landau potentials, predicting topological defects like cosmic strings and domain walls.

Derivation and performance analysis

Why is the expansion truncated at m⁴? Because near a continuous transition m is small, so higher powers are negligible — provided b > 0 so the quartic term alone stabilizes the curve. The whole structure rests on the assumption that F is an analytic function of m near the transition, which is precisely what fluctuations can spoil.

The key results, in one place:

Quantity	Mean-field result	Exponent
Order parameter (T < Tc)	m ∝ (Tc − T)^(1/2)	β = 1/2
Susceptibility	χ ∝ |T − Tc|^(−1)	γ = 1
Specific heat	finite jump ΔC = a0²Tc/(2b)	α = 0
Order param. vs field at Tc	m ∝ h^(1/3)	δ = 3
Correlation length	ξ ∝ |T − Tc|^(−1/2)	ν = 1/2
3D Ising (real, for comparison)	m ∝ (Tc − T)^0.326	β ≈ 0.326

Mean-field exponents satisfy scaling relations like Rushbrooke's α + 2β + γ = 2 (0 + 1 + 1 = 2 ✓), so Landau theory is internally consistent — it is simply the wrong limit for low-dimensional systems. The reason it ever works is the upper critical dimension: above d = 4, fluctuations are too dilute to matter and mean field becomes exact. Below it, Wilson's renormalization group (1971 Nobel-winning work) systematically corrects Landau theory by integrating out fluctuations scale by scale, recovering the true exponents.

Computationally, Landau theory is essentially free: minimizing a quartic in one variable is a closed-form cubic root, O(1) per temperature. Its value isn't computational speed — it's conceptual compression. A handful of symmetry arguments and two coefficients reproduce the universal qualitative behavior of magnets, superconductors, superfluids, and the vacuum of the Standard Model.