PHYS3036 · Condensed Matter and Particle Physics
Landau Theory of Phase Transitions
Lectures 9–11 of University of Sydney PHYS3036 develop Landau's phenomenological theory: expand the free energy as a power series in the order parameter, let symmetry fix which terms are allowed, and minimise. This distinguishes continuous (second-order) from discontinuous (first-order) transitions — an odd (cubic) term or a negative quartic coefficient forces first order — and yields the mean-field critical exponents. Landau derivations are the backbone of the condensed-matter exam, worked with no formula sheet.
What this chapter covers
- 01Landau free energy: F expanded as a power series in the order parameter; symmetries dictate the allowed terms; the global minimum selects the macrostate
- 02Second-order (continuous): purely even expansion with positive quartic coefficient → order parameter turns on continuously, susceptibility diverges
- 03First-order (discontinuous): a cubic (odd) term or a negative quartic coefficient → order parameter jumps, latent heat, coexistence, super-heating/-cooling
- 04Mean-field critical exponents from the standard F = ½A t m² + ¼B m⁴ − h m: β = 1/2, γ = 1, δ = 3, α = 0 (a jump)
- 05Distinguishing features table: order parameter, susceptibility, entropy/latent heat, correlation length, universality — continuous vs discontinuous
- 06Coexistence condition for a first-order transition: equal free energies of the two competing minima, F(Q*) = F(0)
- 07Landau's assumptions and limits: fluctuations ignored; valid for large systems near (but not too near) Tc; fails for d < 4
A cubic term forces a first-order transition (liquid crystal)
- +1Write α ≡ a(T − Tc). At a first-order transition the ordered minimum Q* is degenerate with the disordered minimum at Q = 0, so BOTH F(Q*) = F(0) = 0 and F′(Q*) = 0 hold. [+1]
- +1Degeneracy condition F(Q*) = 0 with Q* ≠ 0: α Q*² − b Q*³ + c Q*⁴ = 0 ⇒ divide by Q*²: α − b Q* + c Q*² = 0. [+1]
- +1Stationarity F′(Q*) = 0: 2α Q* − 3b Q*² + 4c Q*³ = 0 ⇒ divide by Q*: 2α − 3b Q* + 4c Q*² = 0. [+1]
- +1Eliminate α: from the first relation α = b Q* − c Q*². Substitute: 2(b Q* − c Q*²) − 3b Q* + 4c Q*² = −b Q* + 2c Q*² = Q*(2c Q* − b) = 0 ⇒ Q* = b/(2c). [+1]
- +1Back-substitute for the temperature: α = b Q* − c Q*² = b·b/(2c) − c·b²/(4c²) = b²/(2c) − b²/(4c) = b²/(4c). Since α = a(T* − Tc), T* = Tc + b²/(4ac). [+1]
Key terms
- Landau theory
- A phenomenological theory of phase transitions that expands the free energy as a power series in the order parameter, with symmetry-allowed terms, and locates the equilibrium state at the global minimum.
- Second-order (continuous) transition
- A transition where the order parameter turns on continuously from zero (even Landau expansion, positive quartic term); the susceptibility and correlation length diverge and critical exponents are universal.
- First-order (discontinuous) transition
- A transition where the order parameter jumps (driven by an odd cubic term or a negative quartic coefficient); it has latent heat, phase coexistence and super-heating/-cooling.
- Critical exponent
- A power-law exponent (β, γ, δ, α) describing how a quantity behaves near Tc; mean-field/Landau gives β = 1/2, γ = 1, δ = 3, α = 0.
- Coexistence condition
- At a first-order transition the two competing minima have equal free energy, F(Q*) = F(0), so both phases can coexist at T*.
- Reduced temperature (t)
- The dimensionless distance from criticality, t = (T − Tc)/Tc; Landau results are naturally expressed as powers of t.
Landau Theory of Phase Transitions FAQ
How do I tell a first-order from a second-order transition in Landau theory?
Look at the allowed terms. A purely even expansion with a positive quartic coefficient gives a continuous (second-order) transition: the order parameter grows smoothly from zero. An odd (cubic) term, or a quartic coefficient that turns negative, produces a first-order transition: a second minimum appears away from zero and becomes degenerate with it, so the order parameter jumps and there is latent heat.
Why do you impose two conditions at a first-order transition?
Because at the transition temperature the ordered minimum is exactly as low as the disordered one and is a genuine minimum. That is two statements — equal free energies F(Q*) = F(0) and stationarity F′(Q*) = 0 — and solving them together fixes both the jump Q* and the transition temperature T*. In the second-order case one condition (minimising F) suffices because the minimum grows continuously out of zero.
What are the mean-field critical exponents and why do they recur?
From the standard even Landau free energy you get β = 1/2 (order parameter), γ = 1 (susceptibility), δ = 3 (critical isotherm) and α = 0 (a specific-heat jump). They recur because Landau — and mean-field theory — ignore fluctuations, and they are exact above four dimensions; below d = 4 fluctuations shift the true exponents, which is where the renormalization group takes over.
How central is Landau theory to the PHYS3036 exam?
Very. The condensed-matter past papers are dominated by Landau derivations — minimise a free energy, classify the transition, extract exponents or a jump — worked with no formula sheet. Practise the standard even case and at least one cubic/strain-driven first-order case until they are automatic. Confirm the exam's structure and weight on Canvas and the unit outline.
Exam move
Landau theory is where the condensed-matter marks concentrate, so make the machinery reflexive: write the symmetry-allowed expansion, minimise, test F″ > 0 for a genuine minimum, and classify the transition. Drill both templates — the even F = ½A t m² + ¼B m⁴ giving β = 1/2, γ = 1, δ = 3, α = 0, and a cubic or negative-quartic case giving a first-order jump via the two-condition method (equal free energies plus stationarity). Keep a mental table of continuous-vs-discontinuous features (jump, latent heat, divergent vs finite susceptibility, universality). Because no formula sheet is provided, rehearse these derivations from scratch weekly rather than in a STUVAC scramble; they anchor your quiz and final marks and your WAM. When a minimisation snags, ask Sia to check each algebra line and confirm which extremum is the stable one.
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