11.06.03 · stat-mech-physics / phase-transitions

Landau Theory: Order Parameter, Symmetry Breaking, and the Free Energy Expansion

shipped3 tiersLean: none

Anchor (Master): Kardar, Statistical Physics of Fields (2007), Ch. 3–4; Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (1992), Ch. 5

Intuition Beginner

Landau's great insight was this: near a phase transition, you do not need to know the microscopic details of a system to predict its behaviour. All you need is a single quantity that tracks the transition — the order parameter — and the symmetry of the system.

Consider a ferromagnet. Above the Curie temperature $T_{c}$ , the spins point randomly in all directions. There is no net magnetisation, and the system is symmetric under rotations — every direction looks the same. Below $T_{c}$ , the spins align and a spontaneous magnetisation $m$ appears. The system has chosen a direction, and the rotational symmetry is broken. The magnetisation $m$ is the order parameter: it is zero in the disordered (high-temperature) phase and non-zero in the ordered (low-temperature) phase.

Landau asked: what is the simplest free energy function $F (m)$ that captures this physics? It must satisfy two constraints. First, $F$ must be symmetric under $m \to - m$ (flipping all spins costs no energy when there is no external field). Second, $F$ must have a single minimum at $m = 0$ above $T_{c}$ (disordered) and two minima at $m \neq = 0$ below $T_{c}$ (ordered).

The simplest such function is:

F (m) = a (T - T_{c}) m^{2} + b m^{4},

where $a > 0$ and $b > 0$ are constants. Above $T_{c}$ , the coefficient $a (T - T_{c})$ is positive, so both terms are positive for $m \neq = 0$ . The minimum is at $m = 0$ . Below $T_{c}$ , the coefficient $a (T - T_{c})$ becomes negative, so the $m^{2}$ term wants to push $m$ away from zero. The $m^{4}$ term (always positive) prevents $m$ from running away to infinity. The result is a "double-well" potential with two minima at $m = \pm a (T_{c} - T) / (2 b)$ .

The order parameter grows as the square root of the temperature distance from $T_{c}$ : $m \sim T_{c} - T$ . This is the Landau prediction for the critical exponent $β = 1/2$ , shared by all systems in the same universality class.

The symmetry-breaking perspective is the deepest part of Landau's framework. In the high-temperature phase, the free energy has the full symmetry of the Hamiltonian (e.g., $m \to - m$ for an Ising magnet, rotational symmetry for a Heisenberg magnet). Below $T_{c}$ , the system must pick one of the degenerate minima. The free energy retains the full symmetry (both wells are present), but the physical state does not — the system sits in one well, not both. The symmetry of the Hamiltonian is hidden (spontaneously broken) in the ordered phase.

This idea extends far beyond magnets. In a liquid-gas transition, the density difference between liquid and gas is the order parameter. In a superfluid, the condensate wavefunction is the order parameter. In a superconductor, the Cooper-pair wavefunction plays this role. Landau theory provides a unified description of all these transitions through the order parameter and its symmetry.

An important distinction is between discrete and continuous symmetry breaking. In the Ising model, the symmetry is $Z_{2}$ (flip all spins): the order parameter can point "up" or "down," and there are exactly two equivalent ordered states. In the Heisenberg model, the symmetry is continuous (rotations in three-dimensional space): the magnetisation vector can point in any direction, and the ordered states form a continuous sphere of equivalent possibilities. The number of equivalent states matters for the physics of the ordered phase — it determines the number of soft (low-energy) excitations called Goldstone modes, which we explore in the advanced section.

The Landau approach is sometimes called a phenomenological theory because it does not derive the free energy from a microscopic Hamiltonian. Instead, it constructs the free energy from general principles (symmetry, analyticity) and fits the few remaining parameters ( $a$ , $b$ , $T_{c}$ ) to experiment. This is a strength and a weakness: it makes Landau theory universally applicable, but it also means the theory cannot predict $T_{c}$ or the numerical values of the coefficients from first principles. That requires connecting back to a microscopic model like the Ising model 11.06.01 or the Curie-Weiss model 11.06.02.

Visual Beginner

The Landau free energy $F (m) = a (T - T_{c}) m^{2} + b m^{4}$ has a characteristic shape that changes at $T_{c}$ .

For $T > T_{c}$ : $F$ is a single upward-opening parabola (modified by the $m^{4}$ term). The unique minimum is at $m = 0$ .

For $T = T_{c}$ : $F = b m^{4}$ — a very flat minimum at the origin. Small perturbations produce large responses (critical fluctuations).

For $T < T_{c}$ : $F$ has two minima at $m = \pm m_{0}$ separated by a barrier at $m = 0$ . As $T$ decreases further below $T_{c}$ , the minima move outward and the barrier grows.

Worked example Beginner

A ferromagnet has Landau free energy $F (m) = 2 (T - T_{c}) m^{2} + 3 m^{4}$ (in units where $k = 1$ ), with $T_{c} = 100 K$ .

At $T = 120 K$ (above $T_{c}$ ): The coefficient of $m^{2}$ is $2 (120 - 100) = 40 > 0$ . Both terms are positive for $m \neq = 0$ . The minimum is at $m = 0$ . No spontaneous magnetisation — the paramagnetic phase.

At $T = 100 K$ (exactly $T_{c}$ ): The $m^{2}$ coefficient is zero. $F (m) = 3 m^{4}$ . The minimum is still at $m = 0$ , but the curvature there is zero — the system is "soft" and responds strongly to any perturbation.

At $T = 80 K$ (below $T_{c}$ ): The coefficient of $m^{2}$ is $2 (80 - 100) = - 40 < 0$ . Setting $d F / d m = 2 \cdot (- 40) m + 12 m^{3} = - 80 m + 12 m^{3} = 0$ : either $m = 0$ (now a maximum, not a minimum) or $m^{2} = 80/12 = 20/3$ , giving $m = \pm 20/3 \approx \pm 2.58$ . The system spontaneously magnetises. The magnet chooses one of these two values — it breaks the $m \to - m$ symmetry.

Check your understanding Beginner

Formal definition Intermediate+

The Landau expansion

The Landau free energy is a phenomenological expansion of the Helmholtz free energy in powers of the order parameter $η$ :

F (η, T) = F_{0} (T) + a_{2} (T) η^{2} + a_{4} (T) η^{4} + a_{6} (T) η^{6} + \dots - h η,

where $h$ is the field conjugate to the order parameter (external magnetic field for a ferromagnet, chemical potential difference for a fluid).

Symmetry constraints. The form of the expansion is dictated by the symmetry group $G$ of the disordered phase. For a system with $Z_{2}$ symmetry (Ising universality class), the free energy must be invariant under $η \to - η$ . This eliminates all odd powers of $η$ (when $h = 0$ ): the terms $η$ , $η^{3}$ , $η^{5}$ , and so on are forbidden. For a system with continuous $O (n)$ symmetry, the free energy must be invariant under rotations of the $n$ -component order parameter, so it depends only on $∣ η ∣^{2}$ and higher rotational invariants.

Temperature dependence. The key assumption of Landau theory is that the coefficients $a_{k} (T)$ are smooth (analytic) functions of temperature near $T_{c}$ . The phase transition is driven by the sign change of the quadratic coefficient:

a_{2} (T) = a (T - T_{c}), a > 0.

The quartic coefficient $a_{4} = b > 0$ is taken to be constant (or weakly temperature-dependent). Higher-order terms ( $η^{6}$ and beyond) are neglected near $T_{c}$ where $η$ is small.

The resulting Landau free energy for a system with $Z_{2}$ symmetry is:

F (η) = a (T - T_{c}) η^{2} + b η^{4} - h η, a > 0, b > 0.

Second-order (continuous) transitions

For $b > 0$ and $h = 0$ : when $T$ drops below $T_{c}$ , the quadratic coefficient changes sign, and the minimum of $F$ moves continuously from $η = 0$ to $η = \pm a (T_{c} - T) / (2 b)$ . The order parameter grows as $(T_{c} - T)^{1/2}$ . There is no latent heat — the transition is continuous. This is a second-order phase transition in the Ehrenfest classification.

First-order transitions

If $b < 0$ , the quartic term is destabilising, and the $η^{6}$ term (with $c > 0$ ) must be included to prevent $F \to - \infty$ :

F (η) = a (T - T_{c}) η^{2} + b η^{4} + c η^{6}, b < 0, c > 0.

In this case, the order parameter jumps discontinuously at some temperature $T^{*} \neq = T_{c}$ , and there is a latent heat. This is a first-order phase transition. The sign of $b$ determines whether the transition is first or second order. The point in parameter space where $b = 0$ (so the transition switches from second to first order) is called a tricritical point.

Critical exponents from Landau theory

From the Landau free energy $F = a (T - T_{c}) η^{2} + b η^{4} - h η$ , the four standard critical exponents are:

Exponent	Observable	Landau prediction
$α$	Specific heat $C$	$α = 0$ (jump, not divergence)
$β$	Order parameter $η$	$η \sim (T_{c} - T)^{1/2}$ , $β = 1/2$
$γ$	Susceptibility $χ$	$χ \sim ∣ T - T_{c} ∣^{- 1}$ , $γ = 1$
$δ$	Critical isotherm	$h \sim η^{3}$ , $δ = 3$

Comparison with exact and numerical results

Landau theory gives the same exponents as all mean-field theories (Curie-Weiss, van der Waals, Bragg-Williams). These are exact for the Curie-Weiss model (infinite-range Ising) and for short-range models in dimension $d \geq 4$ . In $d = 3$ , the exact exponents differ substantially:

Exponent	Landau	3D Ising (numerical)	2D Ising (exact)
$α$	0	0.110	0 (log)
$β$	1/2	0.326	1/8
$γ$	1	1.237	7/4
$δ$	3	4.790	15

The discrepancy arises because Landau theory neglects fluctuations of the order parameter. In low dimensions, fluctuations are strong and modify the critical behaviour. The renormalization group 11.07.01 explains when and why Landau theory fails.

Relation between Landau theory and mean-field theory

Landau theory and mean-field theory (as developed in 11.06.02 for the Curie-Weiss and van der Waals models) give the same critical exponents. This is not a coincidence: they are two different routes to the same approximation. Mean-field theory starts from a microscopic Hamiltonian, replaces interactions by their average, and derives a self-consistency equation. Landau theory starts from symmetry, writes the most general free energy consistent with that symmetry, and minimises it. The self-consistency equation of mean-field theory and the minimisation of the Landau free energy are mathematically equivalent near $T_{c}$ .

The difference is one of emphasis and generality. Mean-field theory is tied to a specific microscopic model; Landau theory applies to any system with the same symmetry. The mean-field approach tells you why the quadratic coefficient changes sign at $T_{c}$ (because the interaction energy overcomes the entropy); the Landau approach tells you what follows from that sign change (universality, power laws, scaling relations). Both perspectives are valuable, and the connection between them is the content of the Hubbard-Stratonovich transformation that was introduced in 11.06.02.

Key derivation: Critical exponents from the Landau free energy Intermediate+

Proposition. The Landau free energy $F (η) = a (T - T_{c}) η^{2} + b η^{4} - h η$ with $a, b > 0$ yields the critical exponents $β = 1/2$ , $γ = 1$ , $δ = 3$ , $α = 0$ .

Proof. The equilibrium order parameter minimises $F$ : $\partial F / \partial η = 2 a (T - T_{c}) η + 4 b η^{3} - h = 0$ .

Exponent $β$ ( $h = 0$ , $T < T_{c}$ ). The equation becomes $2 a (T - T_{c}) η + 4 b η^{3} = 0$ . For $η \neq = 0$ : $η^{2} = a (T_{c} - T) / (2 b)$ , so $η = \pm a (T_{c} - T) / (2 b) \sim (T_{c} - T)^{1/2}$ , giving $β = 1/2$ .

Exponent $γ$ ( $h = 0$ , $T > T_{c}$ ). The susceptibility is $χ = (\partial η / \partial h)_{h = 0}$ . Linearise the equilibrium equation in $h$ : $2 a (T - T_{c}) χ + 0 = 1$ (the $4 b η^{3}$ term vanishes at $η = 0$ ). So $χ = 1/ (2 a (T - T_{c})) \sim (T - T_{c})^{- 1}$ , giving $γ = 1$ . For $T < T_{c}$ , evaluating the second derivative at the minimum $η_{0} = a (T_{c} - T) / (2 b)$ gives $χ = 1/ (4 a (T_{c} - T)) \sim (T_{c} - T)^{- 1}$ , so $γ = 1$ on both sides.

Exponent $δ$ ( $T = T_{c}$ ). The equilibrium equation at $T = T_{c}$ is $4 b η^{3} = h$ , so $h \sim η^{3}$ , giving $δ = 3$ .

Exponent $α$ ( $h = 0$ ). The specific heat is $C = - T \partial^{2} F_{min} / \partial T^{2}$ . For $T > T_{c}$ : $η = 0$ and $F_{min} = F_{0} (T)$ , so $C = - T d^{2} F_{0} / d T^{2}$ — no singularity. For $T < T_{c}$ : substituting $η^{2} = a (T_{c} - T) / (2 b)$ into $F$ gives $F_{min} = - a^{2} (T_{c} - T)^{2} / (4 b) + F_{0} (T)$ . The second temperature derivative gives $C = - T d^{2} F_{0} / d T^{2} + T a^{2} / (2 b)$ — again no singularity, but a finite jump $Δ C = T_{c} a^{2} / (2 b)$ at $T_{c}$ . The exponent $α = 0$ (finite discontinuity). $□$

Bridge. The Landau critical exponents are identical to the mean-field exponents derived from the Curie-Weiss and van der Waals models 11.06.02. This is because Landau theory is mean-field theory in a different language: expanding the free energy in powers of the order parameter and minimising is equivalent to solving the mean-field self-consistency equation. The difference is that Landau theory makes the symmetry and universality manifest, while the self-consistency approach hides them in the microscopic details. The Ginzburg criterion 11.06.04 determines when this approach is self-consistent by comparing fluctuation magnitudes to the mean-field order parameter. The renormalization group 11.07.01 then provides the systematic correction for dimensions where fluctuations dominate.

Exercises Intermediate+

Exercise 6 (medium, symbolic).

A system has Landau free energy $F (η) = a (T - T_{c}) η^{2} + b η^{4} + c η^{6}$ with $a > 0$ , $c > 0$ , and $b < 0$ (the first-order case). Find the temperature $T^{*}$ at which the order parameter jumps discontinuously. Express $T^{*}$ in terms of $a$ , $b$ , $c$ , and $T_{c}$ .

Hint

A first-order transition occurs when the free energy of the ordered phase ( $η \neq = 0$ ) becomes equal to the free energy of the disordered phase ( $η = 0$ ). Set $F (η^{*}) = F (0) = 0$ where $η^{*}$ is the non-zero minimum of $F$ .

Answer

The equilibrium condition $\partial F / \partial η = 0$ gives $2 a (T - T_{c}) η + 4 b η^{3} + 6 c η^{5} = 0$ . For $η \neq = 0$ : $a (T - T_{c}) + 2 b η^{2} + 3 c η^{4} = 0$ , which is quadratic in $η^{2}$ :

η^{2} = \frac{- 2 b \pm 4 b ^{2} - 12 c \cdot a ( T - T _{c} )}{6 c} .

The non-zero minimum exists when $T < T_{c} - b^{2} / (3 a c)$ . At the first-order transition, the free energies of the two phases are equal: $F (η^{*}) = 0$ , where $η^{*}$ is the non-zero minimum. Substituting the equilibrium condition into $F (η^{*}) = a (T^{*} - T_{c}) (η^{*})^{2} + b (η^{*})^{4} + c (η^{*})^{6} = 0$ and using $a (T^{*} - T_{c}) = - 2 b (η^{*})^{2} - 3 c (η^{*})^{4}$ :

(- 2 b (η^{*})^{2} - 3 c (η^{*})^{4}) (η^{*})^{2} + b (η^{*})^{4} + c (η^{*})^{6} = - b (η^{*})^{4} - 2 c (η^{*})^{6} = 0.

For $η^{*} \neq = 0$ : $(η^{*})^{2} = - b / (2 c)$ . Substituting back: $a (T^{*} - T_{c}) = - 2 b \cdot (- b / (2 c)) - 3 c \cdot b^{2} / (4 c^{2}) = b^{2} / c - 3 b^{2} / (4 c) = b^{2} / (4 c)$ .

So $T^{*} = T_{c} + b^{2} / (4 a c)$ . Since $b < 0$ , $b^{2} > 0$ and $T^{*} > T_{c}$ . The first-order transition occurs above $T_{c}$ .

Exercise 7 (medium, symbolic).

Verify that the Landau critical exponents satisfy the four Rushbrooke, Widom, Fisher, and Josephson scaling relations: $α + 2 β + γ = 2$ , $γ = β (δ - 1)$ , $γ = (2 - η) ν$ , and $d ν = 2 - α$ (the last two require $ν = 1/2$ and $η = 0$ from mean-field theory). Identify which relation fails in $d < 4$ .

Hint

Substitute the Landau exponents $α = 0$ , $β = 1/2$ , $γ = 1$ , $δ = 3$ , $ν = 1/2$ , $η = 0$ into each relation and check consistency. The Josephson relation involves the spatial dimension $d$ .

Answer

Rushbrooke: $α + 2 β + γ = 0 + 1 + 1 = 2$ . Satisfied.

Widom: $γ = β (δ - 1) = (1/2) (3 - 1) = 1$ . Satisfied.

Fisher: $γ = (2 - η) ν = (2 - 0) (1/2) = 1$ . Satisfied.

Josephson (hyperscaling): $d ν = 2 - α$ , so $d \cdot (1/2) = 2 - 0 = 2$ , giving $d = 4$ . The hyperscaling relation is satisfied only at $d = 4$ — this is the upper critical dimension. For $d < 4$ , the left side is $d /2 < 2$ while the right side is $2$ , so hyperscaling fails with mean-field exponents. This failure signals that fluctuations (neglected by Landau theory) modify the critical behaviour in $d < 4$ . The renormalization group 11.07.01 provides the corrected exponents that do satisfy hyperscaling.

Exercise 8 (hard, symbolic).

Consider an $O (n)$ -symmetric Landau free energy $F (η) = a (T - T_{c}) ∣ η ∣^{2} + b ∣ η ∣^{4}$ where $η$ is an $n$ -component vector. Show that the equilibrium condition gives $∣ η ∣^{2} = a (T_{c} - T) / (2 b)$ for $T < T_{c}$ , with the direction of $η$ undetermined (the Goldstone degeneracy). Compute the number of Goldstone modes for general $n$ and explain why the $n = 1$ (Ising) case is special.

Hint

Write $η$ in polar coordinates: $∣ η ∣$ and $n - 1$ angular variables. The free energy depends only on $∣ η ∣$ , so all angular directions are degenerate. The number of Goldstone modes equals the number of broken symmetry generators: $dim O (n) - dim O (n - 1) = n (n - 1) /2 - (n - 1) (n - 2) /2 = n - 1$ .

Answer

Write $∣ η ∣ = ρ$ . The free energy $F = a (T - T_{c}) ρ^{2} + b ρ^{4}$ depends only on $ρ$ . The equilibrium condition $d F / d ρ = 2 a (T - T_{c}) ρ + 4 b ρ^{3} = 0$ gives $ρ_{0}^{2} = a (T_{c} - T) / (2 b)$ for $T < T_{c}$ .

The direction of $η$ on the sphere $S^{n - 1}$ of radius $ρ_{0}$ is completely free — the free energy is independent of all $n - 1$ angular coordinates. These $n - 1$ flat directions in the free energy landscape are the Goldstone modes. They correspond to long-wavelength fluctuations of the order parameter direction that cost vanishingly little energy as the wavelength goes to infinity.

For the Ising case ( $n = 1$ ): there are $n - 1 = 0$ Goldstone modes. The order parameter is a scalar with only a discrete $Z_{2}$ symmetry ( $η \to - η$ ), and the two minima are isolated points rather than a continuous manifold. There is no flat direction.

For the XY model ( $n = 2$ ): 1 Goldstone mode (the phase angle of the order parameter). For the Heisenberg model ( $n = 3$ ): 2 Goldstone modes. The Mermin-Wagner theorem 11.06.01 states that continuous symmetries cannot be spontaneously broken in $d \leq 2$ at finite temperature — Goldstone modes destroy long-range order in low dimensions.

Lean formalization Intermediate+

The Landau expansion requires: (1) a formal order-parameter type with a symmetry group acting on it; (2) the free energy as a polynomial in the order parameter with coefficients constrained by invariance under the symmetry group; (3) a proof that the polynomial minimum changes character (from unique to degenerate) at $T_{c}$ ; (4) extraction of critical exponents from the asymptotic behaviour near $T_{c}$ . The polynomial algebra and extremum analysis are within Mathlib's scope, but the physical framework (order parameter, symmetry breaking, critical exponents) requires a domain-specific construction. The gap between what Mathlib provides and what a formalisation needs is substantial but well-defined.

Advanced results Master

The Ginzburg-Landau functional

Landau theory treats the order parameter as spatially uniform. To include spatial variations, Ginzburg and Landau (1950) introduced a functional that depends on a spatially varying field $η (r)$ :

F [η] = \int d^{d} r [\frac{κ}{2} ∣\nabla η ∣^{2} + a (T - T_{c}) ∣ η ∣^{2} + b ∣ η ∣^{4}],

where $κ > 0$ is the stiffness (gradient energy) coefficient. The gradient term penalises spatial variations of the order parameter, and its presence distinguishes the Ginzburg-Landau functional from the spatially uniform Landau free energy.

Minimising $F$ gives the Ginzburg-Landau equation:

- κ \nabla^{2} η + 2 a (T - T_{c}) η + 4 b ∣ η ∣^{2} η = 0.

For $T > T_{c}$ : the uniform solution $η = 0$ is stable. For $T < T_{c}$ : the uniform solution $η_{0} = a (T_{c} - T) / (2 b)$ is stable, but there are also non-uniform solutions (domain walls, vortices, and other topological defects) that play an important role in the physics of the ordered phase.

Correlation length

Linearising the Ginzburg-Landau equation around $η = 0$ for $T > T_{c}$ gives the correlation function:

G (r) = ⟨ η (r) η (0)⟩ \sim \frac{e ^{- r / ξ}}{r ^{(d - 2) /2}},

where the correlation length is

ξ = \frac{κ}{2 a ( T - T _{c} )} \sim ∣ T - T_{c} ∣^{- 1/2},

giving the mean-field exponent $ν = 1/2$ . The correlation length diverges at $T_{c}$ , signalling the onset of long-range correlated fluctuations. The number of correlated degrees of freedom in a correlation volume is $ξ^{d} \sim ∣ T - T_{c} ∣^{- d /2}$ .

The upper critical dimension

The Ginzburg-Landau functional can be analysed by separating the field into a uniform (mean-field) part and fluctuations. The mean-square fluctuation of the order parameter in a correlation volume is:

⟨(δ η)^{2} ⟩ \sim \frac{k _{B} T}{κ} ξ^{2 - d} \sim ∣ T - T_{c} ∣^{(d - 4) /2} .

Mean-field theory is self-consistent when fluctuations are small compared to the mean-field order parameter $η_{0}^{2} \sim ∣ T - T_{c} ∣$ :

∣ T - T_{c} ∣^{(d - 4) /2} ≪ ∣ T - T_{c} ∣ ⟹ ∣ T - T_{c} ∣^{(d - 4) /2} ≫ 1.

This is the Ginzburg criterion (developed in detail in 11.06.04). For $d > 4$ : the left side diverges as $T \to T_{c}$ , so mean-field theory is always valid near $T_{c}$ . For $d < 4$ : the left side vanishes as $T \to T_{c}$ , so mean-field theory always fails sufficiently close to $T_{c}$ . The upper critical dimension is $d_{c} = 4$ .

At $d = 4$ : the criterion is marginal. Mean-field theory gives the leading behaviour, with logarithmic corrections. The renormalization group 11.07.01 provides a systematic framework for computing these corrections.

Goldstone modes and continuous symmetry breaking

When the order parameter has a continuous symmetry group $G$ that is broken to a subgroup $H$ below $T_{c}$ , the number of Goldstone modes equals the number of broken generators: $n_{G} = dim G - dim H$ . For $O (n)$ symmetry broken to $O (n - 1)$ : $n_{G} = n - 1$ .

The Goldstone modes are massless excitations (their energy vanishes as the wavevector $q \to 0$ ). In the Ginzburg-Landau framework, they appear as flat directions in the free energy landscape: the free energy depends on $∣ η ∣$ but not on the direction of $η$ in the ordered phase. The dispersion relation for Goldstone modes is $ω (q) \sim ∣ q ∣$ (linear, like phonons) for most systems.

The Mermin-Wagner theorem states that in $d \leq 2$ , Goldstone modes are so strong that they destroy long-range order at any finite temperature. The 2D XY model exhibits a Kosterlitz-Thouless transition (a topological phase transition driven by vortex unbinding) but no true long-range order. The Heisenberg model ( $n = 3$ ) in $d = 2$ has no phase transition at all. The essential mechanism is that the Goldstone mode correlation function $⟨ π_{i} (r) π_{i} (0)⟩ \sim ln (r)$ in $d = 2$ grows without bound, so the ordered-state fluctuations are too large to sustain a fixed direction for the order parameter. In $d = 1$ , the divergence is even stronger ( $\sim r$ ), destroying order at any temperature. Only in $d \geq 3$ do the Goldstone mode correlations remain bounded, allowing true long-range order to persist.

The Goldstone theorem has deep analogues in particle physics. In the standard model, the Higgs field undergoes spontaneous symmetry breaking from $S U (2) \times U (1)$ to $U (1)$ , producing three Goldstone bosons that are "eaten" by the $W^{\pm}$ and $Z$ gauge bosons (giving them mass) and one massive radial mode (the Higgs boson). The mathematical structure is identical to the Ginzburg-Landau functional in superconductivity — a connection made precise by the Anderson-Higgs mechanism.

Limitations of Landau theory

Landau theory has three principal limitations:

Neglect of fluctuations. The order parameter is treated as a classical variable that minimises the free energy. Thermal and quantum fluctuations are ignored. This is a good approximation far from $T_{c}$ (where the correlation length is small) and in high dimensions ( $d > 4$ , where the Ginzburg criterion is satisfied), but fails near $T_{c}$ in $d < 4$ .
Assumption of analyticity. The expansion assumes the free energy is analytic in $η$ near $η = 0$ . This is correct above $T_{c}$ (where $η = 0$ at equilibrium) but is an uncontrolled approximation at $T_{c}$ itself, where the order parameter is small and fluctuations are large. The renormalization group shows that the free energy can develop non-analytic behaviour at $T_{c}$ in $d < 4$ .
Inability to determine $T_{c}$ from first principles. Landau theory is phenomenological: $T_{c}$ and the coefficients $a$ , $b$ are taken as inputs. They can be computed from a microscopic theory (e.g., the Curie-Weiss model gives $T_{c} = J z / k$ ) but not from Landau theory itself.

Synthesis. Landau theory is the phenomenological counterpart of mean-field theory 11.06.02: where mean-field theory starts from a microscopic Hamiltonian and derives an effective self-consistency equation, Landau theory starts from symmetry and constructs the most general free energy consistent with those symmetries. Both give the same critical exponents. The Ginzburg-Landau extension adds spatial gradients, enabling the study of domain walls, vortices, and correlation functions. The Ginzburg criterion 11.06.04 then determines the regime of validity. The renormalization group [11.07.01, 11.07.02] provides the complete framework that supersedes Landau theory by treating fluctuations systematically through scale-dependent coupling constants. The passage from Landau theory to the renormalization group is the central conceptual advance of late twentieth-century statistical physics: it replaces the idea that the free energy is a single analytic function of the order parameter with the idea that the free energy is a scale-dependent object whose form changes as you zoom in on the critical point.

Full proof set Master

Proposition. The Ginzburg-Landau functional $F [η] = \int d^{d} r [(κ /2) ∣\nabla η ∣^{2} + a (T - T_{c}) ∣ η ∣^{2} + b ∣ η ∣^{4}]$ yields the correlation length $ξ \sim ∣ T - T_{c} ∣^{- 1/2}$ and the criterion for the breakdown of Landau theory in dimension $d$ .

Proof. Expand the field around the mean-field value: $η (r) = η_{0} + δ η (r)$ . For $T > T_{c}$ , $η_{0} = 0$ and the quadratic part of $F$ in Fourier space is:

F^{(2)} = \frac{1}{2} \int \frac{d ^{d} q}{( 2 π ) ^{d}} (κ q^{2} + 2 a (T - T_{c})) ∣ δ \tilde{η} (q) ∣^{2} .

The correlation function in Fourier space is:

⟨ ∣ δ \tilde{η} (q) ∣^{2} ⟩ = \frac{k _{B} T}{κ q ^{2} + 2 a ( T - T _{c} )} .

Inverse Fourier transforming gives $G (r) \sim e^{- r / ξ} / r^{(d - 2) /2}$ with $ξ = κ / (2 a (T - T_{c})) \sim ∣ T - T_{c} ∣^{- 1/2}$ .

The total mean-square fluctuation in a correlation volume $ξ^{d}$ is:

⟨(δ η)^{2} ⟩_{vol} = \int_{∣ q ∣ < ξ^{- 1}} \frac{d ^{d} q}{( 2 π ) ^{d}} \frac{k _{B} T}{κ q ^{2} + 2 a ( T - T _{c} )} \sim ξ^{2 - d} \sim ∣ T - T_{c} ∣^{(d - 2) /2} .

Landau theory is self-consistent when this is small compared to $η_{0}^{2} \sim ∣ T - T_{c} ∣$ : $∣ T - T_{c} ∣^{(d - 4) /2} ≫ 1$ , which requires $d > 4$ as $T \to T_{c}$ . $□$

Proposition (Goldstone's theorem, Landau theory version). For a Ginzburg-Landau functional with $O (n)$ symmetry, spontaneous symmetry breaking from $O (n)$ to $O (n - 1)$ produces $n - 1$ massless (Goldstone) modes.

Proof. The ordered state satisfies $∣ η ∣ = η_{0} \neq = 0$ with arbitrary direction. Choose $η_{0} = η_{0} \overset{e}{^}_{1}$ (ordering along the first component). Write $η = (η_{0} + σ) \overset{e}{^}_{1} + \sum_{i = 2}^{n} π_{i} \overset{e}{^}_{i}$ where $σ$ is the radial (Higgs) mode and $π_{i}$ are the angular (Goldstone) modes. The quadratic part of $F$ in Fourier space is:

F^{(2)} = \frac{1}{2} \int \frac{d ^{d} q}{( 2 π ) ^{d}} [(κ q^{2} + m_{σ}^{2}) ∣ \tilde{σ} ∣^{2} + i = 2 \sum n κ q^{2} ∣ \tilde{π}_{i} ∣^{2}],

where $m_{σ}^{2} = 8 b η_{0}^{2} = 4 a (T_{c} - T) > 0$ is the mass of the radial mode. The Goldstone modes $π_{i}$ have no mass term ( $κ q^{2}$ only), so their correlation function $⟨ \tilde{π}_{i} (q) \tilde{π}_{i} (- q)⟩ = k_{B} T / (κ q^{2})$ diverges as $q \to 0$ . There are $n - 1$ such modes. $□$

Connections Master

11.06.01 The Ising model is the microscopic foundation; Landau theory is the phenomenological framework that extracts universal behaviour from the symmetry of the order parameter. The exact 2D Ising solution confirms Landau's symmetry arguments but gives different exponents, demonstrating the role of fluctuations.
11.06.02 Mean-field theory (Curie-Weiss, van der Waals) and Landau theory give identical critical exponents. Landau theory is the more general framework: it applies to any system with a known order parameter and symmetry, while mean-field theory requires a specific microscopic Hamiltonian.
11.06.04 The Ginzburg criterion determines the regime of validity of Landau theory by comparing fluctuation magnitudes to the mean-field order parameter. The criterion identifies $d_{c} = 4$ as the upper critical dimension for the Ising universality class.
11.07.01 The renormalization group generalises Landau theory by introducing scale-dependent coupling constants. The Gaussian fixed point of the RG reproduces Landau theory; the Wilson-Fisher fixed point provides the corrected exponents for $d < 4$ .
11.07.02 Block-spin renormalization of the Ginzburg-Landau Hamiltonian yields the RG flow equations. The Gaussian fixed point (Landau theory) is stable for $d > 4$ and unstable for $d < 4$ , where a new (Wilson-Fisher) fixed point takes over.

Historical and philosophical context Master

Lev Landau introduced his theory of phase transitions in 1937, during one of the most creative periods of his career (he was 29 years old). His insight was that the universal features of phase transitions — the existence of an order parameter, the breaking of symmetry, the power-law behaviour near $T_{c}$ — could be derived from general principles without reference to any specific microscopic model. The key inputs were the symmetry of the disordered phase and the assumption that the free energy is analytic in the order parameter.

Landau's 1937 paper ("On the theory of phase transitions," Zh. Eksp. Teor. Fiz. 7, 19–32) laid out the framework that bears his name. He classified phase transitions by the way the symmetry group changes: if the symmetry group of the ordered phase is a subgroup of the disordered phase, the transition is continuous (second order); if there is no group-subgroup relation, the transition is discontinuous (first order). This classification is more physical than the Ehrenfest scheme (which classifies by the lowest derivative of the free energy that is discontinuous).

In 1950, Vitaly Ginzburg and Landau extended the theory to include spatial gradients, producing the Ginzburg-Landau functional for superconductivity. This extension was crucial for describing the phenomenology of superconductors — the penetration depth, the coherence length, and the distinction between type-I and type-II superconductors — without needing a microscopic theory of superconductivity (the BCS theory would not appear until 1957). The Ginzburg-Landau theory was later shown by Gor'kov (1959) to be a consequence of the BCS theory near $T_{c}$ , confirming the consistency of the phenomenological and microscopic approaches.

Landau's framework has had an influence that extends far beyond condensed matter physics. In particle physics, the Higgs mechanism (spontaneous symmetry breaking of a gauge symmetry) is described by a Ginzburg-Landau-like functional. The distinction between first-order and second-order transitions maps onto the distinction between a discontinuous and a continuous symmetry-breaking event in cosmology. The concept of an order parameter has become central to fields from neuroscience to economics to machine learning.

The limitations of Landau theory — the neglect of fluctuations and the assumption of analyticity — were understood by Landau himself and were the motivation for the development of the renormalization group by Wilson (1971). Wilson's insight was that Landau theory is the first term in a systematic expansion (the epsilon expansion) that can be continued to $d < 4$ , providing accurate critical exponents in any dimension.

Bibliography Master

@article{landau1937,
  author = {Landau, L. D.},
  title = {On the theory of phase transitions},
  journal = {Zh. Eksp. Teor. Fiz.},
  volume = {7},
  pages = {19--32},
  year = {1937}
}

@article{ginzburg1950,
  author = {Ginzburg, V. L. and Landau, L. D.},
  title = {On the theory of superconductivity},
  journal = {Zh. Eksp. Teor. Fiz.},
  volume = {20},
  pages = {1064--1082},
  year = {1950}
}

@book{landau1980,
  author = {Landau, L. D. and Lifshitz, E. M.},
  title = {Statistical Physics, Part 1},
  edition = {3rd},
  publisher = {Pergamon Press},
  year = {1980}
}

@book{kardar2007,
  author = {Kardar, Mehran},
  title = {Statistical Physics of Fields},
  publisher = {Cambridge University Press},
  year = {2007}
}

@book{pathria-beale2011,
  author = {Pathria, R. K. and Beale, Paul D.},
  title = {Statistical Mechanics},
  edition = {3rd},
  publisher = {Academic Press},
  year = {2011}
}

@book{goldenfeld1992,
  author = {Goldenfeld, Nigel},
  title = {Lectures on Phase Transitions and the Renormalization Group},
  publisher = {Addison-Wesley},
  year = {1992}
}

@article{wilson1971,
  author = {Wilson, Kenneth G.},
  title = {Renormalization group and critical phenomena. {I}. {R}enormalization group and the {K}adanoff scaling picture},
  journal = {Phys. Rev. B},
  volume = {4},
  pages = {3174--3183},
  year = {1971}
}

@article{gorkov1959,
  author = {Gor'kov, L. P.},
  title = {Microscopic derivation of the {G}inzburg-{L}andau equations in the theory of superconductivity},
  journal = {Sov. Phys. JETP},
  volume = {9},
  pages = {1364--1367},
  year = {1959}
}

Prerequisites

11.06.01
11.06.02
11.01.01

Used in

11.06.04
11.07.01
11.07.03

Tier anchors

beginner: Schroeder, An Introduction to Thermal Physics (2000), Ch. 8 (Phase transitions); Susskind, Statistical Mechanics: The Theoretical Minimum (2014), Lecture 11
intermediate: Landau & Lifshitz, Statistical Physics, Part 1, 3rd ed. (1980), Ch. 14; Pathria & Beale, Statistical Mechanics, 3rd ed. (2011), Ch. 11
master: Kardar, Statistical Physics of Fields (2007), Ch. 3–4; Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (1992), Ch. 5

References

Landau 1937 — On the theory of phase transitions · Zh. Eksp. Teor. Fiz. 7, 19–32. The foundational paper: the free energy is expanded in powers of an order parameter, with coefficients constrained by symmetry. The expansion yields a universal description of phase transitions independent of microscopic details.
Ginzburg & Landau 1950 — On the theory of superconductivity · Zh. Eksp. Teor. Fiz. 20, 1064–1082. Extension of Landau theory to include spatial gradients, giving the Ginzburg-Landau functional. Nobel Prize in Physics 2003 (Ginzburg).
Kardar — Statistical Physics of Fields (Cambridge, 2007) · Ch. 3 The Landau-Ginzburg Hamiltonian. Ch. 4 Saddle-point approximation, critical exponents, and fluctuations. The Ginzburg criterion and the upper critical dimension.
Pathria & Beale — Statistical Mechanics, 3rd ed. (Academic Press, 2011) · Ch. 11 Phase transitions: critical exponents, scaling, and the renormalization group. Landau theory as a mean-field framework.
Landau & Lifshitz — Statistical Physics, Part 1, 3rd ed. (Pergamon, 1980) · Ch. 14 Phase transitions of the second kind. Critical indices. The order parameter and symmetry breaking.
Goldenfeld — Lectures on Phase Transitions and the Renormalization Group (Addison-Wesley, 1992) · Ch. 5 The Ginzburg-Landau model. Fluctuations, the Ginzburg criterion, and the upper critical dimension.

Estimated time

beginner: 15 min
intermediate: 45 min
master: 80 min