Landau Theory of Phase Transitions

Iron loses its magnetism at exactly 1043 K, carbon dioxide has a critical point at 31 °C, a superconductor switches on at a few kelvin — three transitions in three unrelated materials. Landau's insight, in 1937, was that near any such transition the microscopic details barely matter. What matters is a single slowly varying quantity, the order parameter $m$ , and the symmetry it can break. Write down the free energy as an analytic power series in $m$ , keep only the terms the symmetry allows, let the coefficients depend smoothly on temperature, and minimize. Everything the transition does — its order, its exponents, its phase diagram — falls out of the algebra of that polynomial. The construction has three moves. Identify an order parameter that is zero in the symmetric (disordered) phase and nonzero in the ordered phase. Write the most general free-energy density $f(m,T,h)$ consistent with the symmetry, treating its Taylor coefficients as smooth functions of $T$ . Then find equilibrium by minimizing $f$ over $m$ . A phase transition appears the instant one coefficient — the coefficient of $m^2$ — passes through zero at $T_c$ : the single minimum at $m=0$ splits into a degenerate pair of broken-symmetry minima, and the system must choose one. The consequences are enormous for such elementary mathematics. A symmetric quartic potential gives a continuous transition with a fixed set of mean-field exponents. A symmetry-allowed cubic term, or a negative quartic coefficient, gives a first-order transition with latent heat and metastable phases. Add a gradient energy and $m$ becomes a field: this is Landau–Ginzburg theory, and it predicts the correlation length, domain walls, and — for superconductors — quantized vortices. This article owns the Landau free-energy expansion, the mean-field exponents that follow from it, and the algebraic rule for reading transition order off the polynomial. It is quantitatively exact only above four dimensions, but it is the organizing skeleton on which the entire modern theory of critical phenomena is built.

Key takeaways

Landau theory expands the free-energy density as a power series in a scalar order parameter, keeping only symmetry-allowed terms with smooth temperature-dependent coefficients: $f=f_0+a(T)\,m^2+b\,m^4-h\,m$ with $a(T)=a_0(T-T_c)$ .
Minimizing over $m$ gives the equilibrium state; the transition is triggered when the quadratic coefficient $a(T)$ changes sign at $T_c$ , splitting the minimum at $m=0$ into the broken-symmetry pair $m=\pm\sqrt{-a/2b}$ .
The symmetric quartic ( $b>0$ ) gives a continuous transition with mean-field exponents $\beta=\tfrac12$ , $\gamma=1$ , $\delta=3$ , $\alpha=0$ ; a cubic term or $b<0$ forces a first-order transition with a jump $\Delta m=w/2b$ and latent heat.
Adding a gradient term $\tfrac12 c(\nabla m)^2$ gives Landau–Ginzburg theory: correlation length $\xi\sim|t|^{-1/2}$ , Ornstein–Zernike correlations, and the Ginzburg criterion pinning the upper critical dimension $d_c=4$ .
Key idea: write the free energy as the most general symmetry-allowed analytic polynomial in the order parameter and minimize — the sign change of the quadratic coefficient produces spontaneous symmetry breaking, and the algebra of the polynomial dictates both the order of the transition and its exponents.

The core insight

Expand the free energy in powers of the order parameter, keep only the terms allowed by the system's symmetry, let the coefficients vary smoothly with temperature, and minimize. A sign change of the quadratic coefficient at $T_c$ converts the symmetric minimum into a degenerate pair of ordered minima — spontaneous symmetry breaking — and the order of the transition and its mean-field exponents follow entirely from the algebraic structure of the polynomial.

Where this lives today

Landau theory is the default language of order in condensed matter. Ginzburg–Landau theory remains the working model of superconductivity and superfluidity, describing vortex lattices in type-II superconductors, the mixed state probed in every high-field magnet, and the coherence length that sets the scale of Josephson devices. In soft matter it organizes the phases of liquid crystals through tensor order parameters, and in materials science it underlies phase-field simulations that track solidification fronts and microstructure evolution on a computer. Even where its exponents are wrong, its concepts — order parameter, free-energy landscape, symmetry classification — are the vocabulary in which every result of the renormalization group is stated. The RG did not overthrow Landau theory; it explained exactly when to trust it and how to correct it, making the 1937 polynomial the starting point rather than the endpoint of critical-phenomena theory.

The physical picture

Think of the free energy as a landscape over the order-parameter axis. Above $T_c$ it is a single bowl centred on $m=0$ : the symmetric phase is the unique equilibrium. As $T$ drops, the bottom of the bowl flattens, then buckles upward into a hill, and the minimum splits into two symmetric wells at $m=\pm m_0$ . The system rolls into one of them and breaks the $m\to-m$ symmetry — but which well it picks is arbitrary, because the landscape itself remains symmetric. That buckling is the entire content of a continuous transition, and it happens the moment the curvature at the origin, $\partial^2 f/\partial m^2|_0=2a$ , changes sign. Symmetry does the bookkeeping. If the system is invariant under $m\to-m$ (an Ising magnet in zero field), only even powers survive, the two wells are mirror images, and the transition is generically continuous. Allow a cubic $m^3$ term — as the liquid–gas transition does, since density difference has no up–down symmetry — and the wells become lopsided: one deepens faster than the other, they cross at a finite separation, and the order parameter jumps. That jump is a first-order transition with latent heat. The order of the transition is not extra physics; it is read directly off which powers the symmetry permits.

Figure

Landau free energy f(m) as temperature crosses T_c: a single well above T_c (blue) flattens at T_c (amber), then buckles into a symmetric double well below T_c (red) with degenerate minima at ±m₀ — the pitchfork of spontaneous symmetry breaking

The order of the transition is encoded entirely in the algebra of the polynomial. A minimum that splits smoothly from the origin gives second order; two competing minima of equal depth at a finite separation give first order. You can draw the whole phase diagram by tracking when minima appear, when they swap stability, and when they merge. And because the terms are fixed by symmetry rather than microscopics, systems as different as a magnet, an alloy, and a fluid near its critical point share the same free-energy form — the first hint of universality. The one place Landau theory quietly cheats is analyticity. It assumes $f$ is a smooth polynomial, yet a phase transition is precisely where thermodynamic functions turn non-analytic. Mean-field theory gets away with it when each region feels many neighbours (high dimension, long-range forces) and fails when fluctuations are large (low dimension near $T_c$ ). The punchline plot below shows the price: the mean-field order parameter rises as $(-t)^{1/2}$ , but the true 3D curve rises faster, as $(-t)^{0.326}$ — Landau underestimates how sharply order sets in.

Figure

Punchline: order parameter m below T_c — mean-field β = ½ (blue, m ∝ (−t)^½) versus the true 3D-Ising β ≈ 0.326 (red). Both vanish at T_c, but the real curve is steeper right at the transition, so Landau's analytic polynomial systematically underestimates the ordering near criticality

The core equations

The starting point is the free-energy density itself. For a system invariant under $m\to-m$ in zero field, analyticity plus symmetry plus stability fix the retained terms: even powers only, coefficients smooth in $T$ , and a positive top coefficient so a minimum exists. The field $h$ enters through the explicitly symmetry-breaking term $-hm$ .

f(m,T,h)=f_0(T)+a(T)\,m^2+b\,m^4-h\,m,\qquad b>0

where: $m$ is the order parameter (magnetization per site, density difference, condensate amplitude); $a(T)$ is the quadratic coefficient that vanishes and changes sign at $T_c$ ; $b>0$ is the quartic coefficient, taken smooth and positive; $h$ is the field conjugate to $m$ ; and $f_0(T)$ is the smooth $m$ -independent background. Physically, $a>0$ makes the origin a minimum (disordered); $a<0$ makes it a maximum, forcing the system out to $m\neq0$ (ordered). Near the transition the temperature dependence that matters is entirely in $a(T)$ , linearized about its zero:

a(T)=a_0\,(T-T_c)=a_0\,T_c\,t,\qquad t\equiv\frac{T-T_c}{T_c},\qquad a_0>0

where: $t$ is the dimensionless reduced temperature, $T_c$ is the critical temperature where $a$ vanishes, and $a_0$ is a positive constant. Reading this: $t>0$ (above $T_c$ ) keeps the single symmetric minimum; $t<0$ (below $T_c$ ) tips the origin into a maximum and grows the ordered minima. Equilibrium is the minimization itself — stationarity plus positive curvature:

\left.\frac{\partial f}{\partial m}\right|_{m_0}=2a\,m_0+4b\,m_0^3-h=0,\qquad \left.\frac{\partial^2 f}{\partial m^2}\right|_{m_0}\ge0

where: $m_0$ is the equilibrium order parameter. The cubic stationarity equation is the equation of state of the whole theory — every critical exponent below is a different limit of this one relation. Promoting $m$ to a spatially varying field and adding the cheapest symmetry-allowed gradient term gives the Landau–Ginzburg functional, which is what lets the theory describe correlations and interfaces rather than just a uniform state:

F[m]=\int \mathrm{d}^dx\left[\tfrac12 c\,(\nabla m)^2+a\,m^2+b\,m^4-h\,m\right]

where: $c>0$ is the stiffness penalizing spatial variation, $\nabla m$ is the gradient of the field $m(\mathbf{x})$ , and $d$ is the spatial dimension. This single addition delivers the correlation length $\xi\sim|t|^{-1/2}$ , the Ornstein–Zernike correlation function, and — through the Ginzburg criterion — the boundary of the theory's own validity.

Derivation AThe mean-field exponents from the symmetric quartic

The goal here is to show that minimizing the symmetric Landau polynomial $f=a_0T_c\,t\,m^2+b\,m^4-hm$ produces the complete set of bulk critical exponents $\beta,\gamma,\delta,\alpha$ as pure numbers — independent of the non-universal constants $a_0$ , $b$ , $T_c$ . That independence is the first concrete face of universality: the amplitude of a response depends on the material, but its power law does not. The strategy is to take one limit of the equation of state $2a\,m+4b\,m^3=h$ at a time — zero field below $T_c$ for $\beta$ , linear response for $\gamma$ , the $t=0$ isotherm for $\delta$ , and the minimized free energy for $\alpha$ .

Step 1. Spontaneous order parameter and $\beta$ . Set $h=0$ . The equation of state factors as $m(2a+4b\,m^2)=0$ , so either $m=0$ or $m^2=-a/2b$ . For $t>0$ , $a>0$ and only $m=0$ is real (and it is a minimum). For $t<0$ , $a<0$ gives the real pair $m_0=\pm\sqrt{-a/2b}=\pm\sqrt{a_0T_c/2b}\,(-t)^{1/2}$ . Why the origin flips: below $T_c$ the curvature $2a<0$ makes $m=0$ a local maximum, so the system is expelled to the new minima. The power $\tfrac12$ is the exponent: $m_0\propto(-t)^{1/2}\Rightarrow\beta=\tfrac12$ .

Step 2. Susceptibility and $\gamma$ . Differentiate the equation of state $2a\,m+4b\,m^3=h$ with respect to $h$ : $(2a+12b\,m^2)\,\partial_h m=1$ , so $\chi^{-1}=2a+12b\,m_0^2$ . Above $T_c$ , $m_0=0$ gives $\chi^{-1}=2a=2a_0T_c\,t$ , hence $\chi\sim t^{-1}$ . Below $T_c$ , substitute $m_0^2=-a/2b$ : $\chi^{-1}=2a+12b(-a/2b)=2a-6a=-4a=4a_0T_c|t|$ , hence $\chi\sim|t|^{-1}$ . Why the factor of two: the ordered phase is stiffer, giving a susceptibility amplitude ratio $C_+/C_-=2$ , a genuine mean-field prediction. Both sides give $\gamma=1$ .

Step 3. Critical isotherm and $\delta$ . Sit exactly at $T_c$ , where $a=0$ . The equation of state collapses to $h=4b\,m^3$ . Why the cubic surfaces: the linear term $2am$ vanishes at $t=0$ , promoting the cubic to leading order. Reading $h\sim m^{3}$ gives $\delta=3$ .

Step 4. Specific heat and $\alpha$ . Substitute the minimizing $m_0^2=-a/2b$ back into $f$ (at $h=0$ ): $f_{\min}=a\,m_0^2+b\,m_0^4=-a^2/4b=-(a_0T_c)^2t^2/4b$ for $t<0$ , and $f_{\min}=0$ for $t>0$ . The singular specific heat is $C=-T\,\partial^2 f/\partial T^2$ ; the extra $t^2$ below $T_c$ contributes a constant jump $\Delta C=a_0^2T_c/2b$ at $T_c$ , with nothing above. Why $\alpha=0$ : a finite step (not a power-law divergence) is by definition the exponent $\alpha=0$ — but note this is a jump, unlike the 2D-Ising $\alpha=0$ which is a logarithm.

Step 5. Scaling-relation check. With $(\alpha,\beta,\gamma,\delta)=(0,\tfrac12,1,3)$ the Rushbrooke relation $\alpha+2\beta+\gamma=0+1+1=2$ holds and Widom $\gamma=\beta(\delta-1)=\tfrac12(2)=1$ holds — both as exact equalities. Why they must: any analytic homogeneous free energy satisfies the scaling relations, and Landau's polynomial is exactly such a function. *Verified symbolically in verify_landau-theory-phase-transitions.py (stationarity, $m_0^2=-a/2b$ , $\beta=\tfrac12$ , $\chi^{-1}=2a$ and $-4a$ , amplitude ratio $2$ , $h=4bm^3$ , $\Delta C=a_0^2T_c/2b$ , and all four scaling relations).*

Go deeper · Derivation B · 1 minA cubic term forces a first-order transition

Derivation BA cubic term forces a first-order transition

When the symmetry permits an odd term — as for the liquid–gas order parameter, which has no $m\to-m$ invariance — the free energy carries an $m^3$ contribution, and the character of the transition changes qualitatively. The goal of this derivation is to locate the first-order transition of $f=a\,m^2-w\,m^3+b\,m^4$ (with $w,b>0$ , $a=a_0(T-T_0)$ ) and show that the order parameter jumps discontinuously, producing a latent heat. The method is the coexistence condition: a first-order transition occurs when a nonzero minimum becomes exactly degenerate with the $m=0$ minimum, i.e. $f(m^*)=0$ and $f'(m^*)=0$ simultaneously.

Step 1. Stationary points. $\partial f/\partial m=m\,(2a-3w\,m+4b\,m^2)=0$ . Besides $m=0$ , the bracket supplies two more roots once its discriminant is positive, so three minima can compete.

Step 2. Impose degeneracy. Dividing $f=0$ by $m^2$ gives $a-w\,m^*+b\,m^{*2}=0$ ; the stationarity of that same nonzero root gives $2a-3w\,m^*+4b\,m^{*2}=0$ . Why both conditions: degeneracy with the origin ( $f=0$ ) plus its being a genuine extremum ( $f'=0$ ) is exactly the equal-free-energy coexistence of two phases.

Step 3. Solve the pair. Eliminating between the two equations yields the finite jump and the transition point:

\Delta m=m^*=\frac{w}{2b},\qquad a^*=\frac{w^2}{4b}

At $T=T^*$ where $a=a^*$ , the order parameter jumps discontinuously from $0$ to $w/2b$ . Why this is first order: a discontinuous $m$ means a discontinuous entropy $\Delta s=-\partial(\Delta f)/\partial T\neq0$ , hence a nonzero latent heat $L=T^*\Delta s$ — the defining signature, and the quantity entering the Clausius–Clapeyron relation. The transition sits at $T^*=T_0+w^2/(4a_0 b)$ , above the spinodal $T_0$ where the disordered phase finally loses local stability. Verified symbolically in verify_landau-theory-phase-transitions.py (simultaneous solution of the two coexistence conditions returns $m^*=w/2b$ , $a^*=w^2/4b$ ).

Limiting cases

case	Landau functional	result
Symmetric continuous transition	$f=a_0t\,m^2+b\,m^4$ , $b>0$	second order; $m\propto(-t)^{1/2}$ , exponents $(\alpha,\beta,\gamma,\delta)=(0,\tfrac12,1,3)$
First order via cubic	$f=a\,m^2-w\,m^3+b\,m^4$ , $w>0$	discontinuous jump $\Delta m=w/2b$ , latent heat, transition at $a^*=w^2/4b$
First order via negative quartic	$f=a\,m^2-\lvert b\rvert m^4+c\,m^6$ , $c>0$	must keep $m^6$ ; first order, with a tricritical point at $b=0$
Landau–Ginzburg (spatial)	$+\tfrac12 c(\nabla m)^2$	correlation length $\xi=\sqrt{c/2a}\sim\|t\|^{-1/2}$ , $\nu=\tfrac12$ , $\eta=0$
Above $d_c=4$	any short-range	fluctuations irrelevant; Landau/mean-field exponents exact
Below $d_c=4$	any short-range	fluctuations dominate near $T_c$ ; Landau exponents fail, RG required

Common trapLandau exponents are not the real exponents below four dimensions

The single most common error is to quote $\beta=\tfrac12$ , $\gamma=1$ , $\delta=3$ as the critical exponents of a 3D system. They are not. Landau theory assumes the free energy is an analytic polynomial in $m$ , but the true singularities of a transition come from long-wavelength fluctuations that the theory throws away. In three dimensions the real Ising values are $\beta\approx0.326$ , $\gamma\approx1.237$ , $\delta\approx4.79$ ; in two dimensions $\beta=\tfrac18$ exactly (Onsager). The Ginzburg criterion makes this precise: the ratio of fluctuation amplitude to background order scales as $|t|^{(d-4)/2}$ , which vanishes at $T_c$ only for $d>4$ . Trust Landau exponents quantitatively only above the upper critical dimension $d_c=4$ ; below it, Landau theory gives the right symmetry, order, and phase diagram but the wrong numbers.

Worked examples

Example 1 — Spontaneous magnetization and its onset. Take a ferromagnet described by $f=a_0T_c\,t\,m^2+b\,m^4$ with (in reduced units) $a_0=1$ , $T_c=1$ , $b=\tfrac12$ . Find the equilibrium magnetization at $t=-0.01$ (i.e. 1% below $T_c$ ).

Step 1. Write the minimizer. Below $T_c$ , $m_0=\sqrt{-a/2b}=\sqrt{a_0T_c(-t)/2b}$ .

Step 2. Substitute. $m_0=\sqrt{(1)(1)(0.01)/(2\cdot\tfrac12)}=\sqrt{0.01/1}=\sqrt{0.01}$ .

Step 3. Evaluate. $m_0=\mathbf{0.100}$ . Note the square-root law at work: moving to $t=-0.04$ (4% below $T_c$ ) only doubles $m_0$ to $0.20$ , not quadruples it — the hallmark of $\beta=\tfrac12$ .

Example 2 — The specific-heat jump. For the same magnet, compute the discontinuity in specific heat at $T_c$ .

Step 1. Use the derived result. $\Delta C=a_0^2T_c/2b$ .

Step 2. Substitute. $\Delta C=(1)^2(1)/(2\cdot\tfrac12)=1/1$ .

Step 3. Evaluate. $\Delta C=\mathbf{1.0}$ (reduced units). The specific heat steps up by a finite amount on cooling through $T_c$ and shows no divergence — the mean-field signature $\alpha=0$ . In a real 3D ferromagnet this step is replaced by a weak cusp ( $\alpha\approx0.11$ ), and in 2D by a logarithmic spike.

Example 3 — How badly Landau fails right at the transition. Compare the mean-field and 3D-Ising magnetization at $|t|=10^{-3}$ , using $\beta_{\text{MF}}=\tfrac12$ and $\beta_{\text{Ising}}=0.326$ (equal amplitudes for the comparison).

Step 1. Form the ratio. $\dfrac{m_{\text{MF}}}{m_{\text{Ising}}}=\dfrac{|t|^{0.5}}{|t|^{0.326}}=|t|^{0.174}$ .

Step 2. Substitute. $(10^{-3})^{0.174}=10^{-0.522}$ .

Step 3. Evaluate. The ratio is $\approx\mathbf{0.30}$ : within 0.1% of $T_c$ , mean-field theory predicts an order parameter only about a third of the true value. The disagreement grows as $t\to0$ — precisely the regime where fluctuations, and the renormalization group, take over.

Origins · 1 minLandau theory and how this was discovered

Landau theory

the phenomenological free-energy expansion introduced by Lev Landau (1908–1968) in 1937 and extended to spatial variation with Vitaly Ginzburg; the organizing framework of critical phenomena · order parameter · symmetry-allowed expansion · mean-field exponents · first-order via cubic term

Landau theory is Lev Landau's 1937 answer to a question that had resisted microscopic attack: what is common to all phase transitions? His radical simplification was to ignore the microscopic Hamiltonian entirely and expand the free energy in the order parameter, keeping only what symmetry allows. From that one polynomial he read off the order of the transition, the mean-field exponents, and the mechanism of spontaneous symmetry breaking. In 1950, with Vitaly Ginzburg, he added a gradient term to describe superconductivity — the Landau–Ginzburg functional — which predicted the coherence length, the type-I/type-II distinction, and quantized vortices years before BCS supplied the microscopic derivation, and which earned Ginzburg a share of the 2003 Nobel Prize. Landau himself received the 1962 Nobel Prize for his theory of superfluid helium. The theory's great virtue and its great limitation are the same fact: by treating the free energy as analytic, it captures the universal structure of transitions exactly, yet misses the fluctuation-driven exponents below four dimensions — the gap that Kadanoff and Wilson's renormalization group would finally close.

History

1937

— Landau publishes the order-parameter theory of phase transitions, expanding the free energy in symmetry-allowed powers

1937

— Landau applies the framework to classify transitions by the order of the free-energy derivative that jumps

1944

— Onsager's exact 2D-Ising solution (

\beta=\tfrac18

) reveals that Landau's mean-field exponents are quantitatively wrong below four dimensions

1950

— Ginzburg and Landau introduce the gradient (spatially varying) functional to describe superconductivity

1960

— Ginzburg formulates the criterion for when fluctuations invalidate mean-field theory, defining the upper critical dimension

1966

— Kadanoff's block-spin picture explains how Landau theory is corrected by successive coarse-graining

1971

— Wilson's renormalization group computes the true exponents, recasting Landau theory as the trivial (Gaussian) fixed point

2003

— Ginzburg shares the Nobel Prize for the Ginzburg–Landau theory of superconductivity

Going further

For the canonical treatment see Landau & Lifshitz, Statistical Physics Part 1 (3rd ed.), §142–§148, where the theory is developed alongside its symmetry foundations. Kardar's Statistical Physics of Fields (Ch. 1–3) presents the Landau–Ginzburg functional as the launch point for the renormalization group; Pathria & Beale, Statistical Mechanics (3rd ed.), Ch. 12 gives a compact statistical-mechanics-first account; Sethna's Statistical Mechanics: Entropy, Order Parameters, and Complexity (Ch. 9 & 12) is strong on the order-parameter and symmetry viewpoint; and Goldenfeld's Lectures on Phase Transitions and the Renormalization Group (Ch. 5) carefully develops the Ginzburg criterion. Provenance: all equations are standard Landau/Landau–Ginzburg results, cross-checked against Landau & Lifshitz §142–148, Kardar SPF Ch. 1–3, Pathria & Beale Ch. 12, Sethna Ch. 9 & 12, and Goldenfeld Ch. 5. Verified numerically in verify_landau-theory-phase-transitions.py. Prerequisites: order parameter · continuous phase transitions · spontaneous symmetry breaking. Leads to: critical exponents · correlation length and its divergence · universality and universality classes · scaling hypothesis and scaling relations.

Is the Landau free energy the true free energy of the system?

No — it is a phenomenological expansion, not a first-principles thermodynamic potential. More precisely, minimizing it gives the best mean-field (variational) estimate: the Landau functional is the free energy restricted to configurations with a uniform order parameter, ignoring the correlated fluctuations of the real system. This is why minimizing it is legitimate — the minimum is the equilibrium of the mean-field approximation — but also why its exponents are only exact above four dimensions, where those neglected fluctuations genuinely do not matter.

Where does the coefficient

a(T)=a_0(T-T_c)

come from?

It is the leading Taylor expansion of a smooth coefficient about the temperature where it vanishes. Landau does not derive $a_0$ or $T_c$ from microscopics — they are material parameters. The content of the ansatz is only that $a(T)$ is analytic and changes sign at $T_c$ ; the linear form is the generic first term of that analytic function near its zero. Everything universal (the exponents) is independent of the actual value of $a_0$ , which is exactly why different materials share the same exponents.

Why does a cubic term specifically give a first-order transition?

A cubic term breaks the $m\to-m$ symmetry of the polynomial, so the two wells are no longer mirror images. As temperature drops, a second minimum appears at a finite value of $m$ while the origin is still a minimum; at the transition temperature the two become exactly degenerate, and the system jumps between them. A jump in $m$ means a jump in entropy and volume — latent heat — which is the definition of first order. In a symmetric ( $m\to-m$ ) theory the minima instead grow continuously out of the origin, giving second order.

How does Landau theory connect to the renormalization group?

Landau theory is the renormalization group evaluated at its trivial fixed point. The RG systematically integrates out short-wavelength fluctuations, generating corrections to the Landau coefficients. Above $d_c=4$ those corrections are irrelevant and the flow ends at the Gaussian (Landau) fixed point, so mean-field exponents are exact. Below $d_c=4$ the flow is driven to a new, non-trivial (Wilson–Fisher) fixed point whose exponents differ from Landau's. So the RG does not replace Landau theory; it tells you when Landau theory is exact and computes the corrections when it is not.

What is the upper critical dimension and how does the Ginzburg criterion fix it at 4?

The upper critical dimension $d_c$ is the dimension above which fluctuations of the order parameter are negligible compared with the mean-field background, so Landau exponents are exact. The Ginzburg criterion compares the mean-square fluctuation $\langle\phi^2\rangle$ within a correlation volume to the background $m_0^2\sim|t|$ . Using $\xi\sim|t|^{-1/2}$ , the ratio scales as $|t|^{(d-4)/2}$ . As $t\to0$ this vanishes only if $d>4$ , so $d_c=4$ : above it, mean-field theory is self-consistent; below it, fluctuations grow without bound near $T_c$ and Landau theory breaks down.

What happens when the quartic coefficient

b

can be tuned to zero?

Then the $m^4$ term no longer stabilizes the free energy near the transition, and you must retain a positive $m^6$ term. The special point $b=0$ , $a=0$ is a tricritical point, where a line of continuous transitions meets a line of first-order transitions. Landau theory predicts distinct tricritical exponents ( $\beta_t=\tfrac14$ , $\delta_t=5$ in mean field), realized in metamagnets and in $^3$ He– $^4$ He mixtures. It is a clean example of how retaining a higher term in the same expansion opens a qualitatively new piece of the phase diagram.

Does Landau theory apply to first-order transitions like melting?

Yes, and this is one of its strengths. A first-order transition arises whenever the polynomial has two competing minima of equal depth at a finite separation — from a cubic term, a negative quartic, or the symmetry of the order parameter. Crystallization breaks continuous translational symmetry with density-wave order parameters that admit cubic invariants, which is why melting and freezing are generically first order with a latent heat. Landau theory thus unifies continuous and discontinuous transitions in one framework, reading the distinction off the algebra.