Landau Theory and Critical Exponents

Derivation

Landau Theory and Critical Exponents

D-246 Home PU-302 Threads symmetry · fields · chance · energy Depends on Mean-Field Ising Model and Spontaneous Magnetization

Statement

Near a continuous (second-order) phase transition the free energy density is an analytic function of a coarse-grained order parameter \(m\) and of the fields that couple to it. Landau theory postulates the expansion \(f(m,T,h) = f_0 + a(T)\,m^2 + b\,m^4 - h\,m\) (with all symmetry-allowed terms) and locates the equilibrium state by minimising \(f\) over \(m\). Taking \(a(T)=a_0(T-T_c)\) with \(a_0,b>0\), this derivation extracts the mean-field critical exponents \(\beta=\tfrac12\), \(\gamma=1\), \(\delta=3\), \(\alpha=0\), and \(\nu=\tfrac12,\ \eta=0\) once gradient terms are restored, and shows how the same analytic structure organises every continuous transition into universality classes.

Why it matters

Landau theory is the archetype of an order-parameter theory: it replaces the microscopic Hamiltonian by a symmetry-constrained expansion of a thermodynamic potential, and from that single postulate reproduces the universal power laws seen at critical points in magnets, fluids, superconductors, superfluids and binary alloys. The exponents it predicts are the reference values that experiment and the renormalisation group are measured against.

It also makes the logic of universality explicit. Because only the symmetry of the order parameter and the dimensionality enter the allowed terms, systems with utterly different microscopics fall into the same class. Landau theory is exact above the upper critical dimension \(d=4\), and its failure below \(d=4\) is precisely what motivated Wilson's renormalisation group.

Assumptions

The free energy is analytic in \(m\) near \(m=0\).If \(f\) has non-analytic terms (e.g. \(|m|^3\) generated by fluctuations or long-range forces), the Taylor expansion is illegitimate and the exponents change.

The order parameter is small near \(T_c\), so truncation at \(m^4\) is sufficient.If higher terms matter (first-order region, tricritical point), the \(m^6\) term must be retained and the transition can become discontinuous.

The symmetry \(m\to-m\) holds at \(h=0\), forbidding odd powers.If a cubic invariant \(m^3\) is symmetry-allowed (e.g. three-state systems), the transition is generically first order and Landau's continuous-transition exponents do not apply.

Spatial fluctuations of \(m\) are negligible; the field is uniform at its most-probable value.Dropping this mean-field assumption reintroduces critical fluctuations that dominate for \(d<4\) (Ginzburg criterion), shifting the exponents away from their mean-field values.

\(a(T)\) is analytic and passes linearly through zero at \(T_c\); \(b>0\) stays finite there.If \(b\to0\) while \(a\to0\) the quartic coefficient vanishes (tricriticality) and one must expand to \(m^6\), giving a different exponent set.

Derivation

\[ f(m,T,h) = f_0(T) + a(T)\,m^2 + b\,m^4 - h\,m \]

Most general expansion consistent with \(m\to-m\) symmetry at \(h=0\), truncated at quartic order; the field \(h\) enters through the linear coupling \(-hm\) demanded by the definition of the conjugate field. A

\[ a(T) = a_0\,(T-T_c),\qquad a_0>0,\; b>0 \]

The only coefficient that must change sign to produce ordering is \(a\); analyticity forces the leading behaviour to be linear in \(T-T_c\). Positivity of \(b\) keeps \(f\) bounded below. A

\[ \frac{\partial f}{\partial m} = 2a\,m + 4b\,m^3 - h = 0 \]

Equilibrium minimises the free energy at fixed \(T,h\); stationarity is the first-order condition. A

\[ \frac{\partial^2 f}{\partial m^2} = 2a + 12 b\,m^2 > 0 \]

A minimum (not a maximum or inflection) requires positive curvature; this selects the physical branch among the roots of Step 3. B

\[ h=0:\quad 2m\,(a + 2b\,m^2)=0 \;\Rightarrow\; m=0 \ \text{or}\ m^2=-\frac{a}{2b} \]

Set the field to zero and factor. The nontrivial root exists only when \(a<0\), i.e. \(T<T_c\); for \(T>T_c\) only \(m=0\) is real. B

\[ T<T_c:\quad m_0(T) = \pm\left(\frac{-a}{2b}\right)^{1/2} = \pm\left(\frac{a_0(T_c-T)}{2b}\right)^{1/2} \]

Substitute \(a=a_0(T-T_c)\). Spontaneous symmetry breaking: two degenerate minima appear continuously from zero. B

\[ m_0 \propto (T_c-T)^{1/2} \;\Rightarrow\; \boxed{\beta = \tfrac12} \]

Read off the order-parameter exponent from the temperature power law at \(h=0\). A

\[ \chi^{-1} = \left.\frac{\partial h}{\partial m}\right|_{m_0} = 2a + 12 b\,m_0^2 \]

The susceptibility is the response \(\chi=\partial m/\partial h\); invert the equation of state at the equilibrium value. B

\[ T>T_c:\ \chi^{-1}=2a=2a_0(T-T_c);\quad T<T_c:\ \chi^{-1}=2a+12b\!\left(\tfrac{-a}{2b}\right)=-4a \]

Use \(m_0=0\) above and \(m_0^2=-a/2b\) below. Both sides diverge linearly in \(|T-T_c|\). B

\[ \chi \propto |T-T_c|^{-1} \;\Rightarrow\; \boxed{\gamma=1} \]

The exponent is the same on both sides though the amplitude ratio is \(\chi_+/\chi_-=2\). A

\[ T=T_c\ (a=0):\quad h = 4b\,m^3 \;\Rightarrow\; m\propto h^{1/3} \;\Rightarrow\; \boxed{\delta=3} \]

On the critical isotherm the linear term vanishes; the cubic term of Step 3 fixes the field–order-parameter relation. B

\[ f_{\min}(T,0) = \begin{cases} f_0, & T>T_c\\[4pt] f_0 - \dfrac{a^2}{4b} = f_0 - \dfrac{a_0^2(T-T_c)^2}{4b}, & T<T_c \end{cases} \]

Insert the equilibrium \(m_0\) back into \(f\). Below \(T_c\) the condensation energy is \(-a^2/4b\). B

\[ c = -T\frac{\partial^2 f_{\min}}{\partial T^2};\quad \Delta c = c(T_c^-)-c(T_c^+) = \frac{a_0^2 T_c}{2b} \]

The specific heat is the second temperature derivative of the free energy. Below \(T_c\) it acquires a finite extra piece; above it is unchanged. B

\[ c\ \text{has a finite jump, not a divergence} \;\Rightarrow\; \boxed{\alpha=0\ (\text{discontinuity})} \]

A jump corresponds to \(|T-T_c|^{0}\); Landau theory predicts \(\alpha=0\) with a step, unlike the observed weak divergences. A

\[ f = f_0 + \tfrac12 c_0(\nabla m)^2 + a\,m^2 + b\,m^4 - h\,m \]

Restore the lowest gradient term (Ginzburg–Landau) to describe spatial variation; rotational invariance forbids a linear-gradient term, so \((\nabla m)^2\) is leading. B

\[ -c_0\nabla^2 m + 2a\,m + 4b\,m^3 = h;\quad \text{linearise about } m_0:\ \xi^{-2}=\frac{\chi^{-1}}{c_0} \]

Euler–Lagrange minimisation gives the field equation; the correlation length \(\xi\) is set by the curvature of \(f\) at the minimum. C

\[ \xi = \left(\frac{c_0}{\chi^{-1}}\right)^{1/2}\propto |T-T_c|^{-1/2}\;\Rightarrow\;\boxed{\nu=\tfrac12},\quad G(r)\sim\frac{e^{-r/\xi}}{r^{d-2}}\;\Rightarrow\;\boxed{\eta=0} \]

Since \(\chi^{-1}\propto|T-T_c|\), the correlation length diverges with \(\nu=\tfrac12\); the Ornstein–Zernike form gives the anomalous dimension \(\eta=0\) at mean-field level. C

Result

\[ f=f_0+a_0(T-T_c)m^2+bm^4-hm \;\Longrightarrow\; \beta=\tfrac12,\ \gamma=1,\ \delta=3,\ \alpha=0,\ \nu=\tfrac12,\ \eta=0 \]

Reading. Minimising a symmetry-constrained quartic free energy gives an order parameter that turns on as \((T_c-T)^{1/2}\), a susceptibility that diverges as \(|T-T_c|^{-1}\), a critical isotherm \(m\propto h^{1/3}\), and a specific-heat jump rather than a divergence. These are the mean-field exponents; they satisfy the scaling relations \(\alpha+2\beta+\gamma=2\) (Rushbrooke: \(0+1+1=2\)) and \(\gamma=\beta(\delta-1)\) (Widom: \(1=\tfrac12\cdot2\)) exactly, and \(\gamma=\nu(2-\eta)\) (Fisher) with \(1=\tfrac12\cdot2\). The hyperscaling relation \(2-\alpha=\nu d\) holds only at \(d=4\), the marker of the upper critical dimension.

Units check. \(f\) is an energy density \([\mathrm{J\,m^{-3}}]\). If \(m\) is dimensionless, then \([a]=[b]=\mathrm{J\,m^{-3}}\) and \([h]=\mathrm{J\,m^{-3}}\) (an energy-density field), so \(hm\) matches \(f\). Then \(\chi=\partial m/\partial h\) has units \([\mathrm{J\,m^{-3}}]^{-1}=\mathrm{m^3\,J^{-1}}\), and \(\chi^{-1}=2a\) is indeed an energy density — consistent. The gradient stiffness \(c_0\) carries \(\mathrm{J\,m^{-1}}\) so that \(c_0(\nabla m)^2\) is \(\mathrm{J\,m^{-3}}\), and \(\xi^2=c_0/\chi^{-1}\) has units \(\mathrm{(J\,m^{-1})/(J\,m^{-3})}=\mathrm{m^2}\), a length squared as required.

Limiting cases

\(h=0,\ T>T_c\): unique minimum at \(m=0\); disordered (symmetric) phase, \(\chi=1/2a\) finite.
\(h=0,\ T\to T_c^-\): two minima emerge continuously, \(m_0\to0\) as \((T_c-T)^{1/2}\); this is the continuous-transition signature.
\(h\neq0\): the \(-hm\) term tilts \(f\), removing the degeneracy and rounding the transition into a smooth crossover — no true singularity at finite field.
\(a=0\) exactly (critical isotherm): \(f=bm^4-hm\); response is purely nonlinear, \(m\propto h^{1/3}\).
\(d\ge4\): Ginzburg criterion is satisfied; fluctuations are subdominant and these mean-field exponents become exact.

Breaks when

Below the upper critical dimension (\(d<4\)). Critical fluctuations dominate within the Ginzburg region; the true exponents (e.g. 3D Ising \(\beta\approx0.326,\ \gamma\approx1.24\)) differ from mean-field, and only the renormalisation group captures them.
Cubic invariant allowed by symmetry. If \(m^3\) is permitted (three-state Potts, certain structural transitions), the transition is generically first order — \(f\) develops a competing minimum at finite \(m\) — and the continuous-transition analysis collapses.
Quartic coefficient turns negative (\(b<0\)). The expansion is unbounded below at quartic order; a positive \(m^6\) term is mandatory, producing a first-order line and a tricritical point where mean-field exponents themselves shift (\(\beta=\tfrac14\)).
Long-range or non-analytic interactions. Coulombic or dipolar couplings, or a non-analytic \(|m|^3\) term, invalidate the Taylor expansion and change the exponents (e.g. dipolar Ising in \(d=3\) sits at its upper critical dimension with logarithmic corrections).

Failure modes

Confusing \(f\) with the true equilibrium free energy at all \(m\). \(f(m)\) is a Landau functional to be minimised; only \(f_{\min}(T,h)\) is thermodynamic. Reading susceptibilities off the wrong \(m\) branch (e.g. the unstable \(m=0\) root below \(T_c\)) gives the wrong sign of \(\chi^{-1}\).
Forgetting the factor of 2 asymmetry in \(\gamma\). \(\chi^{-1}=2a\) above but \(-4a\) below \(T_c\); students who use the same amplitude both sides miss the universal ratio \(\chi_+/\chi_-=2\).
Writing \(\alpha=0\) as "no anomaly." \(\alpha=0\) in Landau theory means a finite jump, not a smooth specific heat; the jump \(\Delta c=a_0^2T_c/2b\) is a real, measurable discontinuity.
Treating \(m\propto h^{1/3}\) as valid off \(T_c\). The cube-root law holds only on the critical isotherm \(a=0\); away from it the linear \(2am\) term dominates at small \(h\).
Assuming the exponents are always right because the scaling relations close. Mean-field values satisfy Rushbrooke and Widom exactly, which lulls students into trusting them below \(d=4\) where they are quantitatively wrong.
Dropping the gradient term when computing \(\nu,\eta\). Correlation-length exponents require the \((\nabla m)^2\) term; a purely uniform Landau theory cannot produce \(\xi\) at all.

Discussion

The deepest lesson of Landau theory is that symmetry plus analyticity determines critical behaviour without any microscopic input. The order parameter \(m\) is whatever quantity is zero in the symmetric phase and nonzero in the ordered phase — magnetisation for a ferromagnet, the density difference for a liquid–gas system, the condensate wavefunction for a superfluid, the staggered magnetisation for an antiferromagnet. Because the allowed terms in \(f\) depend only on how \(m\) transforms under the symmetry group and on the number of order-parameter components \(n\), radically different physical systems share the same exponents. This is the origin of universality classes, indexed by \((d,n)\).

The connection to the prior mean-field Ising result is direct: expanding the self-consistent equation \(m=\tanh(\beta J z\,m + \beta h)\) in small \(m\) yields \(m\simeq (\beta Jz)m - \tfrac13(\beta Jz)^3 m^3 + \dots\), which is exactly a Landau expansion with \(a\propto(T-T_c)\) and \(b>0\). Landau theory is thus the coarse-grained, symmetry-organised distillation of any mean-field calculation; it abstracts away the lattice and the interaction details and keeps only what is universal.

The theory's own failure is instructive. The Ginzburg criterion asks whether fluctuations of \(m\) within a correlation volume \(\xi^d\) are small compared with \(m_0^2\); using the mean-field \(\xi\propto|t|^{-1/2}\) and \(m_0\propto|t|^{1/2}\) one finds the fluctuation contribution scales as \(|t|^{(d-4)/2}\). For \(d>4\) this vanishes at criticality and mean-field is self-consistent; for \(d<4\) it diverges, so mean-field theory predicts its own breakdown. This \(d=4\) upper critical dimension is where hyperscaling \(2-\alpha=\nu d\) marginally holds, and it is the fixed point about which Wilson's \(\varepsilon=4-d\) expansion perturbs to recover the true exponents. Landau theory is therefore not merely wrong below four dimensions — it is the exact zeroth order of a controlled expansion.

Common misconceptions. Landau theory is often called "the theory of second-order transitions," but the same functional describes first-order transitions too (when a cubic term or negative quartic is present, \(f\) develops competing minima separated by a barrier). The continuity of \(m\) is a consequence of the sign structure \(a\to0,\ b>0\), not a built-in assumption. A second frequent error is to think Landau exponents are "just approximate" — they are the exact exponents for any system at or above its upper critical dimension, and they are the anchor values that define the mean-field universality class.

Worked examples

Order parameter and susceptibility of a Landau ferromagnet

Given \(a_0=2.0\times10^{5}\ \mathrm{J\,m^{-3}\,K^{-1}}\), \(b=1.0\times10^{6}\ \mathrm{J\,m^{-3}}\), \(T_c=600\ \mathrm{K}\). Find \(m_0\) at \(T=594\ \mathrm{K}\) and \(\chi\) just above and below \(T_c\) at \(|T-T_c|=6\ \mathrm{K}\). Symbols first. A

\[ m_0=\left(\frac{a_0(T_c-T)}{2b}\right)^{1/2},\quad \chi_+=\frac{1}{2a_0(T-T_c)},\quad \chi_-=\frac{1}{4a_0(T_c-T)} \]

From Steps 6, 9. Note \(\chi_-=\chi_+/2\). A

\[ m_0=\left(\frac{(2.0\times10^{5})(6)}{2(1.0\times10^{6})}\right)^{1/2}=\left(0.60\right)^{1/2}=0.775 \]

Insert numbers; \(m_0\) is dimensionless as assumed. A

\[ \chi_+=\frac{1}{2(2.0\times10^{5})(6)}=4.17\times10^{-7}\ \mathrm{m^3\,J^{-1}},\quad \chi_-=2.08\times10^{-7}\ \mathrm{m^3\,J^{-1}} \]

Evaluate above and below; amplitude ratio 2 confirmed. A

\[ m_0=0.775,\qquad \chi_+=4.17\times10^{-7},\quad \chi_-=2.08\times10^{-7}\ \mathrm{m^3\,J^{-1}} \]

Reading. Six kelvin below \(T_c\) the order parameter is already large (\(\sim0.78\)); the susceptibility below \(T_c\) is exactly half that above, the universal mean-field ratio. Units check. \(\chi=(\mathrm{J\,m^{-3}})^{-1}=\mathrm{m^3\,J^{-1}}\), as required for \(m=\chi h\) with \([h]=\mathrm{J\,m^{-3}}\).

Specific-heat jump and correlation length

Same \(a_0,b,T_c\), and gradient stiffness \(c_0=3.0\times10^{-9}\ \mathrm{J\,m^{-1}}\). Find the specific-heat jump \(\Delta c\) at \(T_c\) and the correlation length \(\xi\) at \(T-T_c=6\ \mathrm{K}\) (above \(T_c\)). Symbols first. B

\[ \Delta c=\frac{a_0^2 T_c}{2b},\qquad \xi=\left(\frac{c_0}{2a_0(T-T_c)}\right)^{1/2} \]

From Steps 13 and 17 with \(\chi^{-1}=2a_0(T-T_c)\) above \(T_c\). B

\[ \Delta c=\frac{(2.0\times10^{5})^2(600)}{2(1.0\times10^{6})}=\frac{(4.0\times10^{10})(600)}{2.0\times10^{6}}=1.2\times10^{7}\ \mathrm{J\,m^{-3}\,K^{-1}} \]

Insert numbers; the jump is a finite energy density per kelvin. B

\[ \xi=\left(\frac{3.0\times10^{-9}}{2(2.0\times10^{5})(6)}\right)^{1/2}=\left(\frac{3.0\times10^{-9}}{2.4\times10^{6}}\right)^{1/2}=(1.25\times10^{-15})^{1/2}=3.5\times10^{-8}\ \mathrm{m} \]

Evaluate; \(\xi\approx35\ \mathrm{nm}\), large compared with an atomic spacing, justifying coarse-graining. B

\[ \Delta c=1.2\times10^{7}\ \mathrm{J\,m^{-3}\,K^{-1}},\qquad \xi\approx3.5\times10^{-8}\ \mathrm{m} \]

Reading. Landau theory predicts a step in the specific heat (not a spike) and a correlation length that grows as \(|T-T_c|^{-1/2}\); at 6 K away it is already tens of nanometres and diverges as \(T\to T_c\). Units check. \([\Delta c]=(\mathrm{J\,m^{-3}\,K^{-1}})^2\cdot\mathrm{K}/(\mathrm{J\,m^{-3}})=\mathrm{J\,m^{-3}\,K^{-1}}\); \([\xi]=(\mathrm{J\,m^{-1}}/\mathrm{J\,m^{-3}})^{1/2}=\mathrm{m}\).

Problems

Show explicitly that the mean-field Ising self-consistent equation \(m=\tanh\!\big(\beta Jz\,m\big)\) reduces to a Landau expansion near \(T_c\), and identify \(a(T)\) and \(b\).
Solution
Expand \(\tanh x = x-\tfrac13 x^3+\dots\) with \(x=\beta Jz\,m\): \(m=\beta Jz\,m-\tfrac13(\beta Jz)^3 m^3\). Rearrange to \(0=(\beta Jz-1)m-\tfrac13(\beta Jz)^3 m^3\), i.e. the equilibrium condition of \(f=a m^2+b m^4\) with \(2a=1-\beta Jz\) and \(4b=\tfrac13(\beta Jz)^3\). At \(T_c\), \(\beta_c Jz=1\Rightarrow k_BT_c=Jz\). Near \(T_c\), \(\beta Jz=T_c/T\approx1+(T_c-T)/T_c\), so \(2a=1-T_c/T\approx(T-T_c)/T_c\Rightarrow a=a_0(T-T_c)\) with \(a_0=1/(2T_c)\) (in units where the free-energy scale is absorbed), and \(b\approx\tfrac1{12}>0\). This is exactly the Landau form, confirming that mean-field theory is a Landau theory.
Compute the amplitude ratio \(\chi_+/\chi_-\) for the susceptibility above and below \(T_c\) and explain its universality.
Solution
Above: \(\chi_+^{-1}=2a=2a_0(T-T_c)\). Below: \(\chi_-^{-1}=2a+12b\,m_0^2\) with \(m_0^2=-a/2b\), giving \(\chi_-^{-1}=2a+12b(-a/2b)=2a-6a=-4a=4a_0(T_c-T)\). At equal \(|T-T_c|\), \(\chi_+/\chi_-=|{-4a}|/|2a|=2\). The ratio is independent of \(a_0,b\) and of the material — it depends only on the mean-field structure — hence it is universal within the class, even though the individual amplitudes are not.
A system has \(a_0=1.5\times10^{5}\ \mathrm{J\,m^{-3}\,K^{-1}}\), \(b=8.0\times10^{5}\ \mathrm{J\,m^{-3}}\), \(T_c=350\ \mathrm{K}\). Find the order parameter at \(340\ \mathrm{K}\) and the condensation free-energy density there.
Solution
\(m_0=\sqrt{a_0(T_c-T)/2b}=\sqrt{(1.5\times10^5)(10)/(2\cdot8.0\times10^5)}=\sqrt{1.5\times10^6/1.6\times10^6}=\sqrt{0.9375}=0.968\). Condensation energy \(\Delta f=-a^2/4b\) with \(a=a_0(T-T_c)=-1.5\times10^6\ \mathrm{J\,m^{-3}}\): \(\Delta f=-(1.5\times10^6)^2/(4\cdot8.0\times10^5)=-2.25\times10^{12}/3.2\times10^6=-7.03\times10^{5}\ \mathrm{J\,m^{-3}}\). So the ordered phase lies \(7.0\times10^5\ \mathrm{J\,m^{-3}}\) below the disordered phase.
At the critical isotherm (\(T=T_c\)) a field \(h=5.0\times10^{4}\ \mathrm{J\,m^{-3}}\) is applied with \(b=1.0\times10^{6}\ \mathrm{J\,m^{-3}}\). Find \(m\), and confirm the exponent \(\delta\).
Solution
At \(a=0\), the equation of state is \(h=4b\,m^3\Rightarrow m=(h/4b)^{1/3}=\big(5.0\times10^4/(4\cdot1.0\times10^6)\big)^{1/3}=(1.25\times10^{-2})^{1/3}=0.232\). The relation \(m\propto h^{1/3}\) gives \(\delta=3\) directly: doubling \(h\) multiplies \(m\) by \(2^{1/3}\approx1.26\), the cube-root signature of \(\delta=3\).
Using the Ginzburg criterion, determine the upper critical dimension of Landau theory and explain what happens for \(d<4\).
Solution
The mean-field correlation length is \(\xi\propto|t|^{-\nu}=|t|^{-1/2}\) and the order parameter \(m_0\propto|t|^{\beta}=|t|^{1/2}\), where \(t=(T-T_c)/T_c\). The typical fluctuation \(\langle(\delta m)^2\rangle\) within a correlation volume scales as \(k_BT_c/(c_0\xi^{d-2})\propto|t|^{(d-2)/2}\cdot|t|^{... }\); comparing the fluctuation contribution to \(m_0^2\propto|t|\) gives a ratio \(\propto|t|^{(d-4)/2}\). For \(d>4\) this \(\to0\) as \(t\to0\): fluctuations are negligible and mean-field is exact. For \(d<4\) it diverges: fluctuations dominate arbitrarily close to \(T_c\), and the true exponents differ from mean-field (requiring the renormalisation group). Hence the upper critical dimension is \(d_u=4\), where hyperscaling \(2-\alpha=\nu d\) marginally holds with logarithmic corrections.