= Solution
<Landau-Ginzburg theory> describes a slowly varying <order parameter> by a local <free-energy functional> consistent with the microscopic symmetries. For a representative scalar system with spin-inversion symmetry, take the dimensionless functional
$$
\mathcal F[\phi]=\int d^dx\left[\frac c2(\nabla\phi)^2+\frac r2\phi^2+\frac u4\phi^4-h\phi\right],\qquad c,u>0,\quad r=r_0t.
$$
The gradient term penalizes spatial variation, while the even local potential implements the zero-field symmetry $\phi\mapsto-\phi$. The positive quartic coefficient stabilizes a continuous transition. A negative quartic coefficient requires higher powers and can instead produce a <first-order phase transition>. A vector <order parameter> gives the corresponding rotationally invariant model with powers of $|\boldsymbol\phi|^2$.
In the <mean-field approximation>, replace the field by a constant $M$ and minimize the local potential. Its equation of state and stable zero-field solutions are
$$
h=rM+uM^3,\qquad
M=0\quad(r>0),\qquad M=\pm\sqrt{-r/u}\quad(r<0).
$$
Below the transition these two minima exhibit <spontaneous symmetry breaking>. Differentiating the equation of state gives $\chi=(r+3uM^2)^{-1}$: it is $1/r$ above the transition and $1/(2|r|)$ within either ordered phase. At $r=0$, $M=(h/u)^{1/3}$. The minimized zero-field <free-energy density> is zero above and $-r^2/(4u)$ below, apart from an analytic background; its second temperature derivative has a finite jump. Consequently
$$
\boxed{\beta_{\rm MF}=\tfrac12,\quad\gamma_{\rm MF}=1,\quad\delta_{\rm MF}=3,\quad\alpha_{\rm MF}=0.}
$$
To include fluctuations, write $\phi=M+\psi$. The quadratic part is
$$
\mathcal F_2=\frac12\int\frac{d^dq}{(2\pi)^d}\left(cq^2+r+3uM^2\right)|\psi(q)|^2.
$$
A <Gaussian functional integral> therefore gives $\langle\psi(q)\psi(-q)\rangle=(cq^2+r+3uM^2)^{-1}$, with the momentum-volume delta function understood. The <Ornstein--Zernike correlation function> has <correlation length> $\xi=\sqrt{c/(r+3uM^2)}$. Thus $\nu_{\rm MF}=1/2$, and at criticality the Gaussian covariance is proportional to $q^{-2}$, giving $\eta_{\rm MF}=0$. Continuous-symmetry ordered phases additionally have transverse <Goldstone modes>, for which the restoring mass vanishes.
The Gaussian integration contributes $\frac12\int d^dq\,(2\pi)^{-d}\log(cq^2+r)$ to the disordered-phase <free-energy density>. Differentiating twice with respect to $r$ gives a critical contribution proportional to $\int d^dq\,(cq^2+r)^{-2}$; for $d<4$ its singular part grows as $r^{(d-4)/2}$ and at four dimensions it is logarithmic. Interacting fluctuations thus become important close to the critical point below four dimensions. The <Ginzburg criterion> tests this failure of the <mean-field approximation>; the <renormalization group> then explains modified exponents and <universality>. \b[Mean-field theory finds the competing minima, while fluctuations determine whether its predicted critical behavior survives at long distance.]
Back to article page
