Ginzburg criterion

The Ginzburg criterion requires order-parameter fluctuations inside a correlation volume to be small compared with the squared mean-field order parameter. Its failure marks the fluctuation-dominated critical region.

For the dimensionless scalar Landau-Ginzburg theory with stiffness $c$, quadratic coefficient $r<0$ and quartic term $u{\varphi }^4/4$, the mean-field approximation gives ${\varphi }_0^2=|r|/u$ and correlation length $\xi \asymp (c/|r|{)}^{1/2}$. Restricting the Gaussian Fourier transform covariance $(c{q}^2+2|r|{)}^{-1}$ to $|q|\lesssim {\xi }^{-1}$ estimates the variance of the field averaged over a correlation volume as $c^{-d/2}|r{|}^{(d-2)/2}$, up to a dimension-dependent constant. Dividing by ${\varphi }_0^2$ proves the displayed Ginzburg criterion ratio. It diverges below four dimensions, vanishes above four and is marginal at four. Short-distance fluctuations renormalize microscopic coefficients and should not be confused with this long-distance critical test.

The critical region is the range of control parameters near a thermodynamic critical point where correlated order parameter fluctuations invalidate a simple mean-field approximation. For short-range scalar quartic theory below four dimensions, the dimensionless Ginzburg criterion ratio grows as $g{\xi }^{4-D}$ in fixed microscopic units. The range where this ratio is of order one or larger is fluctuation dominated. Its width depends on nonuniversal microscopic coefficients even though the limiting critical exponents can be universal.

For a scalar quartic interaction $g{\varphi }^4/4!$, the tadpole relation $R=r_{\Lambda }+(g/2)I_D(R)$ can be subtracted at the critical point. When $I_D(0)$ exists, $A(T-T_C)=R[1+(g/2)J_D(R)]$, with $J_D(R)=\int d^{D}p(2\pi {)}^{-D}/[p^2(p^2+R)]$. This integral is finite as $R\to 0$ only above four dimensions and is logarithmic at four. A self-consistent one-loop relation is not an exact interacting critical-exponent formula.

For the scalar quartic one-loop critical-mass subtraction, the subtracted tadpole is $I_D(0)-I_D(R)=R{J}_D(R)$. Above four dimensions $J_D(0)=K_D{\Lambda }^{D-4}/(D-4)$ is finite. At four dimensions $J_4=(K_4/2)\log [({\Lambda }^2+R)/R]$. For $2<D<4$, $J_D\sim K_D{R}^{(D-4)/2}{\int }_0^{\infty }{q}^{D-3}(1+q^2{)}^{-1}dq$, which diverges as $R\to 0$. Hence a smooth linear thermal mass survives this test only above the ordinary upper critical dimension four. The massless subtraction is infrared divergent for $D\le 2$ and cannot prove absence of an interacting transition there.

For $R>0$, a spherical ultraviolet cutoff gives the following exact examples of the one-loop critical-mass subtraction integral:The areas of the unit spheres in dimensions three, four and five are $4\pi$, $2{\pi }^2$ and $8{\pi }^2/3$. Dividing by $(2\pi {)}^D$ reduces the integrals to constants times ${\int }_0^{\Lambda }dp/(p^2+R)$, ${\int }_0^{\Lambda }pdp/(p^2+R)$ and ${\int }_0^{\Lambda }{p}^{2}dp/(p^2+R)$ respectively. Elementary integration proves the formulas. As $R\to 0$, the first has a power divergence, the second a logarithmic divergence, and the third a finite limit. These examples make the ordinary upper critical dimension four visible without treating a self-consistent one-loop equation as an exact interacting critical exponent prediction.

At an upper critical dimension, the leading Ginzburg criterion ratio has zero thermal power rather than a negative one. It does not vanish asymptotically, but its magnitude can depend on the coupling. The criterion alone does not establish a power-law divergence or new critical power exponents. Running marginal interactions generally produce logarithmic corrections; the ordinary quartic and tricritical sextic scalar theories are examples.

For a tuned multicritical even Landau potential with leading positive $M^{2n}$ term, $M^2\sim |t{|}^{1/(n-1)}$ and $\xi \sim |t{|}^{-1/2}$. Long-wavelength Gaussian fluctuations in a correlation volume have variance proportional to ${\xi }^{2-D}$. Their ratio to $M^2$ therefore scales as $|t{|}^{(D-2)/2-1/(n-1)}$, vanishing only for $D>2n/(n-1)$. At equality the criterion is marginal, requiring further fluctuation analysis.