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The Ginzburg criterion tests whether fluctuations on the scale of one correlation volume are small compared with the squared equilibrium order parameter. In the scalar quartic Landau-Ginzburg theory of part (a), the ordered mean-field value is $M^2=|r|/u$, and its longitudinal correlation length is proportional to $(c/|r|{)}^{1/2}$.

Average the fluctuation field over a region of size $\xi$. Its smoothing factor selects momenta of order ${\xi }^{-1}$ or smaller. The Gaussian correlation function then gives, up to a positive dimension-dependent constant,The last step follows directly by setting $q=\sqrt{|r|/c}k$. Dividing by $M^2$ proves the correlation-volume derivation of the scalar Ginzburg ratio:For $d<4$, the ratio diverges as the transition is approached, so there is a fluctuation-dominated critical region of a phase transition in which mean-field critical exponents cannot be trusted. With $r=r_{0}t$, an estimate of its width isin fixed microscopic units. Its numerical coefficient depends on the averaging convention and microscopic normalization.

For $d>4$, the ratio tends to zero and the long-distance mean-field approximation becomes self-consistent. Four is therefore the ordinary quartic upper critical dimension. At $d=4$ this test is marginal: its thermal power is zero, and the running quartic interaction produces logarithmic corrections. The marginal Ginzburg criterion alone cannot decide those logarithms. Short-distance contributions to an unaveraged variance renormalize the coefficients; they should not be mistaken for the long-wavelength fluctuations used in this criterion. Nor does the criterion by itself prove that a proposed ordered phase exists below its lower critical dimension.

The relevant boundary here is the upper critical dimension: above it the critical interactions become negligible at long distances and the ordinary mean-field critical exponents apply; at it there can be logarithmic corrections. This differs from the lower critical dimension, below which a particular kind of ordering may fail to exist.

Normalize the coefficient of $(\nabla \varphi {)}^2$ in the Landau-Ginzburg theory. Under a change of length scale $x=b{x}^{\prime }$, its invariance gives the engineering dimension $x_{\varphi }=(d-2)/2$. The quartic interaction then scales asIt is a relevant operator below four dimensions and an irrelevant operator above four. Thus the ordinary scalar transition has upper critical dimension $d_c=4$. At a tricritical point, where the quartic interaction is tuned away, the sextic coupling scales as $v^{\prime }=b^{6-2d}v$, giving the tricritical upper critical dimension $d_c=3$. This is the upper critical dimension of an even scalar interaction applied to powers four and six.

The Ginzburg criterion makes the reason for failure of the Landau approximation quantitative. In the ordered quartic theory, $M^2=|r|/u$, $V^{\prime \prime }(M)=2|r|$ and ${\xi }^2=K/(2|r|)$. The fluctuation of the order parameter averaged over a correlation volume has the scaleCompared with the squared mean order parameter, the Ginzburg criterion therefore givesFor $d<4$ this ratio grows without bound as the correlation length diverges. The assumption that fluctuations are small relative to the ordered value fails arbitrarily near the transition, and the interacting renormalization-group fixed point determines the critical exponents. For $d>4$ the ratio tends to zero. At $d=4$ scale counting is marginal, and the running quartic coupling produces logarithms.

Similarly, at the tricritical point $M^2\asymp \sqrt{K/v}{\xi }^{-1}$, so the multicritical Ginzburg ratio scales as $(k_{B}T\sqrt{v}/K^{3/2}){\xi }^{3-d}$. The Landau-Ginzburg theory remains a useful effective description below its upper critical dimension; it is its fluctuation-free exponent calculation that fails, not the possibility of including fluctuations in the theory.

Write $m_i={\mu }_i^2$ and $r^2={\varphi }_1^2+{\varphi }_2^2$. The uniform Landau free energy isso stationarity requiresFor $m_1,m_2>0$, the unique ground state is $(0,0)$ and the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ discrete symmetry is unbroken. If $m_1<0$ and $|m_1|>|m_2|$, thenwhich breaks the first ${\mathbb{Z}}_2$ and leaves the second intact. This includes the case in which both masses are negative, because $m_1$ is then the more negative one. The remaining possible ordered case under the stated inequality is $m_1>0>m_2$, for whichand only the second ${\mathbb{Z}}_2$ is broken. The Hessian matrix in each ordered state is positive because the uncondensed direction has squared mass $m_k-m_j>0$, where $m_j$ is the condensed, more negative mass.

Near either continuous transition the nonzero order parameter is proportional to $(-m_i{)}^{1/2}$. Hence the mean-field critical exponents are

The lower critical dimension is the dimension at or below which fluctuations destroy the proposed finite-temperature ordered phase. Here the broken symmetry is discrete, so $d_{\mathrm{l}}=1$. In one dimension a domain wall interpolating between the two signs has finite energy, whereas its possible position gives an entropy growing as $\log L$. Domain walls therefore occur with nonzero density at every positive temperature and split the system into domains of finite typical length. Thus there is no finite-temperature ordered phase in one dimension.

For finite $p=4$, the clock model has a discrete symmetry. Domain walls have finite energy per unit boundary area, so thermal disorder destroys long-range order in one dimension but a finite-temperature ordered phase can exist in two dimensions. Its lower critical dimension is therefore $\boxed{d_{\mathrm{l}}=1}$.

As $p\to \infty$, the permitted angles become continuous and the model becomes the XY model with $O(2)$ symmetry. The Mermin-Wagner theorem forbids spontaneous long-range order at positive temperature in two dimensions, so the lower critical dimension for conventional symmetry breaking is $\boxed{d_{\mathrm{l}}=2}$. The two-dimensional model can nevertheless undergo a Berezinskii–Kosterlitz–Thouless transition between algebraic and exponential correlation decay.