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Use $N_0={\nu }_s$ for the normal-state single-spin density of states per unit energy and volume at the Fermi surface. Write the attraction as $-V/\mathcal{V}$, with $V>0$, and define $\lambda =N_{0}V$.^[ref]

Since $V/\mathcal{V}$ has units of energy, $V$ has units of energy times volume, making $\lambda$ dimensionless. Set ${\Omega }_D=\hslash {\omega }_D$, an energy cutoff using the Debye frequency ${\omega }_D$, rather than a frequency itself. These conventions also fix the meaning of the earlier constant-shell BCS gap equation.

Here is the reduction of the BCS gap equation in these conventions. Substituting the zero-temperature BCS anomalous average givesThe $k$ sum counts one spin species. For $\Delta >0$, cancelling $\Delta$ and replacing this sum by an integral with constant $N_0$ givesThe factor two from the symmetric integral cancels the $1/2$ in the BCS anomalous average; it is not an extra spin degeneracy. Substituting $\xi =\Delta \sinh t$ evaluates the integral as $\mathrm{arsinh}({\Omega }_D/\Delta )$. Solving givesThe first equality is exact within the constant-density of states, sharp-cutoff mean-field approximation; the last is the weak coupling asymptote. Extending the first equality to large $\lambda$ does not make this simplified model an accurate description of a strongly coupled material.

For an original numerical example, choose $\hslash {\omega }_D=20\mathrm{meV}$ and $\lambda =0.25$. The exact constant-shell BCS gap equation gives $\Delta (0)=0.732871\mathrm{meV}$; the weak coupling formula gives $0.732626\mathrm{meV}$. Its fractional underestimate, measured against the exact result, is exactly $e^{-2/\lambda }=e^{-8}\simeq 0.03355\%$. At $\lambda =1$ that error is $e^{-2}\simeq 13.53\%$, so the two formulas are no longer interchangeable. No material parameters have been fitted here.

At nonzero temperature, substitute the finite-temperature BCS anomalous average instead. The same cancellation yieldsAt the superconducting transition temperature take the continuous $\Delta \to 0$ limit. The integrand at $\xi =0$ then has the finite limiting value $1/(2k_B{T}_c)$, so no divergent endpoint needs to be inserted into a numerical quadrature. For $k_B{T}_c\ll {\Omega }_D$, the resulting integral gives $k_B{T}_c\simeq 1.134{\Omega }_D{e}^{-1/\lambda }$ and $2\Delta (0)/(k_B{T}_c)\simeq 3.528$. These ratios concern the isotropic weak coupling model.^[ref]

At zero temperature and weak coupling, the BCS gap equation gives $\Delta \simeq 2\Lambda e^{-1/({\nu }_F|V|)}$. Its exponential scale agrees with the renormalization-group scale of the Cooper instability.

An incompressible, equal-site-volume mean-field approximation for two polymers of degrees of polymerization $N_A,N_B$. Here $f$ is mixing free energy per site and $\varphi$ is the volume fraction of species $A$. For a composition-independent Flory interaction parameter, local stability requires $1/(N_A\varphi )+1/[N_B(1-\varphi )]-2\chi >0$. The thermodynamic critical point is ${\varphi }_c=\sqrt{N_B}/(\sqrt{N_A}+\sqrt{N_B})$ and ${\chi }_c=(N_A^{-1/2}+N_B^{-1/2}{)}^2/2$.

For spins $s_i\in \{-1,0,1\}$ with ferromagnetic nearest-neighbour exchange and no single-ion anisotropy, a homogeneous mean-field approximation givesThe self-consistency equation is $S=2\sinh h/(1+2\cosh h)$ with $h=\beta (zJS+B)$. Its small-$h$ expansion is $2h/3-h^3/9+O(h^5)$. Hence $k_B{T}_c=2zJ/3$ and, with $t=T/T_c-1$, the nonzero branches satisfy $S^2=(8/3)(-t)+O(t^2)$. The critical exponent is the mean-field value $1/2$, not an exact two-dimensional exponent. At zero source the finite-system equilibrium magnetization is zero by spin inversion symmetry; the broken branches describe the thermodynamic limit with a phase selected by a vanishing source.

For ferromagnetic exchange $J$ and coordination $z$, the mean-field approximation has $f=zJ{M}^2/2-k_{B}T\log [1+2e^{-g/(k_{B}T)}\cosh (zJM/(k_{B}T))]$. With $K=zJ/(k_{B}T)$ and $q=2e^{-g/(k_{B}T)}/[1+2e^{-g/(k_{B}T)}]$, its quadratic and quartic coefficients vanish at $q=1/3$, $K=3$. The sextic coefficient of $f/(k_{B}T)$ is $9/40>0$. Hence $T_{\mathrm{TCP}}=zJ/(3k_B)$ and $g_{\mathrm{TCP}}=k_B{T}_{\mathrm{TCP}}\log 4$; these are not exact finite-dimensional lattice values.

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.

In dissociative adsorption, an arriving molecule splits into bound fragments. For the idealized process $A_2+2\ast \rightleftharpoons 2A\ast$, let $\ast$ denote a vacant adsorption site and let $\theta$ count sites occupied by $A$. A mean-field approximation with independent vacancies takes the forward rate proportional to $P(1-\theta {)}^2$ and the reverse rate proportional to ${\theta }^2$. Equating them yieldswith a positive coefficient $K$ of inverse-pressure units that includes the chosen pair-counting convention. This square-root dependence illustrates why the intact-molecule Langmuir isotherm cannot simply be assumed for splitting molecules. Requirements for neighboring vacancies or interactions can further change the mean-field approximation.

For a scalar Landau-de Gennes free energy with $\bar{b}>0$, $\bar{c}<0$, and coupling $-\chi E^2\lambda$ with $\chi >0$, the critical endpoint of a quartic Landau free energy is reached at the displayed field strength and ${\bar{a}}_c=3{\bar{c}}^2/(8\bar{b})$, ${\lambda }_c=-\bar{c}/(4\bar{b})$. It terminates the field-biased first-order phase transition when both field and quadratic coefficient can be tuned. This is a quartic small-field mean-field approximation; higher terms may alter its location.

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.

The Landau-de Gennes free energy is a rotational symmetry preserving Landau free energy of the nematic order parameter. One three-dimensional normalization through fourth order isThe cubic invariant generically produces a first-order phase transition. Its absence in two dimensions is an algebraic identity, not a guarantee that a mean-field approximation captures all fluctuations.

At a magnetic compensation temperature, $M_A=M_B\ne 0$: an ordered ferrimagnet has zero net magnetization. Such cancellation need not occur in every material.^[ref]

For an original stipulated example, let $t=T/T_C$, with $0\le t\le 1$, and prescribe sublattice magnitudes per common volume byTake A along $+z$ and B along $-z$, retaining a fixed sublattice orientation convention. These are toy functions, not measured curves or a self-consistent mean-field approximation. Their difference factors exactly:At $t=0$, $M_z=-M_0/2$. At $t=1/3$, both magnitudes equal $2M_0/3$ and $M_z=0$: this is compensation with finite order. At $t=1/2$, $M_z=M_0/8>0$. At $t=1$, both magnitudes vanish: that endpoint represents the stipulated Curie temperature, not another compensation point. For $t>1$ stipulate both ordered magnitudes zero; do not extrapolate the polynomials into negative sublattice magnitudes.

The sign change refers to the net component relative to the fixed A direction. A freely reorienting specimen in an applied magnetic field need not display this signed branch directly. The example isolates the cancellation mechanism; it predicts neither a material's transition temperature nor its response to magnetic domains or an applied magnetic field.

Mean-field theory replaces fluctuating interactions with a self-consistent average field. For example, an Ising spin feels a field determined by the average neighboring spin magnetization; a self-consistency equation approximates the interacting Ising model. Neglecting correlated fluctuations can fail near a continuous phase transition. The mean-field approximation in Landau theory is a particular use of this general framework.

The mean-field approximation neglects products of Ising spin deviations, replacing each Ising spin's neighbours by the common spin magnetization. With coordination number of a lattice $q$, the effective field is $h+qJm$. The self-consistency equation equates the assumed mean Ising spin with its independent-spin expectation in that effective field. It predicts $k_B{T}_c=qJ$ and the usual quartic mean-field critical exponents, while neglecting spatially correlated fluctuations.

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 functionalThe gradient term penalizes spatial variation, while the even local potential implements the zero-field symmetry $\varphi \mapsto -\varphi$. 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 $|\mathbf{\varphi }{|}^2$.

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 areBelow the transition these two minima exhibit spontaneous symmetry breaking. Differentiating the equation of state gives $\chi =(r+3u{M}^2{)}^{-1}$: 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

To include fluctuations, write $\varphi =M+\psi$. The quadratic part isA Gaussian functional integral therefore gives $⟨\psi (q)\psi (-q)⟩=(c{q}^2+r+3u{M}^2{)}^{-1}$, with the momentum-volume delta function understood. The Ornstein--Zernike correlation function has correlation length $\xi =\sqrt{c/(r+3u{M}^2)}$. Thus ${\nu }_{\mathrm{MF}}=1/2$, and at criticality the Gaussian covariance is proportional to $q^{-2}$, giving ${\eta }_{\mathrm{MF}}=0$. Continuous-symmetry ordered phases additionally have transverse Goldstone modes, for which the restoring mass vanishes.

The Gaussian integration contributes $\frac{1}{2}\int d^{d}q(2\pi {)}^{-d}\log (c{q}^2+r)$ to the disordered-phase free-energy density. Differentiating twice with respect to $r$ gives a critical contribution proportional to $\int d^{d}q(c{q}^2+r{)}^{-2}$; 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. Mean-field theory finds the competing minima, while fluctuations determine whether its predicted critical behavior survives at long distance.

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 uniform stationarity equation and local stability condition areFor $u>0$, $h=0$, and small negative $r$, the stable nonzero solution satisfies $M^2=-r/u+O(r^2)$. Thus the order parameter vanishes continuously as $r\uparrow 0$. The equilibrium Landau free energy has leading singular contribution $f_{\min }=-r^2/(4u)$ below the transition and zero above it. Its first thermal derivative is continuous, while its second derivative has a finite jump: the mean-field approximation predicts a continuous phase transition with no latent heat. The magnetic susceptibility follows by differentiating the stationarity equation:It diverges on approaching the ordinary thermodynamic critical point, and the quadratic Landau scalar correlation length behaves as $\xi \sim \sqrt{\kappa /|r|}$, with a different amplitude on the two sides.

For $u<0$, a nonzero stationary point at $h=0$ has $r+u{M}^2+v{M}^4=0$. Substitution into $V(M)-V(0)$ gives $-u{M}^4/4-v{M}^6/3$. Setting this to zero yieldsThe finite jump of $M$ makes this a first-order phase transition. For a thermal path $r=r(T)$ with $r^{\prime }(T)\ne 0$ and the other coefficients fixed, $\partial f_{\min }/\partial T=r^{\prime }(T)M^2/2$ jumps, giving an entropy discontinuity and nonzero latent heat. The Landau free energy itself remains continuous. The disordered minimum loses stability at $r=0$; the two nonzero stationary solutions merge at $r=u^2/(4v)$. These spinodal points bound metastability, but neither replaces the equilibrium phase coexistence condition.

Writing $r=at$ with $a>0$ near an ordinary thermodynamic critical point, the zero-field equation gives $M\sim (-t{)}^{1/2}$. At $r=0$, small $h$ obeys $h=u{M}^3+O(M^5)$, hence $M\sim \mathrm{sgn}(h)|h{|}^{1/3}$. The ordinary mean-field critical exponents are therefore $\beta =1/2$ and $\delta =3$. These are Landau theory predictions, not dimension-independent values for an interacting theory below its upper critical dimension.

At the tricritical point, both the quadratic and quartic even coefficients vanish, while the sextic coefficient remains positive. Two even control parameters must therefore be tuned, in addition to setting the symmetry-breaking conjugate field to zero. The continuous phase transition line and first-order phase transition line meet, the coexistence order parameter jump shrinks to zero, and the tricritical wings end there. The balance governing the equation of state changes from quartic to sextic stabilization.

Along the tricritical thermal path $u=0$, $r=at$, $h=0$, minimizing $V$ gives $M^4=-r/v$ on the ordered side. At $r=u=0$, the equation of state is $h=v{M}^5$. ConsequentlyThese mean-field critical exponents require the tuned tricritical path; a path with fixed $u>0$ instead has the ordinary exponents derived above. The corresponding equilibrium Landau free energy is $f_{\min }=-(-r{)}^{3/2}/(3\sqrt{v})$ below the transition. Its first thermal derivative is continuous but its second derivative diverges, giving the tricritical mean-field approximation a heat-capacity exponent $\alpha =1/2$. Fluctuations are negligible asymptotically only above the tricritical upper critical dimension three; at three dimensions logarithmic corrections can accompany these powers.

An order parameter distinguishes the equilibrium phases and captures the relevant broken symmetry. For an Ising model, it is the spontaneous magnetization: its sign changes under spin reversal, and it is zero in the symmetric phase. Spontaneous magnetization means taking the thermodynamic limit before sending the external field to zero; the finite-volume zero-field mean can vanish even in an ordered phase.

In Landau-Ginzburg theory, a slowly varying real scalar field $\varphi (x)$ represents the coarse-grained local order parameter. A useful local free-energy density gives the functionalAt zero field, spin-reversal symmetry excludes odd powers. The gradient term penalizes rapid spatial variation. The field $h$ is the field conjugate to an order parameter. The mean-field approximation replaces the fluctuating scalar field by the equilibrium minimizer; a uniform minimizer has $\varphi =M$ and obeys $rM+u{M}^3+v{M}^5=h$.

Write $\beta =1/(k_{B}T)$, with $k_B$ the Boltzmann constant. For independent Ising spins, the one-site partition functions on the two sublattices are $z_B=2\cosh \beta (h+g)$ and $z_W=2\cosh \beta (h-g)$. Differentiating each logarithm with respect to its local field givesAt $h=0$, the two sublattices experience opposite fields, so spin reversal combined with sublattice exchange leaves the Hamiltonian invariant and gives $M_W=-M_B$.

For the antiferromagnetic Ising model, take $N$ sites, equal sublattice sizes and $q=2D$ neighbors per site. There are $Nq/2$ bonds counted once. The two-sublattice Ising mean-field approximation expands each bond asand drops only the product of fluctuations. Summing the remaining terms givesThe printed constant omits $N$; for a total Hamiltonian with unnormalized spin sums, this factor is necessary. Applying the independent-spin calculation now givesThe mean-field approximation is favored by large coordination number: averaging many neighbors reduces local relative fluctuations. The size of critical fluctuations still has to satisfy the Ginzburg criterion; high dimension is a reason for validity, not permission to discard every correlation in every regime.

Antisymmetric solutions and the transition. At $h=0$, putting $M_B=-M_W=M$ makes both equations reduce to $M=\tanh \beta (g+qJM)$. A zero-field continuous transition concerns $g=0$ as well: nonzero $g$ explicitly selects the staggered sign and rounds this critical bifurcation. Put $K=\beta qJ$. The graphical intersections of $M$ and $\tanh (KM)$ give only $M=0$ for $K\le 1$. For $K>1$ there are additionally two nonzero intersections $\pm M_{\ast }$: on $M>0$, $\tanh (KM)-M$ is strictly concave, initially increases, and is negative at $M=1$, so there is exactly one positive root. Spin reversal supplies the negative root. The origin becomes unstable in the staggered direction, whereas the nonzero roots have $K(1-M_{\ast }^2)<1$ and are stable. Expanding about the bifurcation givesThus $M_{-}=(M_B-M_W)/2$ turns on continuously, with $k_B{T}_c=qJ$ and order-parameter critical exponent $1/2$.

Uniform solutions. At $g=0$, imposing $M_B=M_W=M$ instead gives $M=\tanh \beta (h-qJM)$. The function $M-\tanh \beta (h-qJM)$ has derivative $1+\beta qJ{\mathrm{sech}}^2\beta (h-qJM)>0$, is negative at $M=-1$ and positive at $M=1$, and consequently has exactly one zero for every positive temperature. At $h=0$ this zero is $M=0$. Antiferromagnetic exchange opposes uniform polarization, so the uniform branch has no ferromagnetic ordering bifurcation. This does not remove the antisymmetric solutions below $T_c$, or exclude transitions of staggered order in the full problem at nonzero uniform field. In particular, uniqueness on the diagonal $M_B=M_W$ is not uniqueness of all solutions.

Free energy and stability. The appropriate off-shell two-sublattice Ising variational free energy is the energy minus temperature times the independent-spin entropy. DefineThere are $Nq/2$ bonds and $N/2$ spins of each kind, so the free-energy density per site isSince $s^{\prime }(m)=\mathrm{atanh}m$, stationarity givesand the analogous equation for $M_W$; these are exactly the two mean-field self-consistency equations. On a stationary solution, the identity $s(m)=m\mathrm{atanh}m-\ln (2\cosh (\mathrm{atanh}m))$ gives the equivalent on-shell expressionThe entropy form must be used to compare off-shell variations; treating this latter auxiliary expression as the variational potential would give spurious stability conclusions.

Using $s(m)=-\ln 2+m^2/2+m^4/12+O(m^6)$ and $M_B=M_{+}+M_{-}$, $M_W=M_{+}-M_{-}$ givesThis includes the requested quadratic expansion and first-order field terms. The positive quartic coefficient verifies a continuous phase transition in $M_{-}$ when the staggered quadratic coefficient changes sign at $k_{B}T=qJ$. The uniform quadratic coefficient remains positive.

Susceptibilities and a printed error. On a stable zero-field ordered branch let $M_B=-M_W=M$ and $a=\beta (1-M^2)$. Since the derivative of $\tanh x$ is $1-{\tanh }^{2}x$, differentiating the two equations with respect to the uniform field givesTheir sum and difference yieldThe difference equation is $(1-qJa){\chi }_{-}=0$; stability implies $qJa<1$ for $T<T_c$. Symmetry also explains the zero result: a uniform perturbation changes the two sublattice magnetizations equally to first order, and hence does not linearly change the staggered magnetization. This ${\chi }_{-}$ is a cross response to $h$, not the magnetic susceptibility to the staggered field $g$.

The displayed ${\chi }_{+}$ in the source instead contains $1+M^2$. It cannot follow from differentiating the stated equations. A decisive limiting check is $T\to 0$: $1-M^2$ vanishes exponentially and the actual uniform response tends to zero, whereas the printed expression tends to $1/(qJ)$. Above $T_c$, $M=0$ and the same linear system givesFor comparison, the actual staggered susceptibility is $\partial M_{-}/\partial g=a/(1-qJa)$, becoming $1/(k_{B}T-qJ)$ above $T_c$. Its divergence, rather than a divergence of the uniform response, detects antiferromagnetic criticality.