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The correlation-length critical exponent is defined by $\xi \sim |t{|}^{-\nu }$. At a renormalization-group fixed point, its reciprocal is the positive eigenvalue of the temperature-like scaling direction.

The quartic interaction $g_1{\varphi }_1^4+g_2{\varphi }_2^4+\lambda {\varphi }_1^2{\varphi }_2^2$ includes the Gaussian fixed point, single and double Ising model Wilson-Fisher fixed points, and an O(N) model fixed point with $N=2$. A rotation of two decoupled identical Ising models produces a second coordinate representation of their renormalization-group fixed point.

The critical surface is the stable manifold of a critical renormalization-group fixed point. Tuning every relevant direction places a theory on this surface.

For dimensionless nearest-neighbour coupling $K$ and diagonal coupling $L$, the second cumulant in checkerboard decimation of the square-lattice Ising model generates $K^2$ for each pair of retained neighbours of an eliminated centre. A retained diagonal pair shares two centres, and an axial pair at distance $2a$ shares one. The displayed two-coupling recursion neglects $K^4$, $K^{2}L$, and $L^2$, treating $L$ as order $K^2$. Its finite nontrivial renormalization-group fixed point is $(K,L)=(1/3,1/9)$. Higher orders generate additional operators, so the closure is a property of the truncation.

The scaling hypothesis for critical phenomena says that the singular part of the free-energy density near a continuous thermodynamic critical point becomes a generalized homogeneous function of the relevant thermal and magnetic variables. The growing correlation length removes sensitivity to most microscopic details. Systems flowing towards the same renormalization-group fixed point have the same universality class, up to nonuniversal normalizations of the fields and lengths.

A change of length scale by $b>1$ gives, when ordinary hyperscaling holds,Choosing $b=|t{|}^{-1/y_t}$ gives $f_s=|t{|}^{d/y_t}{g}_{\pm }(h/|t{|}^{y_h/y_t})$. More generally the specified homogeneous form uses exponents $2-\alpha$ and $\Delta$ without presupposing hyperscaling. Its thermodynamic derivatives determine the critical exponents. Analytic backgrounds are separated from this singular part.

Throughout the calculation, fractional powers of the thermal variable mean $|t|$, with different scaling functions above and below the transition. The spontaneous magnetization exponent uses the ordered branch $t<0$ with $h\to 0^{+}$ after the thermodynamic limit. Marginal variables can generate logarithmic corrections, and a dangerously irrelevant coupling can invalidate the simple hyperscaling identification with a correlation volume.

The renormalization group organizes how a statistical system changes when observed at progressively larger length scales. Start with its partition function and a microscopic cutoff $\Lambda$. Split the Fourier transform field into modes below $\Lambda /b$ and modes in the shell $\Lambda /b<|q|<\Lambda$. Integrate the shell modes, rescale coordinates $x^{\prime }=x/b$ to restore the cutoff, and rescale the field to keep a chosen gradient normalization. This is a renormalization-group transformation. It preserves the partition function when the generated interactions and additive free energy terms are retained; truncating the resulting operator expansion is an approximation.

A renormalization-group fixed point has an invariant dimensionless interaction structure. The continuous flow of a nearby scaling field $v_i$ takes the linearized formA positive $y_i$ defines a relevant operator, a negative $y_i$ an irrelevant operator, and a zero $y_i$ a marginal operator requiring nonlinear analysis. The continuous-flow eigenvalues $y_i$ should not be confused with the discrete multipliers $b^{y_i}$. To reach a critical point, tune the relevant thermal variable and set the relevant symmetry-breaking source to zero. The resulting flow stays on the critical surface.

Microscopically different systems whose remaining couplings approach the same fixed point share a universality class. Irrelevant perturbations lose their influence on long-distance exponents. Relevant directions explain why varying the temperature or field moves the system away from criticality. The thermal scaling field obeys $t^{\prime }=b^{y_t}t$, while the rescaled correlation length is ${\xi }^{\prime }=\xi /b$. Taking $b=|t{|}^{-1/y_t}$ therefore gives $\nu =1/y_t$.

If the field has scaling dimension $x_{\varphi }=(d-2+\eta )/2$, the source term $\int h\varphi$ has eigenvalue $y_h=d-x_{\varphi }=(d+2-\eta )/2$. The singular free-energy density then obeyswhen the hyperscaling relation is valid. This derives the scaling hypothesis for critical phenomena and the exponent relations of Question 2 from the fixed-point picture.

At the Gaussian fixed point, keeping $\int (\nabla \varphi {)}^2$ invariant gives $x_{\varphi }=(d-2)/2$. The quadratic coupling has $y_r=2$ and the quartic coupling has $y_u=4-d$. Thus quartic interactions are relevant below four dimensions, marginal at four, and irrelevant above four. Below four, in dimensions where an ordinary continuous short-range scalar transition exists, an interacting Wilson-Fisher fixed point governs that transition. Above four, the Gaussian fixed point gives mean-field powers, but the quartic term remains necessary to stabilize the ordered state: it is a dangerously irrelevant coupling, explaining the failure of naive hyperscaling. The renormalization group connects scale invariance, universality, relevant tuning parameters and fluctuation corrections in one framework.

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 ${\beta }_T=(k_{B}T{)}^{-1}$ for inverse temperature to distinguish it from the order-parameter critical exponent. A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ is a nonnegative conditional distribution of coarse spins given the fine spins, normalized byFor deterministic blocking it can impose a majority rule or block average, with a Dirac delta replacing the sum constraint for continuous coarse variables. Each block occupies $b^D$ original cells, so the new lattice spacing and number of sites areThe physical volume $N{a}^D$ is unchanged. After $p$ blockings, $a_p=b^{p}a$ and $N_p=b^{-pD}N$.

Define the coarse statistical Hamiltonian by the exact Boltzmann-weight identityThe operator family must include all generated interactions for this renormalization-group transformation to be exact; restricting it to a few couplings is an approximation. The scalar $g(u)$ has energy units and is the contribution per original site that is independent of ${\sigma }^{\prime }$. Restoring the original constant term $NC$ giveshenceThis convention measures $g$ per original site; defining a constant per new site would move a factor $b^D$ into its definition.

The identity-operator contribution to renormalization-group free energy $C_p$ records the configuration-independent normalization generated by the eliminated spins. It includes their contribution to free energy, including entropy and measure factors, and need not be independent of temperature. It has no effect on normalized probabilities of the retained spins, but is indispensable for the total partition function and its temperature derivatives. Iterating its recursion givesThus $N_p{C}_p$ is the original constant $N{C}_0$ plus the accumulated free-energy contributions of the eliminated degrees of freedom.

To obtain the additive free-energy recursion under blocking, use energy-valued free energy per current lattice site,The blocking kernel normalization impliesSubstituting the recursion for $C^{\prime }$ cancels $C$, givingA second step gives $f_0(u_0)=b^{-2D}{f}_0(u_2)+g(u_0)+b^{-D}g(u_1)$, and induction provesFor dimensionless free energy $-\log Z/N$, multiply $F,f_0,g,C$ throughout by ${\beta }_T$.

The singular free-energy density must be distinguished from the exact per-site representative $f_0$. If $a(u)$ is a regular background and $f_s=f_0-a$, thenThus the same displayed inhomogeneous recursion holds with $f_s,g_s$ after subtraction. When a suitable analytic background solves $a(u)=g(u)+b^{-D}a(Ru)$, it leaves a homogeneous singular recursion. Analytic subtraction of an inhomogeneous renormalization recursion explains this separation. A renormalization-group free-energy resonance can obstruct analytic subtraction and produce a logarithm. The original function $g$ originates in the short-distance spins that were summed out, rather than in an extra external interaction.

Near a renormalization-group fixed point $u_{\ast }$, linearize the transformation:An eigenvector of $\mathsf{A}$ with eigenvalue $b^{y_i}$ defines a scaling coordinate $t_i$ with $t_i^{\prime }=b^{y_i}{t}_i$ to leading order. A relevant operator has $y_i>0$: its coupling grows towards large distances, so it must be tuned to reach the fixed point. An irrelevant operator has $y_i<0$ and dies away; a marginal operator has $y_i=0$ and requires higher-order flow analysis. An operator of scaling dimension $x_i$ has $y_i=D-x_i$.

The coarse spins also require a multiplicative normalization. In continuum notation, writewhere ${\varphi }_{<}$ is the retained field and $\eta$ is its anomalous dimension. This restores the normalization of the critical long-distance connected correlation function. Since a uniform field conjugate to an order parameter enters through $\int d^{D}xH\varphi$, it has eigenvalue exponent $y_h=D-x_{\varphi }=(D+2-\eta )/2$.

Let $t$ be the thermal relevant coordinate with exponent $y_t>0$. After regular backgrounds have been removed, the leading singular free-energy density obeysfor a large total scale factor $B=b^p$. Choosing $B=|t{|}^{-1/y_t}$ gives the scaling hypothesis for critical phenomenaTwo temperature derivatives give the heat-capacity critical exponent; one conjugate-field derivative gives the order-parameter critical exponent; two such derivatives give the magnetic-susceptibility critical exponent. Scaling the correlation length gives $\xi (t)=B\xi (B^{y_t}t)$. Consequently,At $t=0$, instead choose $B=|H{|}^{-1/y_h}$; differentiating gives $\delta =y_h/(D-y_h)$. These scaling relations for critical exponents assume that irrelevant variables can be set to their fixed-point values without singular consequences. A dangerously irrelevant coupling or logarithmic correction requires keeping the corresponding dependence.

For the Gaussian field theory, integrate out the Fourier modes in $\Lambda /b<|q|<\Lambda$ and retain ${\varphi }_{<}$. The modes are independent at zero external field, so this Gaussian thinning transformation changes the normalization by a Gaussian determinant and leaves a quadratic statistical Hamiltonian for the retained modes. Restore the cutoff withIn the integrated statistical Hamiltonian, the volume element contributes $b^D$, each gradient contributes $b^{-1}$, and each field contributes $b^{-(D-2)/2}$. Thus the coefficient ${\kappa }^{-1}$ of the gradient energy is unchanged, whileIn particular $y_t=2$, $y_h=(D+2)/2$ and $\eta =0$. The Gaussian critical exponents and the Gaussian order-parameter scaling index therefore give the requested indicesThe sign convention $+h\varphi$ corresponds to the physical conjugate field $H=-h$ and does not affect these powers.

There are physical qualifications to these Gaussian indices. The strictly quadratic Gaussian field theory is stable for positive $m^2$ and has no stable ordered phase for negative $m^2$; its purported spontaneous $\beta$ is therefore the formal Gaussian order-parameter scaling index. A stabilizing quartic term gives an actual ordered phase. Below four dimensions that term is relevant and changes the exponents; above four dimensions it is a dangerously irrelevant coupling and gives the actual mean-field $\beta =1/2$, rather than the Gaussian hyperscaling value.

The heat capacity power can also be checked directly. Apart from constants, the zero-field Gaussian free-energy density isIf $m^2\propto t$, its singular second temperature derivative is controlled by $\int d^{D}q(q^2+\kappa m^2{)}^{-2}$. For $0<D<4$, rescaling $q=\sqrt{\kappa m^2}k$ gives a power $t^{D/2-2}$, as above. At $D=4$ there is a logarithm, and for $D>4$ an analytic background dominates the heat capacity. The negative formal Gaussian $\alpha$ then describes the subleading singular piece, with logarithmic resonances at appropriate even dimensions. This is the Gaussian specific-heat infrared threshold, and it prevents confusing singular indices with the entire observable.

Keeping both gradient coefficients fixed requires $z^2=B{b}^2=B{c}^4$. HenceEquivalently the real-space field normalization is $z_x=z/B=b^{(3-d)/4}$. The transformed thermal and source couplings areThis is the Gaussian fixed point at $t=h=0$, with anisotropic effective volume exponent $D_{\mathrm{eff}}=(d+1)/2$. The stiffnesses remain positive parameters labeling the normalization of coordinates and fields. A strictly positive $t$ is a relevant perturbation away from the renormalization-group fixed point, not itself a critical renormalization-group fixed point.

The two correlation lengths also follow directly by balancing the kernel: $K{q}_{\parallel }^2\sim t$ gives ${\xi }_{\parallel }\sim (K/t{)}^{1/2}$, while $L{q}_{\perp }^4\sim t$ gives ${\xi }_{\perp }\sim (L/t{)}^{1/4}$. Their different exponents are consistent with the required anisotropic scaling. An omitted scalar quartic interaction has scaling eigenvalue $4-D_{\mathrm{eff}}=(7-d)/2$, so the quadratic renormalization-group fixed point alone does not establish the interacting smectic's critical behavior below physical dimension seven.

The renormalization group compares descriptions at successively longer length scales. First eliminate short-wavelength degrees of freedom, then rescale coordinates and fields to restore the reference cutoff. This changes the couplings, including interactions and additive free-energy terms generated during elimination. A renormalization-group fixed point is a scale-invariant description. Linearized directions growing under coarse graining are relevant, shrinking directions are irrelevant, and zero-eigenvalue directions are marginal and require higher-order analysis.

A diverging correlation length means that many coarse-graining steps can approach a renormalization-group fixed point before a relevant perturbation becomes large. Systems with different microscopic details but the same renormalization-group fixed point and relevant directions share universal critical behavior. For the quadratic model below, Fourier modes decouple, so eliminating a high-momentum shell produces a Gaussian determinant but does not couple the remaining modes. The anisotropic gradient kernel requires different length rescalings in the parallel and transverse directions. This is an anisotropic renormalization group, not an isotropic rescaling of all $d$ coordinates.

The scaling hypothesis for critical phenomena describes the singular part of the equilibrium free-energy density, rather than its analytic background. For an isotropic system in spatial dimension $D$, with thermal scaling field $t$ and conjugate field $h$, writeThis follows from coarse graining near a renormalization-group fixed point: a correlation volume is enlarged by $b^D$, while the relevant thermal and field perturbations grow with their eigenvalues. Choosing $b=|t{|}^{-1/y_t}$ givesThe two functions describe the two sides of the transition. Nonuniversal metric factors multiplying $t,h,f_s$ must be allowed when comparing microscopic models. Exponents and suitably normalized scaling functions or amplitude ratios characterize a universality class.

Differentiating the free-energy density gives the order parameter $M=-{\partial }_h{f}_s$, magnetic susceptibility $\chi ={\partial }_{h}M$ and singular heat capacity proportional to $-{\partial }_t^2{f}_s$. ThusThe subscript distinguishes the order-parameter exponent from inverse temperature. At $t=0$, choose $b=|h{|}^{-1/y_h}$ to obtain $M\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$, with $\delta =y_h/(D-y_h)$. These derivatives imply the Rushbrooke scaling relation $\alpha +2{\beta }_{\mathrm{mag}}+\gamma =2$ and the Widom scaling relation $\gamma ={\beta }_{\mathrm{mag}}(\delta -1)$. Equivalently, the equation of state has $h=|t{|}^{\Delta }{\mathcal{H}}_{\pm }(M/|t{|}^{{\beta }_{\mathrm{mag}}})$, with $\Delta ={\beta }_{\mathrm{mag}}\delta ={\beta }_{\mathrm{mag}}+\gamma$.

The correlation length scales as $\xi \sim |t{|}^{-\nu }$, $\nu =1/y_t$. At criticality the connected correlation function behaves as $G(r)\sim r^{-(D-2+\eta )}$, giving $y_h=(D+2-\eta )/2$. Integrating $G$ over a correlation volume gives $\chi \sim {\xi }^{2-\eta }$ and hence the Fisher scaling relation $\gamma =(2-\eta )\nu$. The hyperscaling relation $2-\alpha =D\nu$ expresses one singular free-energy contribution per correlation volume.

These laws organize measurements: plotting $M/|t{|}^{{\beta }_{\mathrm{mag}}}$ against $h/|t{|}^{\Delta }$ collapses near-critical curves onto the appropriate scaling function. A finite system replaces the divergent length by its size $L$; the corresponding prediction is $f_s=L^{-D}\mathcal{F}(t{L}^{y_t},h{L}^{y_h})$ under the same scaling assumptions. Finite size rounds a sharp thermodynamic singularity.

The hypothesis has a domain of validity. At a renormalization-group fixed point one generally needs $f_s=b^{-D}{f}_s(b^{y_t}t,b^{y_h}h,b^{y_u}u,\dots )$, including other scaling fields. Ordinary irrelevant couplings can be set to zero in a nonsingular limiting scaling function. A dangerously irrelevant coupling cannot: above the upper critical dimension of scalar quartic theory, the quartic term is irrelevant at the Gaussian point but still stabilizes the ordered phase. Consequently naive hyperscaling fails there even though other exponent relations can remain valid. Marginal variables can produce logarithms, anisotropic systems require the appropriate volume exponent, and the Berezinskii–Kosterlitz–Thouless transition has an essential correlation-length singularity instead of a finite power-law $\nu$. Scaling theory relates observables through a small number of fixed-point dimensions; it does not make analytic backgrounds, marginal logarithms or dangerously irrelevant variables disappear.

The Berezinskii–Kosterlitz–Thouless transition is a defect-unbinding transition of a two-dimensional system with a compact continuous phase, such as the short-range XY model. At long wavelengths, smooth spin waves have energywhere ${\rho }_s$ is the phase stiffness. Gaussian averaging givesThus smooth fluctuations already destroy true nonzero magnetization in an infinite two-dimensional system, in accordance with the Mermin-Wagner theorem. The low-temperature phase instead has quasi-long-range order, with algebraically decaying correlations. The stiffness in its asymptotic exponent is the renormalized long-distance stiffness, not merely the microscopic coupling.

The phase is periodic modulo $2\pi$, so it also permits topological defects with integer winding $n$. A phase vortex has $|\nabla \theta |=|n|/r$, and integration givesA singly charged phase vortex can be placed in roughly $(L/a{)}^2$ locations, giving entropy $2k_B\log (L/a)$. This energy-entropy balance suggests that isolated vortices become favorable when the effective stiffness falls to $2k_{B}T/\pi$. Below the transition, vortex-antivortex binding predominates; above it their unbinding screens the logarithmic interaction and produces a finite correlation length. A single-vortex estimate identifies the mechanism but does not include screening by smaller pairs.

The dilute-vortex vortex-pair renormalization flow incorporates this screening. With $K={\rho }_s/(k_{B}T)$ and dimensionless core fugacity $y$, one conventional leading-order normalization givesThe first equation says polarizable phase vortex pairs reduce stiffness. The second compares the phase vortex's position scaling dimension two with its logarithmic energy cost $\pi K$. Numerical coefficients in $y$ depend on its definition, but the zero-fugacity threshold $K=2/\pi$ is invariant. For $K>2/\pi$, sufficiently small fugacity flows to zero, leaving a line of Gaussian spin-wave renormalization-group fixed points with varying algebraic exponent. On the other side of the separatrix, fugacity grows and the dilute approximation eventually ceases to apply.

Near the endpoint set $X=2-\pi K$ and $Y=4\pi y$. To leading order $X^{\prime }=Y^2$ and $Y^{\prime }=XY$, so $X^2-Y^2$ is invariant. The critical separatrix approaches $(0,0)$. Just on the disordered side this invariant is $-{\epsilon }^2$, proportional to the reduced temperature deviation for a generic path through the transition. Hence $X^{\prime }=X^2+{\epsilon }^2$, whose integration gives a time in logarithmic scale of order $1/\epsilon$. Stopping when the fugacity becomes order one yieldswith nonuniversal $B>0$. This derives the essential singularity and explains why assigning a finite ordinary correlation-length exponent is inappropriate.

At the transition the limiting stiffness from below has the universal stiffness jumpAbove $T_c$, the thermodynamic long-distance stiffness vanishes and correlations decay exponentially. The singular free-energy contribution is of order ${\xi }^{-2}$ and is exponentially small in $1/\sqrt{\tau }$, so this infinite-order transition is unlike a Landau transition with a discontinuous order parameter or an ordinary power-law heat-capacity singularity. The renormalization mechanism is developed in Kosterlitz's original vortex analysis. The transition separates bound vortices and algebraic order from unbound vortices and short-range order, while neither finite-temperature phase has conventional spontaneous XY magnetization.

The Landau-Ginzburg theory for a scalar order parameter supplements a local symmetry-allowed potential with a gradient penalty. For a ${\mathbb{Z}}_2$ symmetry at zero field, a stable quartic model iswith $r=a(T-T_c)$ near a mean-field transition. Equilibrium fluctuations have weight $e^{-F/(k_{B}T)}$. In the Landau approximation, minimize the functional. A uniform stationary state obeys $r\varphi +u{\varphi }^3=h$. At zero field the minimum is $\varphi =0$ for $r>0$ and $\varphi =\pm \sqrt{-r/u}$ for $r<0$. The latter pair is spontaneous symmetry breaking of the discrete symmetry.

The uniform free-energy density at its minimum is zero above the transition and $-r^2/(4u)$ below, up to regular backgrounds. Differentiating the equation of state gives ${\chi }_{+}=1/r$ and ${\chi }_{-}=1/(2|r|)$. At $r=0$, $\varphi =(h/u{)}^{1/3}$. Gaussian fluctuations around the minimum have inverse covariance $(K{q}^2+r)/(k_{B}T)$ above and $(K{q}^2+2|r|)/(k_{B}T)$ below. Their correlation lengths are ${\xi }_{+}=(K/r{)}^{1/2}$ and ${\xi }_{-}=(K/(2|r|){)}^{1/2}$. These results give the usual mean-field critical exponentsThe heat capacity has a finite jump rather than a power-law divergence. Negative quartic coefficient requires higher stabilizing terms and can produce a first-order transition; a vanishing quartic coefficient requires a tricritical analysis. Thus the assumed positivity of $u$ matters.

The Ginzburg criterion tests whether this saddle description is self-consistent. Below the transition compare fluctuations of the order parameter averaged over a correlation volume with its squared mean-field value ${\varphi }_0^2=|r|/u$. Modes with $q\lesssim {\xi }^{-1}$ dominate this coarse-grained fluctuation estimate:where the positive dimensionless $A_D$ depends on the averaging window and the factor of two used in $\xi$. Dividing by ${\varphi }_0^2$ givesIt is the fluctuation of a correlation-volume average, not the full cutoff-dominated microscopic variance, that should be compared with the mean-field order parameter.

For $D<4$, $\mathcal{G}$ diverges as $r\to 0$, so the asymptotic critical region is fluctuation dominated. Away from it, mean-field theory can still work whenFor $D>4$, the long-wavelength ratio tends to zero near criticality, consistently with Gaussian control and mean-field thermodynamic exponents. At $D=4$, quartic coupling is marginal; scale-dependent logarithms require a renormalization-group analysis rather than interpreting a constant bare ratio as a proof of no corrections. This identifies four as the upper critical dimension of the short-range scalar quartic model.

The Landau-Ginzburg theory is valuable because symmetries restrict a manageable list of local operators, the gradient term describes correlations and interfaces, and the Ginzburg criterion identifies when neglected fluctuations become decisive. Below the upper critical dimension an interacting renormalization-group fixed point, rather than the bare saddle, determines universal exponents. For continuous order parameters the manifold of ordered minima also has Goldstone modes, and lower-critical-dimension fluctuations require separate attention. Mean-field minimization supplies an approximation; the Ginzburg ratio decides whether its neglect of fluctuations is justified.

Write ${\beta }_T=(k_{B}T{)}^{-1}$ to distinguish inverse temperature from an order-parameter critical exponent. For the Ising model, define the full two-point correlation function $G(r)=⟨{\sigma }_0{\sigma }_r⟩$ and its connected correlation function $G_c(r)=G(r)-M^2$ in a translation-invariant phase. Away from a thermodynamic critical point, the large-distance envelope decays exponentially, possibly multiplied by a power of distance:This defines the correlation length in physical distance units when the limit exists. At a continuous phase transition, $\xi \to \infty$ and the critical connected correlation function typically has algebraic decay $G_c(r)\sim |r{|}^{-(D-2+\eta )}$, where $\eta$ is the anomalous dimension. Below the transition the full $G(r)$ tends to $M^2$, so the decay definition must use $G_c$.

Let $S={\sum }_r{\sigma }_r$ and $F=-{\beta }_T^{-1}\log Z$ be the extensive Helmholtz free energy. Holding the field-independent constant $C$ and other physical couplings fixed,Because the Hamiltonian contains $-hS$, ${\partial }_h\log Z={\beta }_T⟨S⟩$ and ${\partial }_h^2\log Z={\beta }_T^2(⟨S^2⟩-⟨S{⟩}^2)$. Translation invariance then proves the correlation-function susceptibility sum rule:In the ordered zero-field phase these statements refer to a selected pure thermodynamic phase; averaging equally over opposite magnetizations introduces the separate spin-mixture contribution to zero-field susceptibility.

A normalized blocking kernel $K_b({\sigma }^{\prime },\sigma )\ge 0$ satisfies ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$. For example, assign one block spin to each block of $b^D$ microscopic spins using majority sign, with equal probabilities for a tie. Define the blocked Hamiltonian, retaining the entire generated operator space, byThe field-independent contribution $N{g}_E$ belongs to the identity operator. With $N^{\prime }=N{b}^{-D}$ andsumming over ${\sigma }^{\prime }$ proves $Z(u,C,N)=Z(u^{\prime },C^{\prime },N^{\prime })$ exactly. Iterating gives $N_p=N{b}^{-pD}$ and the same partition function with $(u_p,C_p,N_p)$. Observables at large scales are reproduced by appropriately blocked observables and sources. Truncating the generated interactions makes this an approximation; dropping the additive constant already destroys the equality of free energies even when normalized spin expectations remain correct.

The extensive Helmholtz free energy is unchanged: $F(u_0,C_0,N)=F(u_p,C_p,N_p)$. In contrast, its value per site changes because the number of sites changes. Define the reduced zero-constant free-energy density $f_0(u)={\beta }_{T}F(u,0,N)/N$ in the thermodynamic limit. The preceding equality yieldsand iteration proves the additive free-energy recursion under blocking:Equivalently $C_p=b^{pD}[C_0+{\sum }_{j=0}^{p-1}{b}^{-jD}{g}_E(u_j)]$. Thus the inhomogeneous source records the identity-operator contribution to renormalization-group free energy from eliminated short-distance degrees of freedom.

To express this recursion for the singular part, write $f_0=a+f_s$, where $a$ is the regular analytic background. Then the same displayed recursion holds with $f_s$ andThis is the analytic subtraction of an inhomogeneous renormalization recursion. A purely analytic source can often be absorbed into $a$, after which it does not control the leading singular powers. It is not legitimate simply to discard an analytic source in the full free energy, where it can dominate the regular background. Moreover an analytic monomial $c{t}^m{h}^n$ contributesIf $m{l}_t+n{l}_h=D$, the sum is $p$ and hence generates a logarithm when $p\sim -\log |t|/(l_t\log b)$: a renormalization-group free-energy resonance. Such a term cannot be discarded when describing that logarithmic singularity. Neglect of the inhomogeneous term in the leading power-law scaling therefore assumes that its regular part has been subtracted, that any remaining source only changes finite scaling amplitudes, and that there are no relevant resonances or marginal logarithms. Omitting an irrelevant operator further requires absence of a dangerously irrelevant coupling, whose vanishing limit can make a scaling function singular.

A renormalization-group fixed point obeys $R_b{u}_{\ast }=u_{\ast }$. Linearization gives $\delta u^{\prime }=A\delta u$; in eigenvector coordinates $w_i^{\prime }=b^{l_i}{w}_i$. Positive $l_i$ label relevant operators, negative $l_i$ label irrelevant operators, and zero $l_i$ labels a marginal operator requiring nonlinear analysis. The critical surface is the stable manifold obtained by tuning all relevant scaling fields to zero. A repulsive renormalization-group trajectory leaves the fixed point along a relevant direction as the observation length increases. Locally a typical zero-field section has $dw/d\ell =-\omega w$, $dt/d\ell =l_{t}t$, $\ell =\log b$, so the critical surface $t=0$ attracts along $w$, while trajectories with either sign of $t$ depart from it. A magnetic scaling field adds another relevant direction.

Under the stated homogeneous-scaling assumptions and after fixing irrelevant fields at their limiting values, two relevant scaling fields obey $t^{\prime }=b^{l_t}t$, $h^{\prime }=b^{l_h}h$. These fields are analytic coordinates near the renormalization-group fixed point, proportional to the physical reduced temperature and conjugate field to leading order. The singular free-energy density consequently obeysHere $F_s$ denotes the singular free energy per original site or unit volume, with regular dimensional factors absorbed; the extensive singular part is $N{F}_s$. Iterate and choose $b^p=|t{|}^{-1/l_t}$ to obtain the scaling hypothesis for critical phenomena:The sign labels the two sides of the transition: $f_{+}$ describes $t>0$, the disordered side, while $f_{-}$ describes $t<0$, the ordered side. The two functions need not coincide; $f_{-}$ has the zero-field one-sided derivative appropriate to spontaneous magnetization. The numbers $l_t,l_h$ are scaling eigenvalues in logarithmic form: the corresponding discrete RG multipliers are $b^{l_t},b^{l_h}$.

The correlation length obeys $\xi (t,h)=b\xi (b^{l_t}t,b^{l_h}h)$. The same scale choice gives $\nu =1/l_t$. Differentiating $F_s$ once and twice with respect to the conjugate field, and twice with respect to temperature, givesAt $t=0$, instead choose the scale $b^p=|h{|}^{-1/l_h}$. Then $F_s(0,h)\sim |h{|}^{D/l_h}$ and $M\sim \mathrm{sgn}(h)|h{|}^{D/l_h-1}$, so $\delta =l_h/(D-l_h)$. Thus differentiation and scale matching establishThe first is the Widom scaling relation and the second the hyperscaling relation. The hyperscaling relation requires precisely the regular homogeneous scaling used here. Above the ordinary upper critical dimension, the quartic coupling is dangerously irrelevant in the ordered phase and this derivation cannot discard it: the mean-field critical exponents $\nu =1/2$, $\alpha =0$ do not obey $\alpha =2-D\nu$ for $D>4$. At marginal dimensions logarithmic factors likewise require care beyond the displayed pure powers.