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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.

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.

Write $\mathfrak{f}(u)$ for the physical free energy per site excluding the identity term, so $F/N=C+\mathfrak{f}(u)$. Blocking sends $u$ to $u^{\prime }=R_{b}u$ and the identity coefficient to $C^{\prime }=b^{D}C+c(u)$. Since the total partition function is unchanged and $N^{\prime }=N/b^D$,The function $g$ records the field-independent energy generated by eliminated modes, including their entropy and normalization. It is the identity-operator contribution to renormalization-group free energy; it must be tracked to recover the complete thermodynamics, even though it does not change normalized spin probabilities.

Iterating the additive free-energy recursion under blocking givesWhen a regular background $A(u)$ solves $A(u)=b^{-D}A(u^{\prime })+g(u)$, the singular remainder $f_s=\mathfrak{f}-A$ satisfies the homogeneous scaling equation $f_s(u)=b^{-D}{f}_s(u^{\prime })$. Thus the full free energy contains an inhomogeneous contribution, while the background-subtracted singular part is homogeneous. A finite initial identity energy $C_0$ adds a regular normalization, not a new magnetization singularity.

Absorb $\beta$ into the nonconstant couplings and write the corresponding dimensionless action as $H_u$. Choose its additive-constant convention so that one blocking step hasThe function $g(u)$ is the identity-operator contribution to renormalization-group free energy: it records the extensive constant generated by eliminating short-distance degrees of freedom. Summing over ${\sigma }^{\prime }$ givesFor the dimensionless free-energy density $\Phi (u)=-N^{-1}\ln Z(u,0,N)$, this is the inhomogeneous recursionFor the explicit constant $C$ in the original partition function, the corresponding new constant obeys $\beta C^{\prime }=b^D[\beta C+g(u)]$; $C$ and $g$ are not additional interactions between spins.

Iterating the recursion $n$ times yieldsAn analytic background $a(u)$ satisfying $a(u)=g(u)+b^{-D}a(Ru)$ absorbs the regular contribution. The remaining singular part $f_s=\Phi -a$ obeysThis separation explains both the scaling of the singular part and the role of the inhomogeneous term; simply dropping the generated constant would incorrectly change the full free energy.

Use ${\beta }_{\mathrm{th}}=1/(k_{B}T)$ for inverse temperature, to distinguish it from the order-parameter exponent. In a translation-invariant pure thermodynamic phase, the connected correlation function isThe subtraction is essential below the transition. Its exponential correlation length is defined by ${\xi }^{-1}=-{\lim }_{r\to \infty }{r}^{-1}\log |G(r)|$, with direction specified if the lattice is anisotropic. Away from criticality, $G(r)$ has a tail proportional to $e^{-r/\xi }$ times an algebraic prefactor. In the massive continuum scalar approximation that prefactor is $r^{-(D-1)/2}$; at criticality the exponential cutoff disappears and $G(r)\asymp r^{-(D-2+\eta )}$ instead. The large-distance limit here is $r\gg \xi$, as the PDF shows.

Let $F_{\mathrm{tot}}=-{\beta }_{\mathrm{th}}^{-1}\log Z$. With the energy convention $-h{\sum }_r{\sigma }_r$, the magnetization per site and magnetic susceptibility areWriting $S={\sum }_r{\sigma }_r$, direct differentiation of the partition function gives ${\partial }_h⟨S⟩={\beta }_{\mathrm{th}}(⟨S^2⟩-⟨S{⟩}^2)$. Thus the correlation-function susceptibility sum rule isIn physical continuum coordinates the lattice sum becomes $a^{-D}\int d^{D}rG(r)$, so omitting the site-density factor would change the normalization. A derivative with respect to the dimensionless source ${\beta }_{\mathrm{th}}h$ instead absorbs the explicit inverse-temperature factor.

An exact renormalization-group transformation with length factor $b>1$ can be defined by a nonnegative normalized blocking kernel $K_b({\sigma }^{\prime },\sigma )$ satisfying ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$. A deterministic majority rule is one example; randomized tie-breaking preserves its normalization. Define the blocked energy, including its constant, bySumming over ${\sigma }^{\prime }$ proves $Z(u^{\prime },C^{\prime },N^{\prime })=Z(u,C,N)$. All generated operators must be retained for this identity to remain exact; a finite-coupling truncation is an approximation. After $p$ iterations, $N_p=N/b^{pD}$ and the spacing is $a_p=b^{p}a$. Long-distance physics is preserved because this is a marginalization of microscopic probabilities, with observables and sources transformed along with the energy. It integrates short-distance variables rather than changing their effects on retained observables. Physical lengths remain fixed even though the correlation length in units of the new spacing is reduced by $b^p$.

For clarity, now use dimensionless free energy per site $F(u,C)=-N^{-1}\log Z=f(u)+{\beta }_{\mathrm{th}}C$. Exact equality of the partition functions givesA constant microscopic energy shifts every blocked weight by the same factor, so $C^{\prime }=b^{D}C+c(u)$. Substitution at one step yields the additive free-energy recursion under blockingInduction now gives the requested relationThe function $g$ is the identity-operator contribution to renormalization-group free energy: it records eliminated-mode entropy, integration constants and normalization factors. A free-energy function satisfying this inhomogeneous equation need not be purely singular until its analytic background has been subtracted.

At a renormalization-group fixed point, $R_b{u}_{\ast }=u_{\ast }$. Linearize in scaling coordinates $v_i$ so $v_i^{\prime }=b^{{\lambda }_i}{v}_i$. Here ${\lambda }_i$ is a scaling exponent, not the discrete multiplier itself. A positive ${\lambda }_i$ gives a relevant operator, a negative one an irrelevant operator, and zero a marginal operator. The critical surface is the stable manifold: tune every relevant scaling field to zero and the irrelevant coordinates flow toward $u_{\ast }$. A repulsive renormalization-group trajectory leaves along a relevant direction; it can still have irrelevant coordinates contracting toward that manifold. The plot shows the $h=0$ section when temperature and field are relevant: $t=0$ is the critical line in this section, while an independent nonzero field is another direction away from criticality.

The inhomogeneous term cannot simply be discarded from the full free energy. If an analytic background $A(u)$ solves $A(u)=b^{-D}A(R_{b}u)+g(u)$, its subtraction leaves $F_s=f-A$ obeying homogeneous scaling. For example, in linear thermal and field coordinates, a source monomial $g_{mn}{t}^m{h}^n$ is removed byWhen the denominator vanishes, the source is resonant and can instead produce a logarithm. Thus the homogeneous singular equation is justified after analytic subtraction when there is no relevant resonance affecting the leading singularity, no marginal logarithm that has been omitted, and no singular dependence on a dangerously irrelevant coupling. Merely saying that $g$ is regular is insufficient to discard the whole weighted sum without this analysis.

Under these assumptions, with two relevant scaling fields $t,h$, choose $p$ so that $b^{p{\lambda }_t}|t|$ is of order one. The scaling hypothesis for critical phenomena then givesWriting ${\Phi }_{\pm }=f_{\pm }+I_{\pm }$ recovers the PDF's notation:The sign labels the two matching branches $t>0$ and $t<0$, which enter different phases when the flow leaves the critical neighborhood. The constants $I_{\pm }$ retain a branch-dependent identity contribution or a chosen split of the scaling function. They are determined by the same matching problem, not independently adjustable nonuniversal free-energy offsets. An arbitrary analytic offset from $C_0$ belongs to the subtracted background. The formula describes the singular part; adding that background recovers the full physical free energy.

At $h=0$, restore the common metric factors as $F_{s,\pm }=A_f|a_{t}t{|}^{D/{\lambda }_t}{\Phi }_{\pm }(0)$. Their ratio at equal $|t|$ is ${\Phi }_{+}(0)/{\Phi }_{-}(0)$, proving the expected universal singular free-energy amplitude ratio because $A_f$ and $a_t$ cancel. The ratio of full free energies including an arbitrary regular $C_0$ is not universal; zero amplitudes and logarithmic resonances also require separate treatment.

The critical exponents are obtained by rescaling and differentiating, not by identifying them with the discrete multipliers. The correlation length obeys $\xi (t,h)=b^p\xi (b^{p{\lambda }_t}t,b^{p{\lambda }_h}h)$, hence $\nu =1/{\lambda }_t$. Two temperature derivatives of $F_s(t,0)$ give $C_{V,s}\asymp |t{|}^{D/{\lambda }_t-2}$, while one and two field derivatives give the ordered magnetization and susceptibility powers. ThusFor $\delta$, instead set $t=0$ and choose the blocking scale by $h$, obtaining $F_s(0,h)\asymp |h{|}^{D/{\lambda }_h}$ and then differentiate. A nonzero spontaneous ordered-branch derivative is assumed in the formula for $\beta$. In particular the hyperscaling relation is $\boxed{\alpha =2-D\nu }$. Its physical assumption is an order-one singular free energy per correlation volume, with a single diverging length and no dangerous variable changing its volume scaling. Above the upper critical dimension, a stabilizing quartic coupling can invalidate that assumption even though the correlation length still scales.

For the last model, take the quadratic Hamiltonian exactly as visually printed: its source term is $+h\varphi$, outside the half multiplying the gradient and mass terms. With $\kappa >0$ and $m^2>0$, its connected quadratic covariance is proportional toThe pole scale, or the massive Gaussian field correlation tail obtained by its Fourier transform, givesThe uniform source shifts the mean to $⟨\varphi ⟩=-h/m^2$ and does not change the connected covariance. Replacing $h$ by $-h$ matches the earlier magnetic source convention without changing any scaling index.

Define a Gaussian thinning transformation by retaining momenta $|q|<\Lambda /b$, integrating the Gaussian shell, and rescalingThe gradient coefficient is unchanged. Counting volume and field powers in the remaining action givesShell integration contributes only the additive determinant term. Thus ${\lambda }_t=2$ when the thermal variable is proportional to $m^2$, and ${\lambda }_h=(D+2)/2$. Inserting these into the homogeneous index formulas gives the requested Gaussian critical exponentsThe specific-heat power can also be verified without hyperscaling: two mass-squared derivatives of $\frac{1}{2}\int d^{D}q(2\pi {)}^{-D}\log ({\kappa }^{-1}{q}^2+m^2)$ produce an integral proportional to $\int d^{D}q(q^2+\kappa m^2{)}^{-2}$. For $0<D<4$, rescaling $q=\sqrt{\kappa }mp$ gives $(m^2{)}^{D/2-2}$; at $D=4$ the divergence is logarithmic, with power index zero. At $D=2$ the determinant itself has an additive $m^2\log m^2$ resonance, although its heat-capacity power remains one.

There is a genuine limitation to the last printed request. The Gaussian order-parameter scaling index ${\beta }_G$ is formal: a pure quadratic theory has $⟨\varphi ⟩=0$ at $h=0,m^2>0$, and at $m^2<0$ the uniform energy $m^2{\varphi }^2/2$ is unbounded below, so its partition function diverges. It therefore has no stable ordered branch from which a spontaneous-magnetization exponent can be measured. The thinning derivation establishes the index requested by the PDF, but not a nonexistent ordered phase. A stabilizing quartic interaction is relevant for $D<4$ and generally changes these Gaussian indices to the interacting critical ones.

Take the connected correlation functionFor a massive phase its large-distance form is an algebraic prefactor times $e^{-|r|/\xi }$, defining the exponential correlation length $\xi$. To specify the normalization of magnetic susceptibility, add $-h{\sum }_r{\sigma }_r$ to the energy and differentiate the mean spin. Translation invariance gives the correlation-function susceptibility sum ruleUsing the dimensionless source ${\beta }_{\mathrm{th}}h$ removes the prefactor ${\beta }_{\mathrm{th}}$. Below the transition, use a selected pure phase rather than a mixture of the two magnetizations, which would have a macroscopic disconnected fluctuation.

Choose a nonnegative blocking kernel $K_b({\sigma }^{\prime },\sigma )$ normalized by ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$. It may enforce a block average, a majority rule or a chosen rescaled coarse spin. Define the transformed weight bySumming this identity over coarse spins shows exact preservation of the partition function. The geometrical changes are $a^{\prime }=ba$ and $N^{\prime }=N/b^D$. A coordinate rescaling subsequently restores the reference cutoff if desired. The transformation is exact only if all generated operators are retained; a finite-coupling truncation is an approximation.

Define the coupling-independent term generated when $C=0$ byRestoring the original $C$ gives the identity-operator contribution to renormalization-group free energy:The parameter $C_p$ is an accumulated additive energy per coarse site. It multiplies the identity operator and leaves normalized spin probabilities unchanged, but it must be kept when calculating the absolute free energy. It records the eliminated-mode entropy and the normalization of their statistical measure.

Use an energy-per-site convention: $F(u,C)=-\log Z(u,C,N)/({\beta }_{\mathrm{th}}N)$ and $f(u)=F(u,0)$, so $F=f+C$. Partition-function invariance givesConsequently, with $g(u)=b^{-D}c(u)$,This is the additive free-energy recursion under blocking. In a dimensionless free-energy convention multiply $f,C,c,g$ by ${\beta }_{\mathrm{th}}$; the same recursion holds. The inhomogeneous term comes from the generated identity operator, not from a change in the spin-dependent couplings alone.

The affine equation naturally applies to the unsubtracted free energy. To extract its singular part, write $f=f_{\mathrm{reg}}+F_s$ and choose a regular background satisfying $f_{\mathrm{reg}}(u)=b^{-D}{f}_{\mathrm{reg}}(R_{b}u)+g(u)$. Subtraction then gives $F_s(u)=b^{-D}{F}_s(R_{b}u)$. This analytic subtraction of an inhomogeneous renormalization recursion is justified when the eliminated finite-shell contribution is analytic near the fixed point and admits a regular subtraction. It can fail in a resonant direction, producing additive logarithms. The simple homogeneous formulas also assume the remaining irrelevant operators are not dangerously irrelevant couplings, and that marginal operators do not produce unaccounted logarithmic corrections. Calling the quantity in the affine equation “singular” does not make $g$ vanish automatically.

Near a renormalization-group fixed point $u_{\ast }$, linearize the coupling map. In scaling coordinates $v_i$, write $v_i^{\prime }=b^{{\lambda }_i}{v}_i$ to leading order. Here $b^{{\lambda }_i}$ is a discrete multiplier, whereas ${\lambda }_i$ is its logarithmic scaling exponent. A relevant operator has ${\lambda }_i>0$ and grows under coarse-graining; an irrelevant operator has ${\lambda }_i<0$ and decays. A marginal operator needs nonlinear analysis. Tuning the relevant fields to zero places the system on the critical surface. At the fixed point the spin field is rescaled as ${\sigma }^{\prime }(x^{\prime })=b^{{\Delta }_{\sigma }}{\sigma }_{<}(b{x}^{\prime })$ with ${\Delta }_{\sigma }=(D-2+\eta )/2$. Its conjugate source therefore has ${\lambda }_h=D-{\Delta }_{\sigma }=(D+2-\eta )/2$. This accounts for both coupling and spin rescaling; the field normalization is part of the transformation, not a change in the physical microscopic spin length.

Assume the thermal field $t=(T-T_c)/T_c$ and magnetic field $h$ are the only remaining relevant directions. The homogeneous scaling hypothesis for critical phenomena givesChoose the stopping scale with $b^{p{\lambda }_t}|t|=1$. Thenwhere $f_{+}$ is the branch for $t>0$ and $f_{-}$ the branch for $t<0$. Nonuniversal metric factors can be absorbed into the definitions of $t,h$ and these functions. Correlation lengths transform as $\xi (t)=b^p\xi (b^{p{\lambda }_t}t)$, so $\nu =1/{\lambda }_t$. Differentiating the singular free energy twice with respect to temperature gives $C_s\propto |t{|}^{D/{\lambda }_t-2}$, and hence the hyperscaling relationTwo source derivatives similarly give $\gamma =(2{\lambda }_h-D)/{\lambda }_t$. A single source derivative gives the ordered-branch exponent $\beta =(D-{\lambda }_h)/{\lambda }_t$, and setting $t=0$ and scaling $h$ to one gives $\delta ={\lambda }_h/(D-{\lambda }_h)$ when the corresponding magnetization branch exists. These show explicitly how the critical exponents follow from the fixed-point scaling eigenvalues.