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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.

Use ${\beta }_{\mathrm{th}}=1/(k_{B}T)$ for inverse temperature to distinguish it from the order-parameter exponent. The decaying two-point correlation function needed for the response identity is connected:Away from the critical point in a short-range Ising model pure phase, the correlation length controls the exponential long-distance falloff, $G(r)\sim A(r)e^{-|r|/\xi }$ with a possible algebraic prefactor. The full two-spin expectation below the transition instead tends to $M^2$. The spin-mixture contribution to zero-field susceptibility explains why a symmetric mixture is unsuitable for defining a finite connected correlation length in the ordered phase.

Let $F=-{\beta }_{\mathrm{th}}^{-1}\log Z$ be the total Helmholtz free energy. Take magnetization and susceptibility per site, soDifferentiating the Boltzmann weights at fixed temperature givesTranslation invariance then yields the correlation-function susceptibility sum ruleTotal magnetization and total susceptibility multiply these expressions by $N$; per physical volume they have an additional factor $a^{-D}$ relative to the per-site convention. These normalizations do not change the critical powers.

For the subsequent scaling requests, let $F_s$ denote singular free energy per site; the fixed extensive normalization is suppressed. The labels $+$ and $-$ denote $t>0$ and $t<0$, with distinct scaling functions and amplitudes on the two sides. Below the transition a one-sided derivative at $h=0$ selects a spontaneous-magnetization branch. Writing $x=h/|t{|}^{\Delta }$, differentiation of the scaling hypothesis for critical phenomena givesThe field-independent $I_{\pm }$ makes no contribution to these field derivatives, although it can change heat-capacity amplitudes. Assuming nonzero leading amplitudes and the indicated pure-phase limits, comparison with the exponent definitions givesThis is the Rushbrooke scaling relation. A temperature-independent critical isotherm requires the large-$|x|$ magnetization scaling function to grow as $\mathrm{sgn}(x)|x{|}^{\beta /\Delta }$: otherwise $M_s=|t{|}^{\beta }{\mathcal{M}}_{\pm }(x)$ would retain a temperature power as $t\to 0$ at fixed small $h$. Therefore $1/\delta =\beta /\Delta$. Since the preceding two equalities also give $\Delta =\beta +\gamma$, the Widom scaling relation follows:For the hyperscaling relation, additionally assume that one correlation volume supplies an order-one singular free-energy contribution and that there is a single isotropic correlation length, with no dangerously irrelevant coupling spoiling this argument. Then $F_s\sim {\xi }^{-D}\sim |t{|}^{D\nu }$. Two temperature derivatives give the singular heat capacity, soHyperscaling is an additional physical assumption, not a consequence of the two-variable equation alone, and generally fails above the upper critical dimension.

Finally the correlation scaling function $f_G(x)$ falls exponentially, up to algebraic factors, for $x\to \infty$ in a massive short-range pure phase. At $x\to 0$ it approaches a finite nonzero value in the scaling regime. Approximating the lattice sum at distances much larger than $a$, and denoting the area of the unit sphere by $S_{D-1}$, givesIf $\eta <2$ and the integral has a finite nonzero limiting value, microscopic distances contribute only a regular background and the leading divergence is ${\xi }^{2-\eta }$. Thus the Fisher scaling relation isThese statements concern leading singular powers; logarithmic corrections and cancellations of amplitudes require separate treatment.