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Partition the lattice into blocks of side $b$ sites, with $b>1$, and retain one coarse spin ${\sigma }_R^{\prime }$ per block. A blocking kernel is a nonnegative conditional weight $K_b({\sigma }^{\prime },\sigma )$ normalized by ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$ for every microscopic configuration. A deterministic majority rule, with fair resolution of ties, is one example.

Define the blocked statistical Hamiltonian and its field-independent constant bySumming over ${\sigma }^{\prime }$ proves exact equality of the old and new partition functions. The lattice spacing becomes $ba$; expressing lengths again in the original units completes the real-space renormalization group step. Blocking generally generates many-body and longer-range interactions, so exactness requires retaining the full operator space and the additive constant. A finite-coupling truncation is a calculational approximation.

A blocking kernel is a conditional probability $K({\sigma }^{\prime }\mid \sigma )$ for replacing the original spins by block spins. It must satisfyfor each original configuration. For example, divide the lattice into blocks of side $b$ and assign each block the sign of its total spin; a tie can be resolved with equal probabilities for the two signs. The product of these block rules is a normalized blocking kernel.

The real-space renormalization group defines the effective Boltzmann weight by summing the old weights against this kernel. Its normalization preserves the full partition function. The blocked lattice has spacing $ba$ and $N^{\prime }=N/b^D$ sites. Rescaling lengths restores the original lattice-spacing convention. An exact transformation generally produces every symmetry-allowed interaction, not just the nearest-neighbor term; retaining only a few couplings is a further approximation.

Use ${\beta }_T=1/(k_{B}T)$ for inverse temperature, to distinguish it from the order-parameter critical exponent $\beta$. In a translation-invariant pure thermodynamic phase, the connected two-point correlation function isFor finite correlation length, at $|r|\gg \xi$ it decays exponentially, possibly with an algebraic prefactor; ${\xi }^{-1}$ is the asymptotic exponential decay rate. Below an ordered transition the pure phase is selected by taking the thermodynamic limit before $h\to 0^{+}$ or $h\to 0^{-}$. A zero-field symmetric mixture of the two ordered phases has a nondecaying connected term and does not describe the usual pure-phase magnetic susceptibility.

Let $\mathcal{F}=-k_{B}T\log Z$ denote extensive physical free energy and $S={\sum }_r{\sigma }_r$. Since the field contributes $-hS$ to the statistical Hamiltonian, differentiation givesExpanding the variance and using translation invariance proves the correlation-function susceptibility sum rule:The sum includes $r=0$. If the source is instead defined as the dimensionless quantity ${\beta }_{T}h$, the corresponding derivative omits the displayed inverse-temperature factor.

A real-space renormalization group step of length factor $b>1$ uses a local blocking kernel $B_b({\sigma }^{\prime },\sigma )\ge 0$ satisfying ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$. For example, a deterministic majority-spin rule assigns one coarse Ising spin to each block; tied blocks can be resolved with equal probabilities. Define the blocked statistical Hamiltonian byThe spacing becomes $a^{\prime }=ba$. All generated interactions, including multispin terms and the field-independent constant, must be retained for an exact transformation. Summing over ${\sigma }^{\prime }$ uses normalization to prove $Z(u^{\prime },C^{\prime },N^{\prime })=Z(u,C,N)$. Iteration gives $N_p=b^{-pD}N$ and $Z(u_p,C_p,N_p)=Z(u_0,C_0,N)$.

The coarse probability law is the exact marginal law of these block variables. Indeed, for any coarse observable $A$, its expected value is the microscopic expected value of ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )A({\sigma }^{\prime })$. A blocking rule that retains the long-wavelength order-parameter mode, with source and observable transformations tracked, therefore retains long-distance response and correlation functions. Their field normalization may change, while their physical length scales and critical singularities do not. Partition-function equality alone would not justify this assertion for an arbitrary kernel that discards all slow variables. Truncating the generated operator family is likewise an approximation.

For the free-energy recursion use dimensionless free energy per site,in the thermodynamic limit. Exact partition-function invariance gives $F(u_0,C_0)=b^{-pD}F(u_p,C_p)$. If one step sends $C$ to $C^{\prime }=b^{D}C+c(u)$, the coefficient $c(u)$ records the identity-operator contribution from eliminated fluctuations and normalization. ConsequentlyRepeated substitution derives the additive free-energy recursion under blocking:In physical energy units, divide this convention by ${\beta }_T$. The additive term is essential for absolute free energies even when it does not affect the leading critical powers.

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. Linearize in scaling coordinates $v_i$ so that $v_i^{\prime }=b^{y_i}{v}_i+O(v^2)$. A relevant operator has $y_i>0$, an irrelevant operator has $y_i<0$, and a marginal operator has $y_i=0$ with behaviour decided by nonlinear terms. Thus the eigenvalues of the discrete linear map are $b^{y_i}$; the sign criterion applies to the generator exponents $y_i$, not to the discrete multipliers. The critical surface is the stable manifold on which every relevant scaling field has been tuned away. Irrelevant coordinates flow to the renormalization-group fixed point along it. A repulsive renormalization-group trajectory leaves the renormalization-group fixed point along an unstable relevant direction, ultimately entering a noncritical phase or another scaling regime.

To separate the singular free-energy density from its regular background, suppose the eliminated-mode contribution $g$ is analytic near the renormalization-group fixed point. Look for an analytic background $a(u)$ satisfying $a(u)=b^{-D}a(R_{b}u)+g(u)$. Then $F_s=f-a$ satisfies $F_s(u)=b^{-D}{F}_s(R_{b}u)$. In linear thermal and field coordinates, an analytic monomial $g_{mn}{t}^m{h}^n$ has background coefficientprovided the denominator is nonzero. This explicitly explains when analytic subtraction of an inhomogeneous renormalization recursion works. The printed inhomogeneous equation may describe the free-energy function before this subtraction, or a singular part defined only up to analytic terms. The inhomogeneous sum need not be small compared with the full free energy: an analytic constant can dominate a vanishing singular contribution. It can be omitted in deriving leading singular powers after regular subtraction, provided no renormalization-group free-energy resonance, marginal logarithm or dangerously irrelevant coupling obstructs the homogeneous critical asymptotics. At a renormalization-group free-energy resonance $m{\lambda }_t+n{\lambda }_h=D$, additive logarithms can remain, so simply dropping the term would miss them.

Suppose there are exactly two relevant scaling fields, thermal $t$ and magnetic $h$, with positive exponents ${\lambda }_t,{\lambda }_h$, and the remaining fields approach a regular limit. Physical reduced temperature and magnetic field agree with these coordinates to leading order, up to nonuniversal metric factors. The homogeneous scaling hypothesis for critical phenomena isas an asymptotic scaling equation. Choose $\ell =|t{|}^{-1/{\lambda }_t}$ to make the thermal argument $+1$ or $-1$. Defining $f_{\pm }(z)=F_s(\pm 1,z)$ givesThe two functions describe respectively $t>0$ and $t<0$: approaching the transition from disordered and ordered sides need not give the same function. In particular the ordered zero-field function has one-sided derivatives associated with spontaneous magnetization. Metric factors multiply this formula in different microscopic normalizations.

The correlation length in Bravais lattice units obeys $\xi (t,h)=\ell \xi ({\ell }^{{\lambda }_t}t,{\ell }^{{\lambda }_h}h)$. At $h=0$, the same choice of $\ell$ gives $\nu =1/{\lambda }_t$. Two thermal derivatives of $F_s(t,0)$ give $2-\alpha =D/{\lambda }_t$. One and two field derivatives give, respectively, $\beta =(D-{\lambda }_h)/{\lambda }_t$ and $\gamma =(2{\lambda }_h-D)/{\lambda }_t$, assuming the relevant branch amplitudes are nonzero. At $t=0$, choose $\ell =|h{|}^{-1/{\lambda }_h}$; then $M\propto \mathrm{sgn}(h)|h{|}^{(D-{\lambda }_h)/{\lambda }_h}$, so $\delta ={\lambda }_h/(D-{\lambda }_h)$. HenceSimple algebra now gives the Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ and the hyperscaling relation $\alpha =2-D\nu$. Also $\beta \delta ={\lambda }_h/{\lambda }_t=\beta +\gamma$, giving the Widom scaling relation. These statements refer to leading singular contributions with the homogeneous two-field scaling assumptions. Above the upper critical dimension, the quartic coupling is dangerously irrelevant because it is still needed to stabilize the ordered phase; ordinary mean-field critical exponents then satisfy Rushbrooke and Widom but generally not hyperscaling. Marginal logarithms also require qualifications at the upper critical dimension.

For the connected-correlation scaling form $G(r)=|r{|}^{-(D-2+\eta )}{f}_G(|r|/\xi )$, the function $f_G(x)$ tends to a finite nonzero value as $x\to 0$ and decays exponentially at large $x$, up to powers of $x$, in a massive short-range phase. The scaling ansatz alone does not fix that prefactor. If the usual massive Gaussian field correlation tail $G(r)\propto r^{-(D-1)/2}{e}^{-r/\xi }$ applies, matching the powers gives $f_G(x)\sim B{x}^{(D-3)/2+\eta }{e}^{-x}$, with a correlation-length-dependent amplitude in $G$ fixed by matching the critical scaling form.

In physical coordinates the lattice sum has site-density factor $a^{-D}$. Its long-distance singular part therefore giveswhere ${\Omega }_{D-1}$ is the area of the unit sphere in $D$ dimensions. For $\eta <2$, finite $f_G(0)$, an integrable exponential tail and a nonzero integral, the last factor tends to a finite nonzero constant as $\xi \to \infty$. Microscopic distances contribute a regular background. Thus ${\chi }_s\propto {\xi }^{2-\eta }\propto |t{|}^{-(2-\eta )\nu }$, proving the Fisher scaling relationEquivalently the field has scaling dimension $(D-2+\eta )/2$, so its conjugate field has ${\lambda }_h=(D+2-\eta )/2$, which gives the same identity from the exponent formulas above.

Use a selected translationally invariant pure thermodynamic phase and define the connected lattice correlation functionAway from criticality its long-distance tail has an exponential factor $e^{-r/\xi }$, possibly multiplied by a power. Equivalently ${\xi }^{-1}=-{\lim }_{r\to \infty }{r}^{-1}\log |G(r)|$ when this limit exists; different directions can have different lengths on an anisotropic lattice. At a thermodynamic critical point the correlation length diverges and the tail becomes algebraic. Differentiate $M=N^{-1}⟨{\sum }_n{\sigma }_n⟩$ with respect to the physical field appearing in the energy Statistical Hamiltonian. With ${\beta }_{\mathrm{th}}=1/(k_{B}T)$, the correlation-function susceptibility sum rule isIn physical continuum coordinates the sum becomes $a^{-D}\int d^{D}rG(r)$. Differentiating with respect to the dimensionless source ${\beta }_{\mathrm{th}}h$ instead removes the explicit inverse-temperature factor. Below the transition, take a one-sided pure-phase limit rather than a macroscopic symmetric mixture.

A normalized blocking kernel $B_b({\sigma }^{\prime },\sigma )$ assigns a probability or a delta constraint to a coarse configuration for each microscopic one, with ${\sum }_{{\sigma }^{\prime }}{B}_b({\sigma }^{\prime },\sigma )=1$. Define the blocked Boltzmann weight bySumming over ${\sigma }^{\prime }$ proves exact equality of the partition functions. Repeated real-space renormalization group steps compose the kernels and integrate out shorter-scale variables. To retain equality one must keep all generated interactions and the field-independent constant; a finite truncation is an approximation. Long-distance observables are mapped through the corresponding coarse observables, not identified blindly with microscopic spins. In physical units,A subsequent coordinate rescaling restores the reference lattice spacing while expressing couplings in cutoff units.

Let $F(u,C)=-({\beta }_{\mathrm{th}}N{)}^{-1}\log Z=C+f(u)$ be free energy per microscopic site. Invariance of $Z$ givesThe constant transforms as $C_1=b^D{C}_0+c(u_0)$. Therefore, with $g(u)=b^{-D}c(u)$, the identity-operator contribution to renormalization-group free energy givesThe second equality follows by substitution and induction. The inhomogeneous term records eliminated-mode entropy, determinants, and normalization factors, all multiplying the identity operator; it is not an interaction with the retained spins. Strictly, this $f$ can contain regular background terms. After an analytic subtraction of an inhomogeneous renormalization recursion, the genuinely singular part satisfies the homogeneous scaling equation, except for possible logarithmic resonances.

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. Linearize in scaling coordinates $s_i$: $s_i^{\prime }=b^{{\lambda }_i}{s}_i$. A relevant operator has ${\lambda }_i>0$, an irrelevant operator has ${\lambda }_i<0$, and a marginal operator has ${\lambda }_i=0$ and needs nonlinear analysis. These ${\lambda }_i$ are logarithmic scaling eigenvalues; the discrete Jacobian eigenvalues are $b^{{\lambda }_i}$, not ${\lambda }_i$. The critical surface is the stable manifold obtained by tuning all relevant coordinates to zero. A repulsive renormalization-group trajectory leaves the fixed point along a relevant direction as the coarse-graining length grows. Stable and unstable manifolds become curved away from linear order.

The left section has $h=0$, one irrelevant coordinate $w$ and a relevant thermal coordinate $t$. Its critical surface is $t=0$ in that section. With two relevant variables, the full critical surface has codimension two: the second section shows repulsion in the $(t,h)$ plane at $w=0$.

Assume an isolated fixed point with two relevant scaling fields, analytic changes from physical controls to $(t,h)$, short-range isotropic scaling, no dangerously irrelevant variable in the thermodynamic scaling function, and no marginal or resonant logarithm. Let ${\lambda }_t,{\lambda }_h>0$ denote their scaling eigenvalues. The homogeneous singular free-energy density obeysChoose a stopping scale $L=b^p\simeq |t{|}^{-1/{\lambda }_t}$ so the renormalized thermal field is order one. Irrelevant fields vanish at this scale. With $s=D/{\lambda }_t$ and $\Delta ={\lambda }_h/{\lambda }_t$,Continuous scale interpolation absorbs the immaterial integer stopping-scale convention. Nonuniversal thermal, field and free-energy metric factors can be included explicitly; the normalized functions depend only on the universality class. The $+$ and $-$ labels refer to $t>0$ and $t<0$, which flow to different sides of the critical surface.

For clarity about the additive terms, let $a(t,h)$ solve $a(t,h)=b^{-D}a(b^{{\lambda }_t}t,b^{{\lambda }_h}h)+g(t,h)$. If $g=\sum g_{mn}{t}^m{h}^n$, its analytic coefficients areprovided the denominators do not vanish. Subtracting $a$ makes the recursion homogeneous. A vanishing denominator is a resonance and generically produces logarithms, which the simple power-law hypothesis excludes. In a convention where the regular field-dependent background has been subtracted, write ${\Phi }_{\pm }=f_{\pm }+C_{\pm }$ and $I_{\pm }(t)=C_0+a(t,0)$. This yields the requested formThe $C_{\pm }$ contain branch-dependent matching contributions; only $f_{\pm }+C_{\pm }$ is physically determined. The $I_{\pm }$ are nonsingular functions analytic through the thermodynamic critical point and represent the same regular Taylor background on the two sides, though either branch may be used to describe them. For the full free energy at nonzero field, the regular background is generally $I(t,h)$, not just $I(t)$. Adding an analytic identity-operator contribution proportional to $h^2$ provides a direct counterexample to an exact unrestricted $I(t)$ formula; it changes no singular exponents. The printed expression is justified for the leading singular scaling form with that analytic field background removed. Confluent corrections from irrelevant fields are also omitted in this leading form.

Differentiate the singular scaling form, with one-sided derivatives in the ordered phase. It givesThe length transforms as $\xi (t,h)=L\xi (L^{{\lambda }_t}t,L^{{\lambda }_h}h)$, so $\nu =1/{\lambda }_t$. Thus the Rushbrooke scaling relation and hyperscaling relation follow by direct substitution:The same reasoning also gives $\gamma =\beta (\delta -1)$. Hyperscaling relation relies on the assumptions above and can fail above an upper critical dimension because of a stabilizing dangerously irrelevant coupling.

For the universal specific-heat amplitude ratio, write $F_s=A_f|a_{t}t{|}^s{\Phi }_{\pm }(0)$ at $h=0$. Two temperature derivatives multiply both branches by the same dimensional and thermal metric factors. Away from logarithmic resonances and exceptional zero amplitudes, $A_{+}/A_{-}={\Phi }_{+}(0)/{\Phi }_{-}(0)$ with the same heat-capacity sign convention on both sides. HenceThe functions and their matched branch constants are tied to the fixed-point theory; the constants are not independent arbitrary experimental parameters. If $\alpha <0$, this means the amplitudes of the subleading singular contribution after subtracting the analytic specific-heat background. A finite background ratio itself need not be universal.

Use dimensionless interaction couplings in the exponent of the Boltzmann weight. In the following formulas $K,L$ denote the physical interaction energies divided by $k_{B}T$ if that factor has not already been absorbed into the displayed Hamiltonian. This convention is required for temperature-independent numerical fixed-point coordinates.

Split the original square lattice into retained sites $E$ with even $x+y$ and eliminated sites $O$ with odd $x+y$, where coordinates are in units of $a$. Every nearest-neighbour bond joins $E$ to $O$, whereas a diagonal next-nearest-neighbour bond stays within one sublattice. Denote retained Ising spins by $s_e$ and eliminated Ising spins by ${\eta }_o$. For fixed retained Ising spins defineThe blocked Boltzmann weight in the checkerboard decimation of the square-lattice Ising model isUse the independent uniform product measure on Ising spins on $O$ to perform a cumulant expansion. Its one-spin mean vanishes, and $⟨{\eta }_o{\eta }_{o^{\prime }}{⟩}_0={\delta }_{o{o}^{\prime }}$. Consequently the first-order $K$ term, the first-order eliminated-sublattice $L$ term, and the mixed $KL$ term vanish. To the requested order,Here the truncation keeps $K^2$ and the linear term in $L$; equivalently count $L$ as order $K^2$. At higher orders the interaction family will generally not close.

Each eliminated site has four retained neighbours, soIts six distinct neighbour pairs comprise four corner pairs at distance $\sqrt{2}a$ and two opposite pairs at distance $2a$. A given retained diagonal pair shares two eliminated centres; a given opposite pair shares only one. Thus the generated pair couplings are $2K^2$ for diagonal pairs and $K^2$ for axial pairs separated by two original spacings. The original $L$ interaction directly contributes to each retained diagonal pair. No single-spin term survives spin inversion symmetry, and no three-spin or four-spin term is present in this second cumulant. This proves closure on exactly two nonconstant interactions at the retained order.

The primitive vectors of the retained square lattice are $a(1,1)$ and $a(1,-1)$. Rotate the axes by $45$ degrees and rescale lengths by $1/\sqrt{2}$; retained diagonal pairs become nearest neighbours, and the axial distance-$2a$ pairs become next-nearest neighbours. With $b=\sqrt{2}$, the leading checkerboard Ising decimation recursion is thereforeThe constant part of the weight is also determined: $4(K^2/2)(N/2)=N{K}^2$, so the original partition function equalsto the same order in the logarithm of the weights. That normalization changes free energy but not the retained-spin interaction operators. If physical energy couplings are kept instead, at fixed ${\beta }_T$ the relations read $K_{\mathrm{energy}}^{\prime }=L_{\mathrm{energy}}+2{\beta }_T{K}_{\mathrm{energy}}^2$ and $L_{\mathrm{energy}}^{\prime }={\beta }_T{K}_{\mathrm{energy}}^2$.

The fixed-point equations are $L_{\ast }=K_{\ast }^2$ and $K_{\ast }=3K_{\ast }^2$. Their only finite real solutions areThe origin is the high-temperature fixed point; its Jacobian matrix is $\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$, with both multipliers zero. The nontrivial candidate critical point is $(1/3,1/9)$. Linearize the discrete renormalization-group transformation there:The characteristic polynomial is ${\lambda }^2-(4/3)\lambda -2/3$, givingEigenvectors may be chosen as $(1,2/(3{\lambda }_{\pm }){)}^T$. The positive eigenvalue greater than one is the relevant direction of a fixed point; the other is irrelevant because its magnitude is less than one. Its negative sign only alternates the sign of the perturbation between steps. For a discrete map the magnitude relative to one decides relevance, not the sign rule used for continuous beta-function eigenvalues.

Let $t$ be the relevant thermal scaling coordinate, so $t^{\prime }={\lambda }_{+}t$ to first order. A correlation length measured in restored lattice units satisfies $\xi (t^{\prime })=\xi (t)/b$. If $\xi (t)\sim |t{|}^{-\nu }$, then ${\lambda }_{+}^{-\nu }=b^{-1}$. Hence the thermal exponent from a discrete renormalization map isThis is the value predicted by the stated low-order real-space renormalization group, whose nontrivial fixed point is evaluated beyond the strictly infinitesimal-coupling limit. It is an approximation, not an exact solution of the full square-lattice Ising model.