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The heat-capacity critical exponent is defined by the singular part $c_{\mathrm{s}}\sim |t{|}^{-\alpha }$ as the reduced temperature tends to zero.

Critical exponents describe leading singular powers near a continuous phase transition. Using the reduced temperature $t=(T-T_c)/T_c$, the standard definitions includeThese are the order-parameter critical exponent, magnetic-susceptibility critical exponent, heat-capacity critical exponent and correlation-length critical exponent. The critical-isotherm exponent is defined by $M(0,H)\sim \mathrm{sgn}(H)|H{|}^{1/\delta }$. The anomalous dimension $\eta$ gives the critical long-distance connected correlation function, $G_c(x)\sim |x{|}^{-(d-2+\eta )}$, when an ordinary scale-invariant critical point exists.

In the Landau approximation, minimize the Landau free energy to obtain the equation of state, differentiate it with respect to the field conjugate to an order parameter to obtain magnetic susceptibility, and differentiate the minimized free energy with respect to temperature to obtain heat capacity. These operations determine $\beta ,\delta ,\gamma ,\alpha$, respectively. Expanding the Landau-Ginzburg theory about its uniform minimum gives the quadratic fluctuation kernel $K{q}^2+V^{\prime \prime }(M)$, whose Fourier transform determines the correlation length and the anomalous dimension. At the ordinary quartic critical point this gives ${\xi }^2=K/V^{\prime \prime }(M)\propto |t{|}^{-1}$ and $\eta =0$.

For $u>0$, $r=0$ gives $H=u{M}^3$ and hence $\delta =3$; the equilibrium free energy $-r^2/(4u)$ gives a finite heat capacity discontinuity, conventionally $\alpha =0$. Together with the explicit calculation below, the ordinary mean-field critical exponents areThese describe neglect of fluctuation corrections, rather than the exact exponents of every low-dimensional Landau-Ginzburg theory.

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.

Critical exponents describe leading singular powers near a continuous phase transition. For reduced temperature $t=(T-T_c)/T_c$, common definitions are the spontaneous order parameter $M\sim (-t{)}^{\beta }$, magnetic susceptibility $\chi \sim |t{|}^{-\gamma }$, singular heat capacity $c_s\sim |t{|}^{-\alpha }$, correlation length $\xi \sim |t{|}^{-\nu }$, and the critical equation of state $M\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$. At criticality the connected correlation function has decay power $D-2+\eta$.

In Landau theory, solve the stationary equation, evaluate its stable minimum, and differentiate with respect to the conjugate variables. For a scalar quartic theory with $r=r_{0}t$, $r_0>0$, the equation of state is $h=rM+u{M}^3$. Its ordered solution and differentiated response give $\beta =1/2$ and $\gamma =1$, calculated explicitly in the unheaded solution below. At $r=0$ the same equation gives $\delta =3$. The minimized zero-field singular potential is zero above the transition and $-r^2/(4u)$ below, giving a heat-capacity jump and $\alpha =0$. Expanding the gradient functional about a stable minimum gives inverse quadratic response $\kappa q^2+U^{\prime \prime }(M)$, so $\xi =\sqrt{\kappa /U^{\prime \prime }(M)}\sim |t{|}^{-1/2}$ and the Gaussian critical decay has $\eta =0$.

These are mean-field critical exponents. Renormalization group methods obtain fluctuation-corrected powers from fixed-point scaling dimensions; they need not equal the Landau values below the upper critical dimension. At a marginal dimension logarithms can supplement the leading powers.

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.

Critical exponents describe leading power-law singularities as the reduced temperature $t=(T-T_c)/T_c$ and the conjugate field tend to zero. Their amplitudes depend on microscopic details, while the exponents are characteristic of a universality class. Define $m_0\sim (-t{)}^{\beta }$ on a selected ordered branch, $\chi \sim |t{|}^{-\gamma }$ at zero field, $C_{V,s}\sim |t{|}^{-\alpha }$ for the singular heat capacity, and $m(t=0,h)\sim \mathrm{sgn}(h)|h{|}^{1/\delta }$. Also define $\xi \sim |t{|}^{-\nu }$ and $G(r,t=0)\sim r^{-(D-2+\eta )}$. In these statements $\beta$ is the order-parameter critical exponent, not inverse temperature.

Assume $r=r_{t}t$ with $r_t>0$ and a nonzero positive quartic coefficient $u$. Minimizing the quartic potential gives $m_0=\sqrt{-r/u}$, hence $\beta =1/2$. Differentiating its equation of state givesso $\gamma =1$ with different amplitudes. At $r=0$, $h=u{m}^3$, giving $\delta =3$. The minimized potential is zero above the transition and $-r^2/(4u)$ below, so its second temperature derivative has a finite jump: $\alpha =0$.

The quadratic fluctuation kernel around the stable uniform minimum is $\kappa p^2+V^{\prime \prime }(m_0)$, so ${\xi }^2=\kappa /V^{\prime \prime }(m_0)$. Since $V^{\prime \prime }$ is $r$ above and $-2r$ below, $\nu =1/2$. Its critical momentum dependence is $p^{-2}$, giving $\eta =0$. Thus the ordinary mean-field critical exponents areThese are predictions of the Landau approximation, whose validity is constrained by the Ginzburg criterion; they need not be the exponents of the fluctuating theory below its upper critical dimension.