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For dimensionless free energy per site $F(u,C)=f(u)+\beta C$, an exact blocking kernel preserves the partition function and gives $F(u,C)=b^{-D}F(u^{\prime },C^{\prime })$. Thus $f(u)=b^{-D}f(u^{\prime })+g(u)$, where $g(u)=\beta [b^{-D}{C}^{\prime }-C]$ is the additive contribution from the eliminated degrees of freedom. Repeating this equality gives the displayed sum. After subtracting analytic backgrounds, the singular free-energy density obeys the homogeneous scaling hypothesis for critical phenomena, subject to marginal or dangerous-variable qualifications.

For a bulk free-energy density $f(\varphi )$, two compositions coexist across a flat interface whenThe first condition equates the chemical potentials, and the second equates the pressure. Equivalently the same straight line of slope $\mu$ is tangent to the graph of $f$ at both compositions. A stable coexistence pair uses a supporting tangent, so the resulting mixture lowers the bulk free energy.

Fix the spatial mean of a scalar order parameter to $M$ and sum the Boltzmann weights of all configurations with that mean. The resulting constrained partition function $Z_M$ defines this free-energy density, with an additive normalization fixed consistently. For an applied conjugate field $h$, $Z(h)=\int dM{e}^{-{\beta }_{T}V[F_c(M)-hM]}$, up to the chosen measure normalization. The thermodynamic limit gives $f(h)={\inf }_M[F_c(M)-hM]$. A homogeneous Landau free energy is an approximation to this constrained potential; phase-separated configurations can convexify it in the full thermodynamic limit.

A field conjugate to a scalar order parameter $m$ contributes $-hm$ to its free-energy density. Differentiating the equilibrium free energy with respect to $h$ gives $-m$, with density or extensive normalization stated consistently. In a ferromagnet the conjugate field is magnetic; for a fluid-density order parameter it is related to a chemical-potential difference. It can explicitly break the symmetry whose spontaneous breaking defines the ordered phase.

For quadratic dispersion $p^2+t$ with no interacting anomalous dimension, the thermal and uniform-source scaling eigenvalues are $y_t=2$ and $y_h=(d+2)/2$. The generic homogeneous singular free-energy density has $2-\alpha =d/2$ and gap exponent $\Delta =(d+2)/4$. These Gaussian-model values below the upper critical dimension are not the exponents of an interacting scalar critical point.

For a quadratic determinant in two effective momentum dimensions, differentiating the free-energy density with respect to the squared mass $R$ gives a term proportional to $\log ({\Lambda }^2/R)$. Integration gives $F_S\propto R\log R$ after subtracting analytic terms. Its thermal power index is still 1, but a strictly homogeneous pure-power expression misses this logarithm. It is a Gaussian determinant resonance, distinct from interaction-generated logarithms at an upper critical dimension.

For a stable massive Gaussian field theory, a finite blocking step yields $F(t,h)=F_{\mathrm{shell}}(t)+b^{-d}F(b^{2}t,b^{(d+2)/2}h)$ for free-energy density. The finite-step shell and measure terms are analytic backgrounds. Generic-dimension singular scaling is homogeneous after subtracting those backgrounds; resonances such as Gaussian free-energy logarithm at effective dimension two need additive logarithmic terms.

On the ordered branch of tricritical crossover scaling, at zero field and $X\to +\infty$, the minimizing square is $a={\psi }^2=(\sqrt{X^2+4}-X)/2\sim 1/X$. Its stationarity relation $Xa+a^2=1$ gives ${\Phi }_{-}(X,0)=-a/4-a^3/12\sim -1/(4X)$. Thus the minimized free-energy density becomes $-r^2/(4q)$ for fixed $q>0$ as $r\uparrow 0$, recovering the ordinary quartic continuous phase transition and its heat-capacity critical exponent $\alpha =0$. Replacing the scaling function by a nonzero constant in this limit would give an incorrect exponent.

Adding $d\varphi$ to a free-energy density adds a constant at fixed total compositional order parameter. In the common-tangent construction for phase coexistence, it shifts the tangent slope by $d$ while preserving its points of contact. Thus coexistence compositions are unchanged, whereas their chemical potential shifts. This does not assert unchanged equilibrium composition at a fixed externally imposed chemical potential.

For the sextic Landau free energy $V=r{M}^2/2+v{M}^6/6-hM$, $v>0$, with $r$ linear in reduced temperature, the equation of state is $h=rM+v{M}^5$. Its ordered zero-field minimum satisfies $M^4=-r/v$, the inverse magnetic susceptibility is $4|r|$ below the transition and $r$ above, and the critical isotherm satisfies $h=v{M}^5$. The minimized singular free-energy density is $-|r{|}^{3/2}/(3\sqrt{v})$ below and zero above. These facts give the displayed mean-field critical exponents, obeying both the Rushbrooke scaling relation and Widom scaling relation. Both the quadratic and quartic coefficients must be tuned; a generic path with a positive quartic coefficient instead has ordinary critical behavior. Fluctuations change the result below the tricritical upper critical dimension three, and marginal corrections can add logarithms at three.

With the dimensionless magnetic field conjugate to the magnetization, $m=-\partial f_s/\partial h$. Set $y=h/|t{|}^{\Delta }$ and differentiate the homogeneous free-energy density:On the ordered branch, the one-sided value $g_{m,-}(0^{+})$ is nonzero, so the order-parameter critical exponent isTo reach the critical isotherm, take $|t|\to 0$ at fixed nonzero $h$. Suppose $g_m(y)\sim A\mathrm{sgn}(y)|y{|}^p$ at large $|y|$. Cancellation of the residual thermal factor requires $2-\alpha -\Delta -p\Delta =0$. Hence $m(0,h)\propto \mathrm{sgn}(h)|h{|}^{(2-\alpha -\Delta )/\Delta }$ andThe singular scaling function determines the nonanalytic critical-isotherm contribution; any analytic response background is subleading for the usual $\delta >1$.

The hyperscaling relation expresses the idea that the singular dimensionless free energy in one correlation volume is of order one. A macroscopic volume $V$ contains of order $V/{\xi }^d$ such correlated regions, soSubstitution of the homogeneous correlation length givesThis has exactly the required homogeneous field variable. Comparing its thermal power with that of the singular free-energy density provesThe argument concerns powers, not equality of the two scaling functions or their amplitudes. It assumes the correlation length supplies the only singular length scale and that no dangerously irrelevant coupling changes the free energy per correlated region. For ordinary short-range quartic theory, hyperscaling holds below the upper critical dimension four, with the usual qualifications for marginal logarithms at four. Above four, mean-field critical exponents $\alpha =0$, $\nu =1/2$ do not satisfy $d\nu =2-\alpha$; the quartic coupling is dangerously irrelevant. The stated homogeneous form of $\xi$ alone therefore does not prove hyperscaling without the correlation-volume assumption.

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.

Landau-Ginzburg theory describes a slowly varying order parameter by a local free-energy functional consistent with the microscopic symmetries. For a representative scalar system with spin-inversion symmetry, take the dimensionless functionalThe gradient term penalizes spatial variation, while the even local potential implements the zero-field symmetry $\varphi \mapsto -\varphi$. The positive quartic coefficient stabilizes a continuous transition. A negative quartic coefficient requires higher powers and can instead produce a first-order phase transition. A vector order parameter gives the corresponding rotationally invariant model with powers of $|\mathbf{\varphi }{|}^2$.

In the mean-field approximation, replace the field by a constant $M$ and minimize the local potential. Its equation of state and stable zero-field solutions areBelow the transition these two minima exhibit spontaneous symmetry breaking. Differentiating the equation of state gives $\chi =(r+3u{M}^2{)}^{-1}$: it is $1/r$ above the transition and $1/(2|r|)$ within either ordered phase. At $r=0$, $M=(h/u{)}^{1/3}$. The minimized zero-field free-energy density is zero above and $-r^2/(4u)$ below, apart from an analytic background; its second temperature derivative has a finite jump. Consequently

To include fluctuations, write $\varphi =M+\psi$. The quadratic part isA Gaussian functional integral therefore gives $⟨\psi (q)\psi (-q)⟩=(c{q}^2+r+3u{M}^2{)}^{-1}$, with the momentum-volume delta function understood. The Ornstein--Zernike correlation function has correlation length $\xi =\sqrt{c/(r+3u{M}^2)}$. Thus ${\nu }_{\mathrm{MF}}=1/2$, and at criticality the Gaussian covariance is proportional to $q^{-2}$, giving ${\eta }_{\mathrm{MF}}=0$. Continuous-symmetry ordered phases additionally have transverse Goldstone modes, for which the restoring mass vanishes.

The Gaussian integration contributes $\frac{1}{2}\int d^{d}q(2\pi {)}^{-d}\log (c{q}^2+r)$ to the disordered-phase free-energy density. Differentiating twice with respect to $r$ gives a critical contribution proportional to $\int d^{d}q(c{q}^2+r{)}^{-2}$; for $d<4$ its singular part grows as $r^{(d-4)/2}$ and at four dimensions it is logarithmic. Interacting fluctuations thus become important close to the critical point below four dimensions. The Ginzburg criterion tests this failure of the mean-field approximation; the renormalization group then explains modified exponents and universality. Mean-field theory finds the competing minima, while fluctuations determine whether its predicted critical behavior survives at long distance.

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.

Use the infinite translation-invariant chain first. Set $\rho =e^{-\kappa }$ and Fourier-transform the exchange matrix:For positive $J,\kappa$ this is strictly positive. Its reciprocal is a linear polynomial in $\cos q$, proving the nearest-neighbour form of the exponentially coupled Ising inverse kernel:Expanding the squared difference givesSince $(\cosh \kappa -1)/\sinh \kappa =\tanh (\kappa /2)$, substitution into part (a) yieldsThis is the exact bulk identity. A finite periodic version uses the periodized coupling $J_{ij}=J{\sum }_{\ell \in \mathbb{Z}}{e}^{-\kappa |i-j+\ell N|}$; its discrete Fourier symbols are the same and the squared differences wrap around the ring. Simply truncating the exponential at open ends gives a different inverse at those ends.

For an open chain of $N\ge 2$ sites with the literal finite matrix $J{\rho }^{|i-j|}$, the open-chain endpoint corrections for an exponential Ising kernel are explicit:The inverse has endpoint diagonal entries $1/[J(1-{\rho }^2)]$, interior diagonal entries $(1+{\rho }^2)/[J(1-{\rho }^2)]$ and adjacent entries $-\rho /[J(1-{\rho }^2)]$. Multiplication by the original geometric matrix verifies every entry. These boundary terms affect boundary amplitudes, but not the bulk free-energy density in the thermodynamic limit at fixed $\kappa >0$.

A first-order phase transition has a discontinuity in a first derivative of the equilibrium free energy, such as entropy or the order parameter; a thermal first-order transition normally has latent heat. At a continuous phase transition the order parameter goes continuously to zero and there is no latent heat, although response functions and higher derivatives of the free energy can be singular.

For the symmetric scalar Landau free energy at zero conjugate field, minimizeIf $u>0$, the sixth-order term is unnecessary near $r=0$. For $r>0$ the minimum is $M=0$, whereas for $r<0$ there are two symmetry-related minima with $M^2=-r/u$ to leading order. The order parameter vanishes continuously as $r\uparrow 0$. The equilibrium free-energy density changes from zero to $-r^2/(4u)$, so its first temperature derivative is continuous and its second derivative has a jump. This is the ordinary mean-field continuous phase transition.

If $u<0$, keep $v>0$ for stability. A nonzero stationary order parameter, with $s=M^2$, obeys $r+us+v{s}^2=0$. For phase coexistence with $M=0$ its free energy must also vanish. Substituting $r=-us-v{s}^2$ into $V$ givesThe nonzero solution of $V=0$ is thereforeThe competing minima exchange stability at positive $r_t$, with a finite jump of the order parameter. This is a first-order phase transition. The metastability region is also visible: nonzero stationary points first appear at the spinodal $r=u^2/(4v)$, and the disordered local minimum loses stability at $r=0$.

For example, holding $u,v$ fixed and taking $r=a(T-T_0)$, the ordered phase has entropy lower than the disordered phase by $a{M}_t^2/2$ at coexistence. Heating through the transition absorbs latent heat per volume $T_{t}a{M}_t^2/2$. More generally the temperature dependence of all coefficients contributes to this entropy difference. With no inversion symmetry, a cubic term can also produce a first-order phase transition, as comparison of the competing minima of the Landau free energy shows.

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.