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Critical exponents describe leading singular powers as the reduced temperature $t$ or conjugate field approaches zero. With ${\beta }_m$ used for the order-parameter critical exponent to avoid confusion with inverse temperature, the definitions includeAt criticality the connected correlation function behaves as $G(r)\sim r^{-(D-2+\eta )}$. These definitions refer to the singular part and allow additional logarithms at marginal dimensions.

For an ordinary scalar Landau free energy, the equation of state is $h=rM+u{M}^3$. Below the transition $M^2=-r/u$, giving ${\beta }_m=1/2$. Differentiating the equation of state gives ${\chi }^{-1}=r+3u{M}^2$: it equals $r$ above and $-2r$ below the transition, so $\gamma =1$. The quadratic fluctuation kernel is $\kappa k^2+r$ in the symmetric phase, giving correlation length $\xi =\sqrt{\kappa /r}$ and $\nu =1/2$. At zero mass the momentum-space correlation function is proportional to $k^{-2}$, so $\eta =0$. The minimized free energy and critical equation of state give $\alpha =0$ and $\delta =3$, as calculated in the root solution.

At the tricritical point, $u=0$ and $h=rM+v{M}^5$. Then $M\sim (-r{)}^{1/4}$, and ${\chi }^{-1}=r+5v{M}^4$ is $r$ above and $-4r$ below the transition. The gradient kernel still gives $\nu =1/2$ and $\eta =0$. ThereforeThese are mean-field critical exponents. A systematic derivation incorporating fluctuations uses the renormalization group, extracts the relevant scaling eigenvalues, and differentiates the singular free-energy density, as in Question 3.

After separating an additive identity term and using dimensionless couplings, let $R$ be one blocking-and-rescaling step. A renormalization-group fixed point satisfies $R(u_{\ast })=u_{\ast }$. Near it, linearization gives scaling coordinates $w_i$ withAt a critical renormalization-group fixed point, no finite characteristic length survives. Since the correlation length measured in lattice units transforms as $\xi (Ru)=\xi (u)/b$, a nonzero scale-invariant critical length must be infinite. A sequence of systems approaching the same fixed point therefore has common long-distance behavior.

The scaling hypothesis for critical phenomena asserts that the singular free-energy density transforms homogeneously under these scaling coordinates. It follows from repeated blocking near the fixed point, once analytic backgrounds are removed and no dangerously irrelevant coupling supplies an additional singular dependence. The temperature and magnetic-field scaling directions set the principal departures from criticality; irrelevant directions label microscopic differences that disappear at long distances.

Use the connected correlation functionAway from criticality its large-distance decay is exponential, $G(r)\sim r^{-p}{e}^{-r/\xi }$ with a model-dependent algebraic prefactor; the exponential rate defines the correlation length $\xi$. At criticality $\xi$ diverges and the decay becomes a power law. Subtracting the disconnected product is essential in the ordered phase: the unconnected correlation function tends to the square of the spontaneous magnetization rather than to zero.

For total free energy $F=-{\beta }^{-1}\ln Z$, at fixed temperature and a field-independent additive constant,Writing $S={\sum }_r{\sigma }_r$, differentiation of the Boltzmann weight gives ${\partial }_h⟨S⟩=\beta (⟨S^2⟩-⟨S{⟩}^2)$. ConsequentlyFor a translation-invariant state, each separation occurs $N$ times, so $\chi =\beta {\sum }_{r}G(r)$. Without translation invariance, the double-sum formula remains the correct relation.

Gaussian calculation and thinning. Interpret the displayed local expression as a Hamiltonian density, to be integrated over space. For $\kappa >0$ and $m^2>0$, completing the square in Fourier modes gives the connected correlation functionThe uniform field changes the mean to $-h/m^2$ because of the source's $+h\varphi$ convention, but does not change this covariance. The nearest pole has inverse length $\sqrt{\kappa m^2}$, equivalently the Green function solves $(-{\nabla }^2+\kappa m^2)G=\kappa \delta /\beta$. ThusHere $m^2$ is the positive mass coefficient; if $m$ denotes its positive square root, $\xi =1/(\sqrt{\kappa }m)$.

For a Gaussian thinning transformation, split modes into $|k|<\Lambda /b$ and $\Lambda /b<|k|<\Lambda$. The modes are independent, so integrating out the shell produces only an additive free-energy density and leaves the low-mode quadratic action unchanged. Restore the cutoff by $x=b{x}^{\prime }$ and $\varphi (x)=b^{-(D-2)/2}{\varphi }^{\prime }(x^{\prime })$. Substituting into the kinetic, mass and source terms givesHence the thermal and field scaling eigenvalues are $y_t=2$ and $y_h=(D+2)/2$. Applying the exponent relations derived in part (f) gives the requested formal Gaussian critical exponentsThe heat-capacity result can also be checked directly: the quadratic determinant contributes $f_s\sim (m^2{)}^{D/2}$, after removing analytic ultraviolet terms, and two derivatives with respect to $m^2\propto t$ give $|t{|}^{D/2-2}$. At $D=2$, the free-energy power is accompanied by a logarithm, $f_s\sim m^2\ln m^2$, whose second derivative still has index $\alpha =1$. At $D=4$ this becomes a logarithmic heat capacity singularity, with index $\alpha =0$, rather than a strictly constant singular term.

There is an important ordered-phase obstruction in a Gaussian scalar model. A purely quadratic potential is unbounded below for $m^2<0$, so the Hamiltonian as printed has no stable ordered phase and no literal spontaneous-magnetization exponent. The displayed ${\beta }_m$ is its Gaussian scaling index, obtained by assigning the field its Gaussian scaling dimension; it is not a construction of a stable ordered equilibrium. For $D<4$, a stabilizing quartic interaction is relevant and generally changes the actual critical fixed point. For $D\le 2$, massless Gaussian infrared fluctuations also obstruct interpreting the formal zero or negative index as an ordinary bounded-spin ordering law. These qualifications do not alter the requested thinning calculation, but are necessary for its physical interpretation.

Use the connected correlation function $G(x)=⟨\varphi (x)\varphi (0)⟩-⟨\varphi {⟩}^2$, so that the constant ordered contribution is removed. Away from a thermodynamic critical point its long-distance decay defines the correlation length: $-{\lim }_{R\to \infty }{R}^{-1}\log |G(R)|={\xi }^{-1}$ when this limit exists. In a massive rotation-invariant phase the exponential decay is typically multiplied by an algebraic factor.

Expand the Landau-Ginzburg theory free energy about a stable uniform minimum $M$, writing $\varphi =M+\delta \varphi$. To quadratic order,integration of the Gaussian field theory gives the Fourier-space connected correlation function $\tilde{G}(q)=k_{B}T/(c+\kappa q^2)$. Thus the Ornstein--Zernike correlation function hasFor an ordinary quartic transition, $M=0$ above the transition and $M^2=-r/u$ below, so $c=r$ and $c=-2r$, respectively. The correlation length diverges as $|T-T_c{|}^{-1/2}$ on both sides, with different amplitudes. At criticality the mass vanishes and the Gaussian correlation function is proportional to $R^{-(D-2)}$ for $D>2$. Interactions can change this power to $R^{-(D-2+\eta )}$.

At a continuous phase transition the equilibrium order parameter tends continuously to its critical value, there is no latent heat, and the correlation length diverges. For $u>0$ at zero field, minimizing $V(M)=r{M}^2/2+u{M}^4/4$ gives $M=0$ when $r\ge 0$ and $M=\pm \sqrt{-r/u}$ when $r<0$. These minima join continuously at $r=0$. The minimized singular free-energy density is zero above and $-r^2/(4u)$ below, so its first thermal derivative is continuous, while its second derivative has a finite jump in mean-field approximation.

A first-order phase transition instead has competing minima whose free energies cross while their order parameters remain distinct. The first derivatives of the equilibrium free energy generally jump; along a temperature-driven crossing this gives latent heat. For the symmetric sextic potential with $u<0,v>0$, let $y=M^2>0$. At phase coexistence, the stationary points and equality of the two free energies requireEliminating $r$ gives $-u{y}^2/4-v{y}^3/3=0$, henceAt coexistence $V(M)=v{M}^2(M^2+3u/(4v){)}^2/6$, proving that all three degenerate minima are global. The equilibrium order parameter jumps from zero to one of the two nonzero values even though $r_{\mathrm{coex}}>0$. The disordered minimum loses local stability at $r=0$, and the nonzero stationary points first appear at $r=u^2/(4v)$. These spinodal points delimit metastability and are not the equilibrium transition line. Cubic terms, when symmetry permits them, provide another common route to a first-order transition.

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.

Write the transverse displacement as $\mathbf{s}$ and take the common line-of-sight interval to be $[0,L]$. The rotation measure is a line integral of the magnetic field, so its correlation isSubstitute the Fourier inversion of the solenoidal isotropic spectral tensor. Keeping the finite observation length gives the exact windowed expressionThe allowed extension of the longitudinal displacement integral to the whole real line is the long-path projection of a magnetic correlation. It applies when $L$ is much larger than the correlation length: equivalently, the squared window becomes $2\pi L\delta (k_z)$ inside the integral. The Dirac delta function sets $k_z=0$, leavingStatistical isotropy makes this a function of $s=|\mathbf{s}|$. The displayed two-dimensional relation is the intended long-path approximation; the preceding windowed formula accounts for finite-length edge effects.

Use the connected correlation function $G(x)=⟨M(x)M(0)⟩-⟨M{⟩}^2$ in a selected pure thermodynamic phase. The exponential correlation length is defined by ${\xi }^{-1}=-{\lim }_{|x|\to \infty }|x{|}^{-1}\log |G(x)|$, when this limit exists in the chosen direction. Linearizing the Landau-Ginzburg theory around a stable uniform minimum $M_0$ givesHence the Landau scalar correlation length is $\boxed{\xi =\sqrt{K/V^{\prime \prime }(M_0)}}$. Away from criticality the massive Gaussian field correlation tail behaves as $|x{|}^{-(D-1)/2}{e}^{-|x|/\xi }$, up to its amplitude. At a continuous transition $V^{\prime \prime }$ tends to zero, so $\xi$ diverges and fields vary on scales much longer than the lattice spacing. This motivates a gradient expansion, but the simultaneous growth of correlated fluctuations is precisely why the Ginzburg criterion must be checked.

A first-order phase transition exchanges distinct global minima of the Landau free energy; the order parameter jumps and a thermal crossing generically has latent heat. A continuous phase transition has minima merging continuously and a diverging correlation length. A local loss of stability at a spinodal point is not the equilibrium coexistence condition.

For the sextic even Landau potential, write $y=M^2$. At a nonzero zero-field stationary point, $r+uy+v{y}^2=0$. To coexist with $M=0$, also require $ry/2+u{y}^2/4+v{y}^3/6=0$. Eliminate $r$ to obtain $y=-3u/(4v)$ and $r=3u^2/(16v)$, which requires $u<0$. On this locusproving that all three equal minima are globally stable. Their curvatures are $3u^2/(16v)$ at zero and $3u^2/(4v)$ at the nonzero minima, so the correlation length is finite when $u<0$ is fixed. For $u>0$, by contrast, the nonzero minimum approaches zero as $r\uparrow 0$, giving the ordinary continuous transition. At fixed $u>0$ and $r<0$, crossing $h=0$ exchanges the two ordered minima discontinuously even though the zero-field thermal transition at $r=0$ is continuous.

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.

Use the arc length $s$ and the unit tangent $\mathbf{t}={\partial }_s\mathbf{r}$ of an inextensible filament. In the high-force state choose its positive longitudinal branch. The constraint givesThe extension is $z={\int }_0^L{t}_{z}ds=L-\frac{1}{2}{\int }_0^L|{\mathbf{t}}_{\perp }{|}^{2}ds+\cdots$. The curvature of a space curve obeys ${\kappa }^2=|{\partial }_s\mathbf{t}{|}^2$. Since ${\partial }_s{t}_z=-{\mathbf{t}}_{\perp }\cdot {\partial }_s{\mathbf{t}}_{\perp }+\cdots$ is second order in small fluctuations, its squared contribution is fourth order. Thus the worm-like chain energy through quadratic order isPositive force suppresses transverse orientation fluctuations, while the filament bending modulus penalizes their spatial variation.

To apply the equipartition theorem without missing either component or a mode-counting factor, expand each component in a real orthonormal basis of eigenfunctions of $-{\partial }_s^2$, $t_{\alpha }(s)={\sum }_q{a}_{\alpha q}{w}_q(s)$, with $\int w_q{w}_{q^{\prime }}ds={\delta }_{q{q}^{\prime }}$ and $\alpha =x,y$. The quadratic energy is $\frac{1}{2}{\sum }_{\alpha ,q}(A{q}^2+f)a_{\alpha q}^2$. Each real quadratic degree of freedom contributes $k_{B}T/2$, soFor a chain long compared with the force-dependent correlation length, the interior is translation invariant and the mode sum per length becomes $\int dq/(2\pi )$. Equivalently, for the Fourier transform ${\tilde{t}}_{\alpha }(q)=\int e^{-iqs}{t}_{\alpha }(s)ds$,The negative and positive wavenumbers here represent the real-field mode count consistently, rather than two independent complex modes. Summing both transverse components gives the high-force worm-like chain elasticitywhere the integral follows by setting $q=\sqrt{f/A}u$ and using ${\int }_{-\infty }^{\infty }(1+u^2{)}^{-1}du=\pi$. ThereforeHere $L_p$ is the three-dimensional zero-force persistence length. The result requires $f{L}_p\gg k_{B}T$, negligible backbone stretching, and a long-chain bulk approximation. It does not describe the eventual bond-stretching regime. For a continuum polymer description, the force-dependent length should also exceed the microscopic cutoff.

For comparison, a three-dimensional freely jointed chain has contour length $L=Nb$ and independent link orientations. At force $f$, a link of polar angle $\vartheta$ has Boltzmann factor $e^{x\cos \vartheta }$ with $x=fb/(k_{B}T)$. Its angular canonical partition function isThe total canonical partition function is $Z_1^N$. Differentiating with respect to force gives the force-extension of a three-dimensional freely jointed chain:ConsequentlyThe extension deficits scale as $f^{-1}$ and $f^{-1/2}$ respectively. Inverting them gives force divergences proportional to the first and second inverse powers of the deficit. Freely jointed link rotations cost no bending energy; the worm-like chain instead has a continuum of coupled bending modes whose contributing wavelengths change with force. Setting a long-wavelength coarse-grained link length $b=2L_p$ can match weak-force statistics, but does not make these high-force laws identical.