Solution
= Solution
Use $\beta_T=1/(k_BT)$ 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> is
$$
G(r)=\langle\sigma_0\sigma_r\rangle-\langle\sigma_0\rangle\langle\sigma_r\rangle=\langle\sigma_0\sigma_r\rangle-M^2.
$$
For 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 $h\to0^+$ or $h\to0^-$. 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 $\mathscr F=-k_BT\log Z$ denote extensive physical <free energy> and $S=\sum_r\sigma_r$. Since the field contributes $-hS$ to the <statistical Hamiltonian>, differentiation gives
$$
M=-\frac1N\frac{\partial\mathscr F}{\partial h}=\frac{\langle S\rangle}N,\qquad \chi=\frac{\partial M}{\partial h}=-\frac1N\frac{\partial^2\mathscr F}{\partial h^2}=\frac{\beta_T}{N}\operatorname{Var}(S).
$$
Expanding the <variance> and using <translation invariance> proves the <correlation-function susceptibility sum rule>:
$$
\boxed{\chi=\beta_T\sum_rG(r).}
$$
The sum includes $r=0$. If the source is instead defined as the dimensionless quantity $\beta_Th$, 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> $B_b(\sigma',\sigma)\geq0$ satisfying $\sum_{\sigma'}B_b(\sigma',\sigma)=1$. 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> by
$$
e^{-\beta_T[H(u',\sigma')+N'C']}=\sum_\sigma B_b(\sigma',\sigma)e^{-\beta_T[H(u,\sigma)+NC]},\qquad N'=b^{-D}N.
$$
The spacing becomes $a'=ba$. All generated interactions, including multispin terms and the field-independent constant, must be retained for an exact transformation. Summing over $\sigma'$ uses normalization to prove $Z(u',C',N')=Z(u,C,N)$. 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 $\sum_{\sigma'}B_b(\sigma',\sigma)A(\sigma')$. 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,
$$
F(u,C)=-\frac1N\log Z(u,C,N)=f(u)+\beta_TC,
$$
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 $C'=b^DC+c(u)$, the coefficient $c(u)$ records the identity-operator contribution from eliminated fluctuations and normalization. Consequently
$$
f(u)=b^{-D}f(u')+g(u),\qquad g(u)=\beta_Tb^{-D}c(u)=\beta_T(b^{-D}C'-C).
$$
Repeated substitution derives the <additive free-energy recursion under blocking>:
$$
\boxed{f(u_0)=b^{-pD}f(u_p)+\sum_{j=0}^{p-1}b^{-jD}g(u_j).}
$$
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 $R_b(u_*)=u_*$. Linearize in scaling coordinates $v_i$ so that $v_i'=b^{y_i}v_i+O(v^2)$. 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.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-51-critical-rg-flow.png]
{title=Local renormalization-group flow in the zero-field slice: attraction along the critical surface and repulsion in the thermal direction}
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 $a(u)=b^{-D}a(R_bu)+g(u)$. Then $F_s=f-a$ satisfies $F_s(u)=b^{-D}F_s(R_bu)$. In linear thermal and field coordinates, an analytic monomial $g_{mn}t^mh^n$ has background coefficient
$$
a_{mn}=\frac{g_{mn}}{1-b^{m\lambda_t+n\lambda_h-D}},
$$
provided 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> is
$$
F_s(t,h)=\ell^{-D}F_s(\ell^{\lambda_t}t,\ell^{\lambda_h}h),\qquad \ell>0,
$$
as an asymptotic scaling equation. Choose $\ell=|t|^{-1/\lambda_t}$ to make the thermal argument $+1$ or $-1$. Defining $f_\pm(z)=F_s(\pm1,z)$ gives
$$
\boxed{F_s(t,h)=|t|^{D/\lambda_t}f_\pm\!\left(\frac{h}{|t|^{\lambda_h/\lambda_t}}\right).}
$$
The 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 $M\propto\operatorname{sgn}(h)|h|^{(D-\lambda_h)/\lambda_h}$, so $\delta=\lambda_h/(D-\lambda_h)$. Hence
$$
\boxed{\nu=\frac1{\lambda_t},\quad \alpha=2-\frac D{\lambda_t},\quad \beta=\frac{D-\lambda_h}{\lambda_t},\quad \gamma=\frac{2\lambda_h-D}{\lambda_t},\quad \delta=\frac{\lambda_h}{D-\lambda_h}.}
$$
Simple 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 $x\to0$ 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 $f_G(x)\sim Bx^{(D-3)/2+\eta}e^{-x}$, 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 gives
$$
\chi_s\simeq\beta_Ta^{-D}\Omega_{D-1}\int_a^\infty r^{1-\eta}f_G(r/\xi)\,dr=\beta_Ta^{-D}\Omega_{D-1}\xi^{2-\eta}\int_{a/\xi}^\infty x^{1-\eta}f_G(x)\,dx,
$$
where $\Omega_{D-1}$ 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 relation>
$$
\boxed{\gamma=(2-\eta)\nu.}
$$
Equivalently 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.