Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 51 / 3

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.

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.

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.

A relevant operator has $y_i>0$, so its coupling grows under coarse-graining and drives the system away from the critical renormalization-group fixed point. Such directions must be tuned to reach the critical manifold. An irrelevant operator has $y_i<0$, so its coupling decays and normally supplies corrections to scaling. For $y_i=0$ the operator is marginal; nonlinear terms in the transformation decide whether its coupling grows or decreases, or whether it labels a line of fixed points.

This classification is a statement about perturbations of a specified fixed point, not an intrinsic label independent of the theory. For example, a quartic scalar interaction has eigenvalue $4-D$ at the Gaussian fixed point, so it is relevant for $D<4$ and marginal at $D=4$. A tricritical scalar point has an extra relevant even direction compared with an ordinary critical point. Although irrelevant couplings usually disappear from leading powers, a dangerously irrelevant coupling can still be needed to stabilize an ordered phase and invalidate naive hyperscaling.

Absorb $\beta$ into the nonconstant couplings and write the corresponding dimensionless action as $H_u$. Choose its additive-constant convention so that one blocking step hasThe function $g(u)$ is the identity-operator contribution to renormalization-group free energy: it records the extensive constant generated by eliminating short-distance degrees of freedom. Summing over ${\sigma }^{\prime }$ givesFor the dimensionless free-energy density $\Phi (u)=-N^{-1}\ln Z(u,0,N)$, this is the inhomogeneous recursionFor the explicit constant $C$ in the original partition function, the corresponding new constant obeys $\beta C^{\prime }=b^D[\beta C+g(u)]$; $C$ and $g$ are not additional interactions between spins.

Iterating the recursion $n$ times yieldsAn analytic background $a(u)$ satisfying $a(u)=g(u)+b^{-D}a(Ru)$ absorbs the regular contribution. The remaining singular part $f_s=\Phi -a$ obeysThis separation explains both the scaling of the singular part and the role of the inhomogeneous term; simply dropping the generated constant would incorrectly change the full free energy.

A finite blocking step integrates degrees of freedom on bounded length scales. Its function $g(u)$ is generally analytic near the critical couplings and contributes to a regular background in the free-energy density. The nonanalytic critical behavior arises from arbitrarily many blocking steps approaching the critical renormalization-group fixed point. After removing the analytic background, the homogeneous recursion determines the leading singular critical exponents, so the inhomogeneous term usually need not be retained when computing those powers.

There are qualifications. If the singular heat capacity has $\alpha <0$, the regular background may dominate the measured total heat capacity, although the singular exponent remains meaningful. At a renormalization-group free-energy resonance, repeated accumulation of analytic terms can generate a logarithm. For example, a term proportional to $t^2$ in $g$ has weight $b^{-jD}{b}^{2j{y}_t}$; when $D=2y_t$, every step contributes the same multiple of $t^2$. The number of steps before $t$ becomes order one is proportional to $-\ln |t|$, giving a $t^2\ln |t|$ singularity. Thus the inhomogeneous term can be ignored for generic leading powers, but not indiscriminately for logarithmic corrections.

Let $t,h$ be the thermal and magnetic scaling fields, with eigenvalues $b^{y_t}$ and $b^{y_h}$. After discarding irrelevant variables whose vanishing is nonsingular, the homogeneous singular free-energy density satisfiesChoose $b=|t{|}^{-1/y_t}$. ThenTwo thermal derivatives give the heat-capacity critical exponent; one and two field derivatives give the order-parameter critical exponent and magnetic-susceptibility critical exponent. The correlation-length critical exponent follows directly from the length rescaling. ThusAt $t=0$, choose $b=|h{|}^{-1/y_h}$. Differentiating $f_s(0,h)\propto |h{|}^{D/y_h}$ gives $M\propto \mathrm{sgn}(h)|h{|}^{(D-y_h)/y_h}$, so $\delta =y_h/(D-y_h)$. The order parameter has scaling dimension $x_M=D-y_h$; therefore its critical connected correlation function has power $r^{-2x_M}$. Comparing with $r^{-(D-2+\eta )}$ yieldsThese formulas express the requested critical exponents in terms of the relevant scaling exponents. Their hyperscaling assumptions must be respected: above an upper critical dimension, a dangerously irrelevant coupling can invalidate the free-energy homogeneity used here. Applying the formulas to the Gaussian thinning transformation in the root solution gives the requested Gaussian indices and their physical qualifications.