Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 46 / 2

Use ${\beta }_{\mathrm{th}}=1/(k_{B}T)$ for inverse temperature to distinguish it from the order-parameter exponent. The decaying two-point correlation function needed for the response identity is connected:Away from the critical point in a short-range Ising model pure phase, the correlation length controls the exponential long-distance falloff, $G(r)\sim A(r)e^{-|r|/\xi }$ with a possible algebraic prefactor. The full two-spin expectation below the transition instead tends to $M^2$. The spin-mixture contribution to zero-field susceptibility explains why a symmetric mixture is unsuitable for defining a finite connected correlation length in the ordered phase.

Let $F=-{\beta }_{\mathrm{th}}^{-1}\log Z$ be the total Helmholtz free energy. Take magnetization and susceptibility per site, soDifferentiating the Boltzmann weights at fixed temperature givesTranslation invariance then yields the correlation-function susceptibility sum ruleTotal magnetization and total susceptibility multiply these expressions by $N$; per physical volume they have an additional factor $a^{-D}$ relative to the per-site convention. These normalizations do not change the critical powers.

For the subsequent scaling requests, let $F_s$ denote singular free energy per site; the fixed extensive normalization is suppressed. The labels $+$ and $-$ denote $t>0$ and $t<0$, with distinct scaling functions and amplitudes on the two sides. Below the transition a one-sided derivative at $h=0$ selects a spontaneous-magnetization branch. Writing $x=h/|t{|}^{\Delta }$, differentiation of the scaling hypothesis for critical phenomena givesThe field-independent $I_{\pm }$ makes no contribution to these field derivatives, although it can change heat-capacity amplitudes. Assuming nonzero leading amplitudes and the indicated pure-phase limits, comparison with the exponent definitions givesThis is the Rushbrooke scaling relation. A temperature-independent critical isotherm requires the large-$|x|$ magnetization scaling function to grow as $\mathrm{sgn}(x)|x{|}^{\beta /\Delta }$: otherwise $M_s=|t{|}^{\beta }{\mathcal{M}}_{\pm }(x)$ would retain a temperature power as $t\to 0$ at fixed small $h$. Therefore $1/\delta =\beta /\Delta$. Since the preceding two equalities also give $\Delta =\beta +\gamma$, the Widom scaling relation follows:For the hyperscaling relation, additionally assume that one correlation volume supplies an order-one singular free-energy contribution and that there is a single isotropic correlation length, with no dangerously irrelevant coupling spoiling this argument. Then $F_s\sim {\xi }^{-D}\sim |t{|}^{D\nu }$. Two temperature derivatives give the singular heat capacity, soHyperscaling is an additional physical assumption, not a consequence of the two-variable equation alone, and generally fails above the upper critical dimension.

Finally the correlation scaling function $f_G(x)$ falls exponentially, up to algebraic factors, for $x\to \infty$ in a massive short-range pure phase. At $x\to 0$ it approaches a finite nonzero value in the scaling regime. Approximating the lattice sum at distances much larger than $a$, and denoting the area of the unit sphere by $S_{D-1}$, givesIf $\eta <2$ and the integral has a finite nonzero limiting value, microscopic distances contribute only a regular background and the leading divergence is ${\xi }^{2-\eta }$. Thus the Fisher scaling relation isThese statements concern leading singular powers; logarithmic corrections and cancellations of amplitudes require separate treatment.

Partition the lattice into blocks of side $b$ sites, with $b>1$, and retain one coarse spin ${\sigma }_R^{\prime }$ per block. A blocking kernel is a nonnegative conditional weight $K_b({\sigma }^{\prime },\sigma )$ normalized by ${\sum }_{{\sigma }^{\prime }}{K}_b({\sigma }^{\prime },\sigma )=1$ for every microscopic configuration. A deterministic majority rule, with fair resolution of ties, is one example.

Define the blocked statistical Hamiltonian and its field-independent constant bySumming over ${\sigma }^{\prime }$ proves exact equality of the old and new partition functions. The lattice spacing becomes $ba$; expressing lengths again in the original units completes the real-space renormalization group step. Blocking generally generates many-body and longer-range interactions, so exactness requires retaining the full operator space and the additive constant. A finite-coupling truncation is a calculational approximation.

A renormalization-group fixed point satisfies $R_b(u_{\ast })=u_{\ast }$. A critical fixed point is scale invariant and has divergent correlation length. The critical surface consists of couplings flowing toward it after relevant variables have been tuned; noncritical flows ultimately leave its neighborhood toward finite-correlation-length phases.

Linearize the renormalization-group transformation near the fixed point and choose scaling coordinates. If the thermal and magnetic coordinates transform as $t^{\prime }=b^{y_t}t$ and $h^{\prime }=b^{y_h}h$, the singular free-energy density obeysChoose $b=|t{|}^{-1/y_t}$ to stop at an order-one thermal coordinate. This gives the scaling hypothesis for critical phenomenaConsequently $\nu =1/y_t$ and $\Delta =y_h/y_t$, with $2-\alpha =D/y_t$ when hyperscaling applies. The fixed point determines universal powers, while microscopic couplings set metric factors and amplitudes. Dangerously irrelevant variables and marginal logarithms can qualify the simple homogeneous form.

Diagonalizing the linearized blocking map gives perturbations $v_i^{\prime }={\lambda }_i{v}_i=b^{y_i}{v}_i$. A relevant operator has $y_i>0$, equivalently $|{\lambda }_i|>1$ for a discrete scaling multiplier: its perturbation grows under repeated coarse-graining and must be tuned to reach the critical fixed point. An irrelevant operator has $y_i<0$ and decays, usually contributing corrections to scaling. A marginal operator has $y_i=0$, so linearization alone is insufficient; nonlinear flow can make it marginally relevant or irrelevant, or exactly marginal.

For an ordinary spin-inversion-symmetric critical point the temperature and magnetic field are relevant. Fixing $h=0$ by symmetry leaves the temperature to tune. At a tricritical point the quartic coupling is another relevant tuning direction. A dangerously irrelevant coupling is an exception to simply discarding small variables: setting it to zero can destabilize the ordered potential or make its scaling function singular, even though it flows toward zero.

Write $\mathfrak{f}(u)$ for the physical free energy per site excluding the identity term, so $F/N=C+\mathfrak{f}(u)$. Blocking sends $u$ to $u^{\prime }=R_{b}u$ and the identity coefficient to $C^{\prime }=b^{D}C+c(u)$. Since the total partition function is unchanged and $N^{\prime }=N/b^D$,The function $g$ records the field-independent energy generated by eliminated modes, including their entropy and normalization. It is the identity-operator contribution to renormalization-group free energy; it must be tracked to recover the complete thermodynamics, even though it does not change normalized spin probabilities.

Iterating the additive free-energy recursion under blocking givesWhen a regular background $A(u)$ solves $A(u)=b^{-D}A(u^{\prime })+g(u)$, the singular remainder $f_s=\mathfrak{f}-A$ satisfies the homogeneous scaling equation $f_s(u)=b^{-D}{f}_s(u^{\prime })$. Thus the full free energy contains an inhomogeneous contribution, while the background-subtracted singular part is homogeneous. A finite initial identity energy $C_0$ adds a regular normalization, not a new magnetization singularity.

Within a finite blocking step, $g(u)$ is generally analytic near the critical couplings because only finitely many microscopic variables have been summed. If its regular contribution can be removed by the analytic subtraction of an inhomogeneous renormalization recursion, derivatives of that background are nonsingular. The critical powers are then determined by the homogeneous singular remainder and the relevant scaling dimensions, not by the normalization of the eliminated modes.

For example, in linear scaling coordinates a source monomial $g_{mn}{t}^m{h}^n$ is removed by choosing a background coefficientprovided the denominator is nonzero. This explicitly demonstrates why the inhomogeneous term normally changes backgrounds and amplitudes rather than critical powers.

The qualification matters: at a renormalization-group free-energy resonance, $m{y}_t+n{y}_h=D$, an analytic subtraction fails and logarithms such as $t^m{h}^n\log |t|$ can be needed. Repeated analytic blocking contributions are not automatically globally analytic at criticality. Even the field-independent $I_{\pm }$ in the supplied scaling form can affect a singular heat-capacity amplitude. Its omission is justified for leading exponents only when the remaining amplitudes are nonzero and the assumed pure power scaling is valid; it is not a license to discard the identity term from an exact free energy.