Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 72 / 2

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.

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 $\delta >1$.

A further derivative gives the magnetic susceptibility within a selected pure thermodynamic phase:Assuming a finite nonzero zero-field amplitude on the branch considered, comparison with $\chi (t,0)\asymp |t{|}^{-\gamma }$ givesTogether with part (a), this proves the Rushbrooke scaling relation $\alpha +2\beta +\gamma =2$ and the Widom scaling relation $\gamma =\beta (\delta -1)$. Below the transition, use the one-sided response of an ordered phase; differentiating through the cusp of the equilibrium free energy at $h=0$ would include phase switching rather than this critical response.

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.

First interpret the displayed critical correlation function as the connected correlation functionAbove the transition at zero field the subtraction is zero; in an ordered phase it is essential. For a dimensionless source coupled to the total magnetization $M=\int d^{d}xm(x)$, differentiating $Z$ twice gives $\chi =V^{-1}(⟨M^2⟩-⟨M{⟩}^2)$. Translation invariance then proves the correlation-function susceptibility sum ruleInsert the scaling form, use spherical coordinates and set $s=r/\xi$:For $\eta <2$, finite nonzero $g(0)$ and an integrable large-$s$ tail, the last integral tends to a finite nonzero constant. Microscopic separations add an analytic response background. Thus the singular magnetic susceptibility obeys ${\chi }_s\asymp {\xi }^{2-\eta }$, and at zero field $\xi \asymp |t{|}^{-\nu }$ gives the Fisher scaling relationA dimensional magnetic field would add a nonsingular inverse-temperature prefactor. Integrating the full unsubtracted ordered-phase correlation function instead would yield a volume-divergent term $V⟨m{⟩}^2$, which is not the intrinsic susceptibility appearing in this exponent identity.