Past exam of the phase transitions and collective phenomena course of the University of Cambridge / 2012 / Paper 1 / 1

Use $s=|t|$, $x=h/s^{\Delta }$ and separate scaling functions above and below the transition. Up to a nonsingular conversion between physical and dimensionless field, magnetization scaling-function matching givesOn the ordered side take $h\to 0^{+}$ after the thermodynamic limit, selecting one pure thermodynamic phase. A finite nonzero $g_{m,-}(0^{+})$ impliesThe critical isotherm must have a finite nonzero limit at fixed small $h$ as $s\to 0$. Thus $g_m(x)\sim C\mathrm{sgn}(x)|x{|}^{1/\delta }$ at large $|x|$. Cancellation of the residual power of $s$ requiresThe absolute value and branch choice matter: spontaneous zero-field magnetization is below $T_c$, whereas the symmetry-unbroken thermodynamic phase has $m(t,0)=0$. The field crossover exponent is not the scaling dimension of the magnetization field.

A further field derivative of the magnetization scaling form givesAt zero field, on a branch with a finite nonzero susceptibility amplitude, comparison with $\chi \sim s^{-\gamma }$ yieldsConsequently $\alpha +2\beta +\gamma =2$ and $\gamma =\beta (\delta -1)$, the Rushbrooke and Widom scaling relations. Regular analytic terms do not change the leading divergent susceptibility.

At fixed zero field, ${\partial }_T=T_c^{-1}{\partial }_t$. Therefore, away from $t=0$ on either branch,Thus the singular heat capacity has exponent $\alpha$. An arbitrary analytic background must be added to obtain the full heat capacity. Vanishing branch amplitudes or the marginal case $\alpha =0$ need care: a finite jump is compatible with this power counting, while a logarithmic singularity requires a logarithmic refinement of the assumed homogeneous form.

Hyperscaling adds the physical assumption that the singular free energy associated with one correlation volume is of order $k_B{T}_c$:At fixed scaled field $x=h/s^{\Delta }$, this predicts $f_s\sim s^{d\nu }$, whereas the given thermodynamic form has power $2-\alpha$. HenceThe product $g_f(x)[g_{\xi }(x){]}^d$ can be a finite dimensionless function of fixed $x$; it need not be constant. The homogeneity of $\xi$ alone is insufficient: the correlation-volume free-energy assumption supplies the connection. Above an upper critical dimension, a dangerously irrelevant coupling can invalidate this argument; at a marginal dimension logarithms may qualify pure powers.

The susceptibility is the integrated connected correlation function, up to a nonsingular field-normalization factor:Above the transition at zero field the subtraction vanishes. In an ordered thermodynamic phase it is essential; integrating the disconnected constant would incorrectly give a volume divergence unrelated to the desired response.

With a microscopic cutoff $a$ and $r=|\mathbf{x}|$, radial integration and $u=r/\xi$ giveIf $g(0)$ is finite, its large-distance tail decays sufficiently rapidly and $\eta <2$, the dimensionless integral approaches a finite nonzero constant. ThusShort-distance contributions give a regular background. This is the Fisher scaling relation; it concerns the fluctuating connected part of the source correlation ansatz.

The annotations on the original plot give $\beta =0.368$ and $\gamma =1.215$. Use those displayed values rather than fitting the linear-axis curve as though its slope were a logarithmic exponent. Exponent inference from a magnetic data collapse identifies $\Delta =\gamma +\beta$.ThereforeThese estimates inherit both the empirical collapse exponents and the assumed scaling identities, including three-dimensional hyperscaling. They are not independent high-precision measurements of five exponents. In particular, the small difference defining $\alpha$ and the subtraction defining $\eta$ amplify uncertainties. No experimental curve or source image is reproduced.