Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 51 / 1 / e / Solution

Critical exponents describe leading singular powers as the reduced temperature $t$ or conjugate field approaches zero. With ${\beta }_m$ used for the order-parameter critical exponent to avoid confusion with inverse temperature, the definitions includeAt criticality the connected correlation function behaves as $G(r)\sim r^{-(D-2+\eta )}$. These definitions refer to the singular part and allow additional logarithms at marginal dimensions.

For an ordinary scalar Landau free energy, the equation of state is $h=rM+u{M}^3$. Below the transition $M^2=-r/u$, giving ${\beta }_m=1/2$. Differentiating the equation of state gives ${\chi }^{-1}=r+3u{M}^2$: it equals $r$ above and $-2r$ below the transition, so $\gamma =1$. The quadratic fluctuation kernel is $\kappa k^2+r$ in the symmetric phase, giving correlation length $\xi =\sqrt{\kappa /r}$ and $\nu =1/2$. At zero mass the momentum-space correlation function is proportional to $k^{-2}$, so $\eta =0$. The minimized free energy and critical equation of state give $\alpha =0$ and $\delta =3$, as calculated in the root solution.

At the tricritical point, $u=0$ and $h=rM+v{M}^5$. Then $M\sim (-r{)}^{1/4}$, and ${\chi }^{-1}=r+5v{M}^4$ is $r$ above and $-4r$ below the transition. The gradient kernel still gives $\nu =1/2$ and $\eta =0$. ThereforeThese are mean-field critical exponents. A systematic derivation incorporating fluctuations uses the renormalization group, extracts the relevant scaling eigenvalues, and differentiates the singular free-energy density, as in Question 3.