Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 43 / 1 / c / Solution

Microscopically different systems can have the same critical exponents and appropriately normalized scaling functions. This universality means that they belong to one universality class when coarse-graining brings them to the same renormalization-group fixed point. Spatial dimension, order parameter symmetry and the range of the interactions affect that classification; changing a lattice spacing or a microscopic coupling generally changes nonuniversal amplitudes and the critical temperature instead.

In the scalar short-range Landau-Ginzburg theory, a positive quartic term and one thermal tuning produce ordinary Ising-type criticality. A generic sextic term does not change that fixed point, but tuning the quartic coefficient to zero exposes a different tricritical point with different exponents. The mean-field approximation itself neglects fluctuations. Its numerical exponents are reliable only when the Ginzburg criterion makes those fluctuations subleading; the ordinary scalar upper critical dimension is four, while the tricritical one is three. At an upper critical dimension logarithmic corrections can remain.